Theory of Mind: A Neural Prediction Problem

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Action understanding as inverse planning
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Citation: Baker, Chris L., Rebecca Saxe, and Joshua B. Tenenbaum. “Action understanding as
inverse planning.” Cognition 113.3 (2009): 329-349.
As Published: http://dx.doi.org/10.1016/j.cognition.2009.07.005
Publisher: Elsevier
Persistent URL: http://hdl.handle.net/1721.1/60852
Version: Author's final manuscript: final author's manuscript post peer review, without
publisher's formatting or copy editing
Terms of use: Attribution-Noncommercial-Share Alike 3.0 Unported

Action Understanding as Inverse Planning
Chris L. Baker, Joshua B. Tenenbaum & Rebecca R. Saxe
Department of Brain and Cognitive Sciences
Massachusetts Institute of Technology
Abstract
Humans are adept at inferring the mental states underlying other agents’
actions, such as goals, beliefs, desires, emotions and other thoughts. We
propose a computational framework based on Bayesian inverse planning for
modeling human action understanding. The framework represents an in-
tuitive theory of intentional agents’ behavior based on the principle of ra-
tionality: the expectation that agents will plan approximately rationally to
achieve their goals, given their beliefs about the world. The mental states
that caused an agent’s behavior are inferred by inverting this model of ra-
tional planning using Bayesian inference, integrating the likelihood of the
observed actions with the prior over mental states. This approach formalizes
in precise probabilistic terms the essence of previous qualitative approaches
to action understanding based on an “intentional stance” (Dennett, 1987)
or a “teleological stance” (Gergely et al., 1995). In three psychophysical
experiments using animated stimuli of agents moving in simple mazes, we
assess how well different inverse planning models based on different goal
priors can predict human goal inferences. The results provide quantitative
evidence for an approximately rational inference mechanism in human goal
inference within our simplified stimulus paradigm, and for the flexible na-
ture of goal representations that human observers can adopt. We discuss
the implications of our experimental results for human action understanding
in real-world contexts, and suggest how our framework might be extended
to capture other kinds of mental state inferences, such as inferences about
beliefs, or inferring whether an entity is an intentional agent.
Introduction
A woman is walking down the street, when suddenly she pauses, turns, and begins
running in the opposite direction. Why? Is she just acting erratically on the way to her
eventual goal? Did she change her mind about where she was going? Or did she complete an
errand unknown to us (perhaps dropping off a letter in a mailbox) and rush off to her next
goal? These inferences, despite their ordinariness, reveal a remarkable aspect of human
cognition: our ability to infer the complex, richly-structured mental states that underlie
others’ actions, given only sparse observations of their behavior.

ACTION UNDERSTANDING AS INVERSE PLANNING 2
Human social interaction depends on our ability to understand and predict other
people’s actions in terms of the psychological states that produce behavior: chiefly, beliefs
and desires. Much like visual perception, action understanding proceeds unconsciously
and effortlessly but is the result of sophisticated computations that effectively solve an ill-
posed, inductive problem, working backwards from sparse data to rich representations of
the underlying causes. Our goal in this paper is to elucidate the computations involved
in human action understanding through a combination of computational modeling and
behavioral experiments. We will describe some of the first models that can explain how
people perform these inferences so successfully, and that can also predict with surprising
quantitative accuracy the judgments that people make.
Vision is often said to be a kind of “inverse graphics”, where graphics describes the
causal physical process by which images are formed from scenes. Similarly, action under-
standing can be characterized as a kind of “inverse planning” or “inverse reinforcement
learning” (Ng & Russell, 2000). Just as computer graphics is based on mathematical mod-
els of image formation, mathematical accounts of planning and reinforcement learning have
been developed by economists, computer scientists, psychologists and neuroscientists (Bell-
man, 1957; Watkins, 1989; Sutton & Barto, 1998; W. Schultz et al., 1997), which provide
rational models of how agents should choose sequences of actions, given their goals, their
prior experience, and their model of the world. Explaining an agent’s actions in terms
of mental states requires inverting a model of its planning process, or inverse planning:
working backwards to infer the desires and beliefs that caused the agent’s behavior.
Formalisms for solving the forward problems of planning and reinforcement learning
are often divided into model-based and model-free approaches (Sutton & Barto, 1998; Doya,
1999; Daw et al., 2005), and there is evidence that the brain has systems corresponding to
both (W. Schultz et al., 1997; Dickinson, 1985). We propose that the same kinds of cognitive
machinery that support learning goal-directed action in the model-based approach – the
ability to build models of the world and plan reward-maximizing sequences of actions over
them – can be used in an inverse direction to infer the goals behind other agents’ observed
behavior.
Philosophers and psychologists have long considered non-formal versions of this pro-
posal in discussions about “belief-desire psychology”. Fig. 1(a) illustrates a typical example:
a folk theory that specifies intentional agents’ beliefs and desires as the causes of their be-
havior (c.f. Dennett, 1987; Wellman, 1990; Perner, 1991; Gopnik & Meltzoff, 1997). Dennett
(1987) argues that this causal relation is governed by the principle of rationality: the ex-
pectation that intentional agents will tend to choose actions that achieve their desires most
efficiently, given their beliefs about the world. At a qualitative level, inverse planning is
simply running the principle of rationality in reverse. Considered as a formal computation,
however, inverse planning is significantly more difficult than forward planning. Just as in
vision (Barrow & Tenenbaum, 1981; Richards et al., 1996), the inverse problem is ill-posed.
Its solution requires strong prior knowledge of the structure and content of agents’ mental
states, and the ability to search over and evaluate a potentially very large space of possible
mental state interpretations. Implementing a formal version of this account, and quantita-
tively evaluating it with human behavioral judgments is the main contribution of our work
here.
Previous experimental evidence suggests that even preverbal infants’ interpretations

(c)
Perceptual
Access
Environment
Belief Goal
Action
Principle of
rational action
Principle of
rational belief
(b)
Environment Goal
Action
Principle of
rational action
(a)
Belief Desire
Action
Principle of
rational action
Situations
General World
Knowledge
General
Preferences
Figure 1. Modeling intuitive theories of intentional action. Diagrams use causal graph notation.
Shaded nodes represent observed variables; unshaded nodes represent latent variables whose values
must be inferred. Dotted boxes indicate the causal relation between variables. In this example,
the observations are an agent’s Action in some Environment and the agent’s Perceptual Access to
the value of the Environment. Given this evidence, the agent’s Belief and Goal must be inferred.
(a) Non-formal “belief-desire psychology” account of folk theories of intentional action. Non-formal
accounts of human action understanding typically assume some version of this causal structure, and
define the causal relation between beliefs, desires and actions in terms of qualitative, context-specific
commonsense rules (e.g. Wellman & Bartsch, 1988; Fodor, 1992). (b) A simplified version of (a)
proposed by Gergely et al. (1995) and Csibra & Gergely (1997) as a model of infants’ early-developing
action understanding competency. We formalize this intuitive theory as Bayesian inverse planning.
The functional form of the causal relation between Environment, Goal and Action is given by rational
probabilistic planning in Markov decision problems, and goal inference is performed by Bayesian
inversion of this model of planning. (c) A generalized framework for human action understanding.
This relates the non-formal account in (a) to the formal model in (b) by sketching how agents’
Beliefs and Desires (or Goals) depend on their Perceptual Access to the Environment, mediated by
their General World Knowledge and General Preferences. The model in (b) is a limiting case of (c), in
which agents are assumed to have complete Perceptual Access to the Environment, constraining their
Beliefs to be equal to the Environment.
of behavior are qualitatively consistent with the inverse planning view (Meltzoff, 1995;
Gergely et al., 1995; Meltzoff, 1988; Gergely et al., 2002; Csibra et al., 2003; Sodian et al.,
2004; Phillips & Wellman, 2005). Six-month-old infants interpret simple human motions
as goal-directed actions, and expect that subsequent behavior will be consistent with these
inferred goals (Woodward, 1998). That is, when actions could be interpreted as a rational
or efficient means to achieve a concrete goal, infants expect the actor to continue to use the
most efficient means to achieve the same goal, even when the environment changes. Gergely,
Csibra and colleagues found that six- to twelve-month old infants extend the same expec-
tations to the novel (and relatively impoverished) movements of two-dimensional shapes
(Gergely et al., 1995; Csibra et al., 1999, 2003). In this context, infants’ inferences were

flexible and productive: given information about any two of the environment, the action
and the goal, infants could infer the likely value of the third. To account for these findings,
Gergely et al. (1995) proposed an early-developing, non-mentalistic version of Fig. 1(a),
shown in Fig. 1(b). On their account, the Environment represents concrete situational con-
straints on the agent’s available actions, such as the agent’s own location and the location of
other agents, objects, or obstacles, and the Goal is some point or entity in the Environment.
Together, the Environment and Goal provide a basis for the agent’s Action under a simple
version of the rationality principle known as the teleological stance. Gergely et al. (1995)
argue that this simplified schema forms the core of a more sophisticated, later-developing
mentalistic theory of intentional action.
This research, along with the essential computational difficulty of action understand-
ing, raises several open questions about how action understanding works in the mind. Can
human action understanding competency be described by formal models, or is our intuitive
psychological knowledge vague and heterogeneous? If action understanding can be formal-
ized, can people’s judgments be explained by models of inverse planning? Does inverse
planning explain people’s judgments better than simple heuristic alternatives? If human
judgments are best explained by inverse planning, what is the form and content of our rep-
resentations of agents’ mental states and actions – the priors that make inductive mental
state inferences possible?
To address these questions, we formalize action understanding as a Bayesian inference
problem. We model the intuitive causal relation between beliefs, goals and actions as
rational probabilistic planning in Markov decision problems (MDPs), and invert this relation
using Bayes’ rule to infer agents’ beliefs and goals from their actions. We test our framework
with psychophysical experiments in a simple setting that allows us to collect a large amount
of fine-grained human judgments to compare with the strong quantitative predictions of our
models.
Specifically, we use the tools of Bayesian inverse planning to formalize the action
understanding schema shown in Fig. 1(b). Inspired by Gergely et al. (1995), we assume
that the agent’s Action depends directly on the Environment and the Goal, without requiring
a separate representation of the agent’s beliefs. To specify the agent’s likely Actions as
a function of the constraints of the Environment and the agent’s Goal, these variables are
encoded within an MDP, and the causal relation between them is computed by a mechanism
for rational planning in MDPs. We assume that the planning relation is probabilistic,
tolerating a certain amount of noise or variability in how agents can execute their plans.
Fig. 1(c) sketches a more general intuitive theory of rational action, intended to
capture various qualitative proposals in the theory of mind literature (e.g. Wellman &
Bartsch (1988); Wellman (1990); Bartsch & Wellman (1995); see also Goodman et al. (2006)
for a related formal account). This schema extends Fig. 1(a) by describing how beliefs
depend on perceptual access to the environment, mediated by general world knowledge, and
how goals depend on general preferences over states of the world. General world knowledge
and preferences are high-level variables that apply across situations, while new beliefs and
goals are generated specifically for each situation. The specific models we work with in this
paper (Fig. 1(b)) correspond to the special case in which agents are assumed to have full
perceptual access to the environment, thereby constraining the contents of their beliefs to
be equal to the environment. A formal implementation of the more general framework in

Fig. 1(c) is beyond our scope here, but in the General Discussion we consider the additional
computational assumptions needed to extend our work in that direction, to allow reasoning
about the unknown contents and origins of agents’ beliefs.
The Bayesian inversion of MDP models of behavior requires strong priors over the
space of agents’ goals. In our framework, the most basic concept of a goal corresponds
to the objective to bring about a particular state of the environment. However, this is
clearly too inflexible to describe the sophisticated kinds of goals that humans can attribute
to other agents, and there are many ways that the basic goal concept can be extended.
As a first step, in this paper we consider two extensions to the most basic goal concept,
which roughly correspond to the explanations of the woman’s behavior in the introductory
vignette: goals that can change over time and goals with more complex content, such as
subgoals along the way to a final goal. We also formulate a simple heuristic alternative
based on low-level motion cues as a limiting case of the changing-goal prior. We describe
Bayesian inverse planning models based on these different goal priors in the Computational
Framework section, and compare how accurately they predict people’s judgments in our
experiments.
Our experiments use a stimulus paradigm of animated displays of agents moving in
simple maze-like environments to reach goal objects, inspired by stimuli from many previous
studies with children and adults (e.g. Heider & Simmel, 1944; Gergely et al., 1995; R. Gel-
man et al., 1995; Scholl & Tremoulet, 2000; Tremoulet & Feldman, 2000; Zacks, 2004; R. T.
Schultz et al., 2003; J. Schultz et al., 2005; Tremoulet & Feldman, 2006). This paradigm
allows fine-grained experimental control of agents’ actions, environment and plausible goals,
and is ideal for both psychophysical experiments and computational modeling. Although
this methodology greatly simplifies real-world action understanding, these kinds of stimuli
evoke a strong sense of agency and the impression of mental states to adults (Heider &
Simmel, 1944; Tremoulet & Feldman, 2000, 2006) (even when adult subjects are instructed
not to make mentalistic interpretations (Heberlein & Adolphs, 2004)), and can lead to the
formation of expectations consistent with goal-directed reasoning in infants (Gergely et al.,
1995; Csibra et al., 1999, 2003). There is evidence that these kinds of stimuli recruit brain
regions associated with action perception in adults (Castelli et al., 2000; R. T. Schultz et al.,
2003; J. Schultz et al., 2005), suggesting a common mechanism with real-world action under-
standing. Further, these stimuli can represent quite complex situations and events (Heider
& Simmel, 1944), with similar abstract structure to more naturalistic contexts. Similarly,
our computational models can be extended to much more general contexts than the simple
scenarios in our experiments, as we will show with several examples in the Computational
Framework section.
We present three experiments, which measure people’s online goal inferences, retro-
spective goal inferences, and prediction of future actions based on previous goal inferences,
respectively. Taken together, our experiments test whether human action understanding
in our experimental domain can be explained by inverse planning. Individually, our ex-
periments probe the space of representations that people apply in action understanding.
Each experiment includes special conditions to distinguish the predictions of inverse plan-
ning models based on different goal priors. By comparing which of these models produces
inferences that match people’s judgments most accurately in each experimental context,
we show how our approach can be used to elucidate the prior knowledge applied in human

action understanding.
Computational Framework
Our computational framework formalizes action understanding as Bayesian inverse
planning: the Bayesian inversion of models of probabilistic planning in Markov decision
problems (MDPs). This section will provide an overview of our framework and its appli-
cation to our experimental stimuli. First, we will describe the encoding of the maze-world
scenarios of our experiments into MDPs. We will also sketch the MDP encoding of several
more realistic environments and contexts than those of our experiments to emphasize the
generality of Bayesian inverse planning principles. Next, we will describe the computations
underlying the mechanism for planning in MDPs. We will then sketch the Bayesian com-
putations involved in inverting MDP models of planning, and give examples of the kinds
of structured goal priors that are required to perform these computations. Finally, we will
compare our framework with previous models of action understanding. Our overview in this
section will be fairly high-level, with the formal details provided in a separate appendix.
Our framework uses MDPs to capture observers’ mental models of intentional agents’
goal- and environment-based planning. MDPs are a normative framework for modeling
sequential decision making under uncertainty, widely used in rational models of human
planning and reinforcement learning (Dayan & Daw, 2008), and in real-world applications
in operations research and other fields (Feinberg & Shwartz, 2002; Puterman, 2005). An
MDP represents an agent’s model of its interaction with its environment. MDPs encode
all relevant information about the configuration of the world and the agent with the state
variable. MDPs also represent the affordances of the environment: what actions the agent
can take and a causal model of how these actions change the state of the world. Finally,
MDPs represent the subjective rewards or costs caused by the agent’s actions in each state.
In our maze-world scenarios, the state includes the location of all obstacles, potential
goals and other objects, and the location of the agent. Agents can take 9 different actions:
Stay, North, South, East, West, NorthEast, NorthWest, SouthEast and SouthWest, except
when these actions are blocked by obstacles. For simplicity, we assume that actions always
lead to their intended movements. The agent’s goal is to achieve a particular state of the
world, and each action is assumed to produce a small cost to the agent until this goal is
reached. Once the agent reaches its goal it is satisfied, and these costs cease. We define
costs to be proportional to the negative distance of the intended movement: actions North,
South, East, and West have costs proportional to −1, and actions NorthEast, NorthWest,
SouthEast and SouthWest have costs proportional to −
√
2. We also define the cost of the
Stay action to be proportional to −1 to capture the desire for continual progress toward
the goal. Formally, these assumptions induce a class of stochastic shortest path problems
(Bertsekas, 2001), implying that rational agents should plan to reach their goal as quickly
and efficiently as possible.
To illustrate the application of MDPs to another domain, consider the game of golf. In
golf, the state is comprised by the current hole, the current score, the current stroke number,
and the position of the ball. Actions must specify club selection and the type of shot to
be attempted. The causal model of the effect of actions reflects the inherent uncertainty
about where a shot ends up, with the outcome of difficult shots being more uncertain than
others. In golf, each shot has a cost of 1, and rational players try to minimize their score.

In addition to golf, many other games can be modeled as MDPs, such as blackjack (Sutton
& Barto, 1998), backgammon (Tesauro, 1994), Tetris (Bertsekas & Tsitsiklis, 1996), and
football (Bertsekas & Tsitsiklis, 1996).
As another example, consider the job of the head chef of a restaurant. Given a menu,
the chef’s goal is to prepare each dish as well and as quickly as possible to maximize customer
satisfaction and restaurant capacity. The state consists of the number of kitchen staff and
the available ingredients, burners, ovens, cooking implements, counter space, et cetera.
Actions include delegation of tasks to kitchen staff, the allocation of cooking resources to
different dishes, and the chef’s own hands-on preparation of dishes. As in golf, the causal
model reflects the uncertainty in preparation time and the performance of staff members.
Relative to golf, cooking has rich logical and hierarchical structure, where every step has
a large number of pre- and post-conditions, and individual actions may enable multiple
subsequent steps. For instance, sending a line cook after two raw eggs furthers both the
goal to cook a frittata as well the goal to prepare a soufflé.
Planning formally describes the way that intentional agents choose actions to achieve
their goals in an MDP. An optimal plan is one that provides the minimum-expected-cost
course of action to the goal from every state. Our models assume that agents choose the
optimal action only probabilistically. This yields a probability distribution over Actions,
given a Goal and the Environment, denoted P(Actions|Goal, Environment), which provides
the functional form of the probabilistic planning relation in Fig. 1(b). We assume that an
agent’s actions are distributed in proportion to the softmax function of the expected value
(negative expected cost) of each available action. The level of determinism in the agent’s
actions is represented by the parameter β: higher β values yield greater determinism (less
noise), and lower β values yield less determinism (more noise). We describe an algorithm
for probabilistic planning based on dynamic programming (Bellman, 1957) in the appendix.
Given an MDP model of goal-directed planning, Bayesian inverse planning computes
the posterior probability of a Goal, conditioned on observed Actions and the Environment,
using Bayes’ rule:
P(Goal|Actions, Environment) ∝ P(Actions|Goal, Environment)P(Goal|Environment). (1)
In this equation, P(Actions|Goal, Environment) is the likelihood of the Goal given ob-
served Actions and the Environment, defined above as probabilistic planning in an MDP.
P(Goal|Environment) is the prior probability of the Goal given the Environment, which sets
up a hypothesis space of goals that are realizable in the environment. Inverse planning
integrates bottom-up information from observed actions and top-down constraints from the
prior to infer the Goal, given observed Actions and the Environment. We describe inverse
planning models based on several different goal priors below.
Inverse planning also enables goal-based prediction of future actions in novel situ-
ations, given prior observations of behavior in similar situations. For simplicity, in this
paper, our goal-based prediction experiment presents scenarios where agents’ environments
and goals remain constant across observations (although our framework handles cases where
the environment changes as well (Baker et al., 2006)). Action prediction averages over the
probability of possible future Actions′, given a Goal and the Environment, weighted by the
posterior over the Goal given previously observed Actions and the Environment from Equa-
tion 1. This is just the posterior predictive distribution over future Actions′ given previous

Actions and the Environment:
P(Actions′
|Actions, Environment) =
X
Goal
P(Actions′
|Goal, Environment)P(Goal|Actions, Environment). (2)
We use Equations 1 and 2 to model people’s judgments in our experimental tasks. In
Experiment 1, we model people’s online goal inferences using an online version of Equation 1.
In Experiment 2, we model people’s retrospective goal inferences using a smoothed version
of Equation 1. In Experiment 3, we model people’s predictions of agents’ future actions,
given observations of their past behavior of varying complexity, using Equation 2.
We formulate several inverse planning models, based on different forms of the prior
over goals, which we refer to as M1, M2 and M3. Each model specifies a family of goal
priors depending on one or a few continuous parameters. These models are surely much
too simple to capture the full range of goals that people can attribute to an intentional
agent, and they simplify in different ways. Each model should be thought of as just a first
approximation to some aspects of people’s goal priors, embodying certain abstract principles
that could be important (along with many others) in structuring our expectations about
intentional action. By comparing the predictions of these models to people’s judgments
from our experiments, we test the extent to which they capture significant dimensions of
people’s prior knowledge, and in what contexts. We sketch the goal priors used by M1,
M2 and M3 below. The formal details of these goal priors are provided in the appendix,
including derivations of Equations 1 and 2 specific to each model.
The first inverse planning model we consider (M1) assumes that a goal refers to a
single state of the environment that an agent pursues until it is achieved. Given a goal,
probabilistic planning produces actions that tend to move the agent closer to the goal state,
but depending on β, the level of determinism, planning can sometimes yield unexpected
actions such as changes in direction or detours. Higher β values will fit actions that follow the
shortest path very well, but will fit noisy action sequences very poorly. Lower β values will
fit most action sequences moderately well, but will not fit any action sequence particularly
closely. For example, in the introductory vignette the best explanation of the woman’s
erratic behavior in terms of M1 is that she tends to pursue her goals in a particularly noisy
manner, or has a low β value. Moreover, the assumption of a noisy (low β) agent is the only
way that M1 can explain paths that deviate from the simplest notion of rational action,
described by a shortest path to a single, fixed goal. Our other models also support such
explanations; formally, they have the same β parameter representing the agent’s level of
determinism. They differ in allowing a broader range of alternative explanations based on
richer representations of agents’ possible goals.
The second inverse planning model we consider (M2) is an extension of M1 that
assumes that agents’ goals can change over the course of an action sequence. This allows
M2 to explain changes in direction or indirect paths to eventual goals, as in the attribution
that the woman in the introductory vignette had changed her mind about where she was
headed. M2 represents the prior probability that an agent will change its goal after an
action with the parameter γ. With γ = 0, goal changing is prohibited, and M2 is equivalent
to M1. With γ close to 0, the model rarely infers goal changes, implying that all past and

recent actions are weighted nearly equally in the model’s goal inferences. With γ close to 1,
the model infers goal changes frequently, and only the most recent actions factor into the
model’s goal inferences. Intermediate values of γ between 0 and 1 interpolate between these
extremes, adaptively integrating or forgetting past information depending on the degree of
evidence for subsequent goal changes.
The third inverse planning model we consider (M3) is an extension of M1 that assumes
that agents can have subgoals along the way to their final goal. For example, M3 can capture
the inference that the woman in the introductory vignette wanted to first complete a task
(such as dropping off a letter) before pursuing her next goal. M3 represents the prior
probability of a subgoal with the parameter κ. With κ = 0, subgoals are prohibited, and
M3 is equivalent to M1. With κ > 0, M3 can infer a sequence of subgoals to explain more
complex behaviors, such as paths with large detours from the shortest path to a final goal.
Finally, we consider a simple alternative heuristic (H) based on low-level motion cues,
inspired by Blythe et al. (1999), Zacks (2004) and Barrett et al. (2005). H looks only at
an agent’s most recent action, rather than a whole sequence of actions, assuming that at
any given time, the agent’s goal is probably the object toward which it had most recently
moved. Including H in our comparisons allows us to test whether the full machinery of
inverse planning is needed to explain human goal inferences, and in particular, the extent
to which temporal integration over agents’ entire paths is an essential feature of these
inferences. For the sake of comparison with our inverse planning models, we formulate H as
a special case of M2 in which agents’ goals can change arbitrarily after every action, i.e. γ
is set to its extreme value of 1. For many action sequences, both in our maze-world settings
and in everyday situations, H makes similar predictions to the inverse planning models.
These are cases that support a single unambiguous goal interpretation through the action
sequence. However, our experiments are designed to include a subset of conditions with
more complex trajectories that can distinguish between H and inverse planning models.
Related Work
Previous computational models of action understanding differ from our framework
along several dimensions. Much of the classic work on action understanding relies on logical
representations of the domain and the agent’s planning process (Schank & Abelson, 1977;
Kautz & Allen, 1986). These approaches use sophisticated, hierarchical representations
of goals and subtasks, such as scripts and event hierarchies, to model the structure of
agents’ behavior, and model goal inference in terms of logical sufficiency or necessity of the
observed behavior for achieving a particular goal. Probabilistic versions of these ideas have
also been proposed, which allow inductive, graded inferences of structured goals and plans
from observations of behavior (Charniak & Goldman, 1991; Bui et al., 2002; Liao et al.,
2004). However, these approaches assume that the distribution over actions, conditioned on
goals, is either available a priori (Charniak & Goldman, 1991; Bui et al., 2002), or must be
estimated from a large dataset of observed actions (Liao et al., 2004). An alternative is to
model the abstract principles underlying intentional action, which can be used to generate
action predictions in novel situations, without requiring a large dataset of prior observations.
Various forms of the rationality assumption have been used to achieve this in both logical
and probabilistic models of action understanding (Kautz & Allen, 1986; Ng & Russell, 2000;
Verma & Rao, 2006). However, these models have not compared against human judgments,

and have not explored the kinds of structured goal representations necessary to explain
human action understanding. In this paper, we integrate probabilistic models of rational
planning with simple structured representations of agents’ goals to model human action
understanding. Although we do not directly test any of the models described above, we
test whether their computational principles can account for human goal inferences in our
experiments.
Experiment 1
Our first experiment measured people’s online goal inferences in response to animated
stimuli of agents moving to reach goal objects in simple maze-like environments. Our stimuli
varied the environmental context, including the configuration of marked goals and obsta-
cles, and varied agents’ paths and the point at which participants’ judgments were collected.
This yielded fine-grained temporal measurements of human goal inferences and their sen-
sitivity to various actions and contexts. We addressed the motivating questions from the
Introduction by comparing how accurately M1, M2, M3 and H predicted participants’ judg-
ments. Comparing models based on different goal priors revealed aspects of the form and
content of the prior knowledge underlying human goal inferences. Comparing the accuracy
of these models with H tested whether people’s judgments in our experimental domain were
best explained as a process of inverse planning or the application of a simple heuristic.
Method
Participants.
Participants were 16 members of the MIT subject pool, 9 female, 7 male.
Stimuli.
Subjects viewed short animations of agents moving in simple mazes. Agents were
represented by small moving circles, and as they moved through the environment, traces of
their trajectories trailed behind them to record their entire movement history as a memory
aid. Each displayed movement sequence paused at a judgment point: a point in the middle
of the agent’s trajectory before a particular goal was achieved, where subjects reported
their online goal inferences. The environment was a discrete grid of squares that agents
could occupy, with dimensions of 17 squares wide by 9 squares high. Agents’ movements
were restricted to adjacent squares, with directions {N,S,E,W,NE,NW,SE,SW}. Known
goals were displayed as capital letters “A”, “B” and “C”, and walls were displayed as solid
black barriers. Animations were shown from an overhead perspective (i.e. looking down
on a room with a wall in the middle). Example stimuli from Experiment 1 are shown in
Fig. 2(a).
Design.
All 36 conditions of Experiment 1 are shown in Fig. 3. Our experimental design
varied three factors: goal configuration, obstacle shape and agent path. There were four
different goal configurations, displayed in columns 1-4 of Fig. 3. Only the location of goal
C changed across conditions; goals A and B were always in the upper and lower right
corners, respectively. There were two different obstacle shapes: “Solid” and “Gap”. Every
environment shown had a wall-like obstacle extending up from the bottom edge. In the

A
B
C
A
B
C
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Experiment 1 Experiment 2
Judgment
point:
7
Judgment
point:
11
Judgment
point:
7
Judgment
point:
11
(a) (b)
Figure 2. Stimulus paradigm for Experiments 1 and 2. Each stimulus presented an animation of
an agent’s path (marked by a dashed line) ending at a judgment point: a pause in the animation
that allowed participants to report their online inferences of the agent’s goal at that point. (a)
Experiment 1: online goal inference task. Subjects rated how likely each marked goal was at the
judgment point. (b) Experiment 2: retrospective goal inference task. Subjects rated how likely each
marked goal was at an earlier judgment point, given by the “+” along the dashed line.
Solid conditions this wall was unbroken, while in the Gap conditions it had a hole in the
middle through which the agent could pass. The first, fourth, and seventh rows of Fig. 3
represent the Solid conditions, while the remaining rows represent the Gap conditions.
Based on the goal configuration and obstacle shape, the agent’s path was generated
by making two choices: first, which goal (A, B or C) the agent was heading toward, and
second, whether the agent went around the obstacle or through it. The second choice only
applied in the Gap conditions; in the Solid conditions the agent could only move around the
obstacle. In Fig. 3, paths are grouped as “A” paths, “B” paths and “C” paths, respectively.
Because of C’s varying location, there were 8 unique C paths, while there were just two
unique A paths and two unique B paths because the locations of A and B were fixed. All
paths started from the same point, marked with an “x” in Fig. 3.
Each condition included a number of trials, which varied the length of the path shown
before a judgment was required. Different conditions queried subjects at different judgment
points, selected at informative points along the paths. Fig. 2(a) displays two stimuli with
judgment points of 7 and 11, respectively, as they were plotted for our subjects. In Fig. 3,
many of the initial trials are identical, and only differ in their eventual destination (e.g.
corresponding trials in rows 1 and 4 of Fig. 3 are identical up to judgment point 10).
Subjects were only shown unique stimuli, and after all redundant conditions were removed,
there were 99 stimuli in total, all represented in Fig. 3.
Procedure.
Participants were given a cover story to establish assumptions about our experimental
scenarios, including the assumption of intentional agency, a model of agents’ environments,
and a hypothesis space of agents’ goals. Participants were told they would be viewing
videos of members of an intelligent alien species collected by scientists, and that each video
displayed a different alien moving toward a different goal in the environment. They were

Goal location
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(through)
“Solid”
“Gap”
(around)
“Gap”
(through)
“Solid”
“Gap”
(around)
“Gap”
(through)
Figure 3. All stimuli from Experiment 1. We varied three factors: goal configuration, obstacle shape
and agent path. There were four goal configurations, displayed in columns 1-4. Path conditions are
grouped as “A” paths, “B” paths and “C” paths. There were two obstacle shape conditions: “Solid”
and “Gap”. There were 36 conditions, and 99 unique stimuli in total.

instructed that aliens could not pass through walls, but that they could pass through gaps
in walls. They were told that after each video, they would rate which goal the alien was
pursuing.
Stimulus trials were ordered with the earliest judgment points presented first to pre-
vent hysteresis effects from showing longer trials before their shorter segments. Trials with
the same judgment points were shown in random order. On each trial, the animation paused
at a judgment point, allowing participants to report their online inferences of the agent’s
goal at that point. Subjects first chose which goal they thought was most likely (or if two
or more were equally likely, one of the most likely). After this choice, subjects were asked
to rate the likelihood of the other goals relative to the most likely goal, on a 9-point scale
from “Equally likely”, to “Half as likely”, to “Extremely unlikely”.
Modeling.
Model predictions take the form of probability distributions over agents’ goals, given
by Equation 1 (specific versions for M1, M2, M3 and H are provided in the appendix). Our
models assumed that all goals were visible, given by the three marked locations in our stim-
uli. M3 assumed there were either 0 or 1 subgoals, which could correspond to any location
in the environment. To put people’s goal inferences on the same scale as model predictions,
subjects’ ratings were normalized to sum to 1 for each stimulus, then averaged across all
subjects and renormalized to sum to 1. The within-subjects normalization guaranteed that
all subjects’ ratings were given equal weighting in the normalized between-subjects average.
Results
We present several analyses of how accurately M1, M2, M3 and H predicted people’s
online goal inferences from Experiment 1. We begin with a qualitative analysis, which
compares subjects’ data and model predictions from several conditions of Experiment 1 to
illustrate the kinds of behavior that people are sensitive to in action understanding, and to
show how closely inverse planning models captured people’s judgments. We then turn to
several quantitative analyses which rigorously support our previous qualitative observations
and address the motivating questions from the Introduction.
Fig. 4 shows examples of our qualitative comparisons between participants’ goal in-
ferences and the predictions of M2. As we will argue quantitatively below, M2 was the
model that best explained people’s judgments in Experiment 1. The conditions in Fig. 4(a)
were selected to highlight the temporal dynamics of subjects’ ratings and model predictions
in response to different observed actions and environments. Each condition in Fig. 4(a)
differs from adjacent conditions by one stimulus feature, which illustrates the main effects
of changing the goal configuration, the obstacle shape and the agent’s path.
Participants’ average ratings with standard error bars from these conditions are shown
in Fig. 4(b). These examples illustrate several general patterns of reasoning predicted by our
models. At the beginning of each trajectory, people tended to be uncertain about the agent’s
goal. As more of the trajectory was observed, their judgments grew more confident (e.g.
compare conditions 1 and 2 of Fig. 4(b)). A comparison of adjacent conditions in Fig. 4(b)
shows how changing stimulus features had specific effects on people’s goal inferences – some
subtle, others more dramatic.
Predictions of M2 using the best-fitting parameters are shown in Fig. 4(c). The model

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P(Goal|Actions)
Model
People
Example conditions
Average
ratings
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Judgment point
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1
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Judgment point
4
Judgment point
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Judgment point
Judgment point
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5
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6
Figure 4. Example conditions, data and model predictions from Experiment 1. (a) Stimuli illus-
trating a range of conditions from the experiment. Each condition differs from adjacent conditions
by one feature—either the agent’s path, the obstacle, or the goal locations. (b) Average subject
ratings with standard error bars for the above stimuli. (c) Predictions of inverse planning model
M2 with parameters β = 2.0, γ = 0.25.
predicted people’s judgments with high accuracy across all conditions, including the effects
of varying key stimulus features of the agent’s environment or path. M2 can be used to
explain why participants’ judgments varied as they did across these conditions, based on
the general expectations about rational action that M2 embodies: agents will tend to move
along the most efficient paths toward their goal, and goals tend to stay constant over time
but occasionally can change.
Conditions 1 and 2 in Fig. 4 differed only at the end of the path, after the agent
had passed the obstacle. Before this point people assigned similar probabilities to goals A
and B, because the trajectory observed was essentially the shortest path toward both goals.
Beyond the obstacle the agent took a step in the direction of goal A (Condition 1) or goal B
(Condition 2) and people’s uncertainty resolved, because the trajectory was now consistent
with the shortest path to only one goal and departed significantly from the shortest path
to the other.
Condition 3 in Fig. 4 differed from Condition 2 only in the presence of a gap in the
obstacle. Relative to Condition 2, this resulted in a significantly lower probability assigned
to goal B early on, as the agent went around the obstacle (see step 7 in particular). This
can be thought of as a counterfactual inference: if the agent had been heading toward B
originally, it would have gone straight through the gap rather than around the obstacle;
this alternative shortcut path was not available in Condition 2. Once the agent had passed
the obstacle and turned toward B, people were very quick to change their judgment to B.
The model explained this as the inference that the agent’s goal, while initially probably A

or C, had now switched to B.
Condition 4 in Fig. 4 reversed the pattern of Condition 3: the agent initially went
through (rather than around) the obstacle and then turned toward A (rather than B).
Before the obstacle, people rated goal B most likely and A about half as likely, because the
agent’s trajectory was along the shortest path to B and followed a less efficient path to A.
Once the agent passed the obstacle and turned toward A, people changed their judgment
to B, which the model explained as a change in the agent’s goal from B to A (or, less likely,
the choice of an inefficient path to reach A).
Conditions 5 and 6 in Fig. 4 changed the position of C. Before the agent passed
through the obstacle, the agent’s trajectory was now along the shortest path to B and C,
which increased the relative probability assigned to C and decreased the probability of A
and B. After the obstacle, the agent either turned toward A (Condition 5) or B (Condition
6). In Condition 5, this conflicted with previous evidence, and people quickly changed their
goal inference to A, which the model explained as a change in goals (or the choice of a less
efficient path). In Condition 6, the movement after the obstacle was consistent with B, the
previously inferred goal, and the probability assigned to B continued to increase.
The basic logic of our quantitative analyses was to compare how accurately different
models predicted people’s judgments using measures of correlation. Our overall quantitative
analysis computed the correlation of each model class with people’s data and assessed the
statistical significance of the differences between these correlations using bootstrap cross-
validation (Cohen, 1995). Bootstrap cross-validation (BSCV) is a technique for model
selection, which measures the goodness-of-fit of models to data while preventing overfitting
and controlling for model complexity. We describe the details of our BSCV analysis further
in the appendix. The average correlations for each model from our analysis are shown in
Table 1. M2 performed best, correlating significantly higher with people’s judgments than
M1 (pBSCV < 0.0001), M3 (pBSCV < 0.0001) and H (pBSCV = 0.032). H performed second
best, correlating significantly higher with people’s judgments than M1 (pBSCV < 0.0001)
and M3 (pBSCV < 0.0001). M3 correlated significantly higher with people’s judgments than
M1 (pBSCV = 0.0004), and M1 performed worst.
M1 M2 M3 H
hri 0.82 (0.017) 0.97 (0.0046) 0.93 (0.012) 0.96 (0.0027)
Table 1: Bootstrap cross-validated correlation of inverse planning models and a simple heuristic
alternative with people’s data from Experiment 1. Numbers in parentheses indicate standard devi-
ation.
To illustrate the pattern of errors for each model, Fig. 5 shows scatter plots of the
correspondence between participants’ judgments and M1, M2, M3 and H using their best-
fitting parameter values. In Fig. 5, M2 has the fewest outliers. M1 has the most outliers,
and M3 and H have a significant number of outliers as well.
These results raised several questions. First, how sensitive was the degree of fit of
each model to the values of the parameters? We compared participants’ goal inferences to
model predictions under a range of different parameter settings. Plots of the correlation
of M1, M2, M3 and H with participants’ data across all tested values of β, γ and κ are
shown in the appendix. These plots show that a robust range of parameters around the

0 0.2 0.4 0.6 0.8 1
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r = 0.83
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People
Figure 5. Scatter plots of model predictions using best-fitting parameter settings (X-axes) versus
people’s online goal inferences (Y-axes) for all Experiment 1 stimuli. Points from the Gap obstacle
condition at judgment points 10 and 11 are plotted in black. These trials account for most of the
outliers of M1, M3 and H.
best-fitting values yielded correlations that were close to the optimum for each model.
Second, what do the best-fitting parameter values tell us about how each model, with
its specific goal representations, explains the range of observed actions? M1 correlated best
with people’s data at low values of β. In trials where the agent did not follow the shortest
path to a single goal, but instead made detours around or through the obstacle, M1 could
only explain these actions as noise, and low β values allowed M1 to filter out this noise
by integrating information more slowly over time. Our other models performed best with
higher β values than M1, because representing more sophisticated goals allowed them to
assume fewer noisy actions. M2 correlated most highly with people’s data at relatively low
values of γ. This allowed M2 to integrate evidence for goals from past actions, inferring
goal changes only rarely, when there was sufficiently strong support for them. M3 correlated
most highly with people’s data at intermediate κ values. This allowed M3 to infer subgoals
when necessary to explain large deviations from the shortest path to the final goal, but to
avoid positing them unnecessarily for paths that had a simpler explanation.
Third, was the pattern of errors of each model class informative? Did any particular
class of trials drive the differences in correlation between models? Given that some of the
differences between models, while statistically significant, were small, do the differences
in model fits reflect genuine differences in the ability of these models to capture people’s
mental representations of agents’ goals? To address these questions, we performed a targeted
analysis of a class of experimental trials that exposed key differences in how the predictions
of M1, M2, M3 and H depended on recent and past information from agents’ paths. The
targeted analysis focused on data from all of the Gap obstacle conditions, at judgment
points 10 and 11. Example conditions are shown in Fig. 6(a), with judgment points 10
and 11 circled. In Fig. 5, the black points represent ratings from the targeted analysis
conditions, which account for all of the outliers of H and most of the outliers of M1 and
M3.
In the targeted conditions, the agent’s recent actions at judgment point 10 were always
ambiguous between goals A and B. However, the entire path provided evidence for either A
or B depending on whether it went around or through the obstacle. In conditions 1 and 2 of
Fig. 6, participants used the information from the agent’s early movements, rating A nearly
twice as likely as B because the agent had gone around, not through the obstacle. M1, M2
and M3 captured this pattern by integrating evidence over the entire path, while H did not,

rating A and B nearly equally because the most recent movements were ambiguous between
the two goals. At judgment point 11, the agent either took a further step that remained
ambiguous between goals A and B, or moved unambiguously toward a particular goal. In
condition 1 of Fig. 6, the agent’s action at judgment point 11 remained ambiguous between
A and B, but participants continued to favor A due to the path history. Again, M1, M2
and M3 predicted this pattern, while H did not. In condition 2 of Fig. 6, the agent’s action
at judgment point 11 strongly supported B – a complete reversal from previous evidence.
Now, participants inferred that B was the goal, weighing the strong recent evidence over
the accumulated past evidence for A. M2 and H matched this pattern, M3 captured it only
weakly, and M1 did not, still favoring A based on all the past evidence against B.
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Judgment point Judgment point
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Figure 6. Targeted analysis of Experiment 1. (a) Example conditions from our targeted analysis.
(b) Participants’ ratings from the above conditions. (c) Predictions of M2 (β = 2.0,γ = 0.25). (d)
Predictions of H (β = 2.5). (e) Predictions of M1 (β = 0.5). (f) Predictions of M3 (β = 2.0, κ = 0.5).

Discussion
Experiment 1 showed that inverse planning models based on simple structured goal
priors can predict people’s online goal inferences very accurately in the maze-world do-
main. M2 predicted participants’ judgments most accurately overall, correlating signifi-
cantly higher with people’s judgments than the alternative models we considered. Although
M2 is a more complex model than M1 or H, with an additional parameter representing the
probability of goal switching, our analysis accounted for this, using model selection tech-
niques to measure the generalization performance of each model while preventing overfitting
and controlling for model complexity.
We probed people’s assumptions about goal switching with two additional analyses.
First, in a targeted analysis of trials where the agent’s recent actions were locally ambiguous
between goals, but globally unambiguous, M2 predicted the pattern of subjects’ judgments
more accurately than alternative models. These targeted trials comprised nearly all of the
largest outliers for the alternative models, which suggested that M2’s representation of the
probability of goal switching was necessary to explain people’s goal inferences in this context.
Second, we found that the best-fitting parameters values for the goal switching prior for
M2 were low, but nonzero, consistent with the notion that the goals of an intentional agent
may switch, but tend to persist across time.
In sum, M2 accounted best for participants’ data from Experiment 1. However,
the lower correlations of M3 and H were mainly driven by a small number of experimental
trials; for the majority of trials, M1, M2, M3 and H all made similar predictions, belying the
essential differences in the way these models parsed actions. Our next experiment looked
for more qualitative differences between these models in a new task, where models that
assumed an agent’s goal could change over time (M2 and H) made qualitatively different
predictions from models that were constrained to infer a single goal (M1 and M3).
Experiment 2
Experiment 2 presented a new task based on retrospective goal inference. Using
stimuli derived from Experiment 1, we showed subjects only paths ending at the longest
judgment points from Experiment 1 and asked them to make retrospective inferences about
agents’ goals at earlier judgment points in the action sequence. Models based on static goals
(M1 and M3) and models based on changing goals (M2 and H) made qualitatively different
predictions in this task. M1 and M3 were constrained to parse an agent’s actions in terms of
a constant, global goal (consisting of a single goal for M1, and possibly containing a subgoal
for M3). This meant that the goal inferences made by M1 and M3 did not change between
judgment points within a particular condition, because the same full path was displayed in
each trial. M2 and H parsed actions in terms of a sequence of goals. M2 was biased to infer
goals that changed infrequently, while H assumed no dependence or consistency between
an agent’s goals at each step. Comparing how accurately M1 and M3 predicted people’s
retrospective goal inferences with the accuracy of M2 and H provided further evidence for
the concept of changing goals in human action understanding.
Method
Participants.

Stimuli.
Stimuli for Experiment 2 were derived from Experiment 1 stimuli as follows. For
each trial of Experiment 1, an Experiment 2 trial first displayed an animation of an agent’s
entire path, up to the longest judgment point from that condition. Then, an intermediate
judgment point along the path, taken from that condition of Experiment 1, was marked
with a red “+”. Subjects then reported their retrospective goal inferences: how likely each
goal was when the agent was at the marked judgment point. Fig. 2(b) shows the appearance
of Experiment 2 stimuli alongside corresponding Experiment 1 stimuli.
Design.
Experiment 2 used the same set of conditions as Experiment 1, all represented in
Fig. 3. Because of this, subjects’ ratings could be compared between experiments to assess
the different effects of online and retrospective goal inference tasks. In each condition,
all but the longest judgment point trials (which provided the endpoint for each displayed
path) and judgment points of 10 (excluded to reduce the total number of stimuli) were
taken from Experiment 1. In Experiment 2, each condition displayed a unique combination
of goal configuration, obstacle shape and complete path, so there was no overlap between
judgment point trials of different conditions as there was in Experiment 1. Experiment 2
had 95 stimuli in total.
Procedure.
Subjects were given a modified version of the cover story from Experiment 1. In
Experiment 2, subjects were again told that they would view videos of members of an
intelligent alien species collected by scientists, but this time, they were told: “the scientists
are not sure how the aliens decide where to go, but the aliens generally move toward goals
in the environment that are labeled for you with capital letters.” They were then told that
after the video of each alien’s movement, a point along the path would be marked, and
they would rate which goal the alien “had in mind” at that marked point. The stimulus
ordering and rating procedure was similar to Experiment 1, except that now subjects rated
the likelihood of different goals at earlier points in agents’ action sequences, rather than at
the end of their action sequences as in Experiment 1.
Modeling.
We modeled Experiment 2 in the same manner as Experiment 1, except now model
predictions were given by retrospective versions of Equation 1 (specific versions for M1, M2,
M3 and H are provided in the appendix).
Results
Our analysis of Experiment 2 paralleled our analysis of Experiment 1, combining
qualitative and quantitative comparisons of how accurately M1, M2, M3 and H predicted
people’s retrospective goal inferences. As we will argue below, models that allow goal
changes, such as M2, best explained people’s judgments in Experiment 2. Fig. 7 shows
qualitative comparisons between participants’ goal inferences and the predictions of M2.
There is a direct correspondence between the conditions shown in Fig. 7 and the conditions

in Fig. 4 from Experiment 1, which allows the effects of the online task of Experiment 1
versus the retrospective task of Experiment 2 to be assessed directly. Fig. 7(a) shows retro-
spective versions of the corresponding conditions in Fig. 4(a). Each condition in Fig. 7(a)
differs from adjacent conditions by one stimulus feature, providing examples of the effect of
changing the goal configuration, the obstacle shape and the agent’s path.
A
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C
3 7 11 13
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(a)
(b)
(c)
P(Goal|Actions)
Model
People
Example conditions
Average
ratings
A
B
C
Judgment point
Judgment point
1
Judgment point
Judgment point
4
Judgment point
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6
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3 7 11 13
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Figure 7. Example conditions, data and model predictions from Experiment 2. (a) Stimuli il-
lustrating a range of conditions from the experiment. These stimuli directly correspond to the
Experiment 1 stimuli in Fig. 4(a). Dashed lines correspond to the movement subjects saw prior
to rating the likelihood of each goal at each judgment point, which are marked by black +’s. (b)
Average subject ratings with standard error bars for the above stimuli. (c) Predictions of inverse
planning model M2 with parameters β = 0.5, γ = 0.65.
Fig. 7(b) shows participants’ average ratings with standard error bars for each of these
conditions. Many of the same patterns of reasoning occurred in our Experiment 2 data as
in Experiment 1. Participants’ inferences again tended to become more certain at judgment
points closer to the end of each path. In general, however, people’s ratings from Experiment
2 reflected greater uncertainty than in Experiment 1. In the task of Experiment 2, people’s
inferences eventually approached the goal that was suggested at the end of the path. These
trends were predicted by M2 and H, but not M1 or M3, which were constrained to make
the same goal inference at each judgment point within a particular condition.
Fig. 7(c) shows the predictions of M2 with the best-fitting parameter values. Com-
paring Fig. 7(b) and (c), the model predicted people’s judgments very accurately for the
examples shown. M2 explained people’s retrospective judgments based on inferring the
sequence of goals that best explained the agent’s path. The agent’s actions near the end of
each path always strongly suggested a particular goal, but earlier actions could be ambigu-
ous, or indicate a different goal. M2 integrated this past and future information to infer the
goal at each judgment point that was most consistent with an agent that took approximately

rational paths toward its current goal, but could occasionally change its goal.
We quantitatively compared how accurately M1, M2, M3 and H predicted partici-
pants’ judgments with a bootstrap cross-validated (BSCV) correlational analysis. The av-
erage correlations of each model with people’s data are shown in Table 2. M2 predicted peo-
ple’s judgments most accurately, correlating significantly higher than M1 (pBSCV < 0.0001),
M3 (pBSCV < 0.0001) and H (pBSCV = 0.0168). H performed second best, correlating sig-
nificantly higher than M1 (pBSCV < 0.0001) and M3 (pBSCV < 0.0001). Finally, M1 and
M3 both performed poorly, and the slightly higher correlation of M3 than M1 was not
significant (pBSCV = 0.44).
M1 M2 M3 H
hri 0.57 (0.016) 0.95 (0.0077) 0.58 (0.019) 0.91 (0.0069)
Table 2: Bootstrap cross-validated correlation of inverse planning models and a simple heuristic
alternative with people’s data from Experiment 2. Numbers in parentheses indicate standard devi-
ation.
Scatter plots of the correspondence between participants’ judgments and M1, M2, M3
and H using the best-fitting parameter values are shown in Fig. 8 to illustrate the pattern of
errors for each model. M2 predicted people’s judgments most accurately, and in Fig. 8, has
no clear outliers from the line-of-best-fit. H predicted people’s judgments less accurately
than M2, and M1 and M3 performed worst and have the most outliers in Fig. 8.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
r = 0.58
M1(β=0.5)
People
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
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1
r = 0.95
M2(β=0.5,γ=0.65)
People
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
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1
r = 0.59
M3(β=1.0,κ=0.95)
People
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
r = 0.92
H(β=2.5)
People
Figure 8. Scatter plots of model predictions using best-fitting parameter values (X-axes) versus
people’s retrospective goal inferences (Y-axes) for all Experiment 2 stimuli.
To assess the dependence of these results on the parameter values, we tested how
accurately M1, M2, M3 and H predicted people’s goal inferences under a range of param-
eter settings. Plots of the correlation of M1, M2, M3 and H with participants’ data across
all tested values of β, γ and κ are shown in the appendix. In Experiment 2, all models
fit subjects’ data best with lower β values than in Experiment 1, indicating that subjects
assumed a noisier agent in Experiment 2. For M2, relatively high γ values yielded the best
correlation, suggesting that subjects were more willing to infer goal changes in Experiment
2 than in Experiment 1. At high γ values, M2 predicted that people’s ratings would depend
strongly on the information provided by agents’ movements close to the retrospective judg-
ment point. H correlated best with people’s judgments with slightly higher β values than
M2, because of the additional flexibility provided by assuming that the agent’s goal could
change arbitrarily at each timestep. Both M1 and M3 performed poorly because they made

the same predictions at all judgment point trials within each condition, while participants’
goal inferences clearly varied at different steps of agents’ paths.
Discussion
Experiment 2 showed that inverse planning models based on priors that allow chang-
ing goals can predict people’s retrospective goal inferences very accurately in the maze-
world domain. In an analysis that controlled for model complexity using bootstrap cross-
validation, M2 correlated significantly higher with participants’ judgments than models that
assumed static goals (e.g. M1 and M3). Although the predictions of M2 and H were qual-
itatively similar, M2 also correlated significantly higher with participants’ judgments than
H.
A further analysis probed the parameter values for each model that best explained
participants’ data. In Experiment 2, the highest correlations for all models occurred at
lower β values than in Experiment 1, indicating that people assumed “noisier” agents in
Experiment 2. This may have been due to several factors. First, our instructions told
subjects that “the scientists are not sure how the aliens decide where to go”, which may
have led them to assume noisier agents. Second, the retrospective task of Experiment 2
may have been more challenging than the online task of Experiment 1, making subjects less
sure of their judgments. The practical consequence of this was that evidence from agents’
movements affected subjects’ inferences less strongly in Experiment 2 than in Experiment 1.
Higher values of γ also yielded better correlations with subjects’ judgments in Experiment
2 (higher κ values technically yielded better correlations as well, but this difference was
negligible). Because the retrospective goal inference task presented a context where people
inferred changing goals, it may have also biased subjects toward responses consistent with
higher values of γ.
The results of Experiment 2 suggested that the concept of changing goals is crucial
for retrospective goal inferences. Representing goal changes allows inferences of past goals
and current goals to differ with sufficient evidence, providing an adaptive forgetting factor
that can weigh recent evidence more strongly than past evidence. Goal switching is just
one way to extend the concept of a single, static goal, but by no means the only way. The
representation of complex goals in M3 is surely also important in some contexts. Experiment
3 explored such a context.
Experiment 3
Experiment 3 probed how a range of different behaviors and situations might lead
people to infer subgoals. Any observed behavior can be explained by an infinite number of
goals, and more complex goals can always fit an observed behavior better than simpler goals,
by positing a sequence of subgoals that follows the observed path arbitrarily closely. For
example, if a person usually takes a fairly direct route home from work, with only occasional
small deviations or apparent detours, they probably have a simple, constant goal: to get
home. However, they could also sometimes have subgoals that cause the deviations: maybe
one day they wanted to stop off at the market on the way home, on another day they
wanted to stop off at the post office, and so on. There is a danger of “overfitting” in
positing these subgoals, as any random deviation could in principle be explained this way.

Stronger evidence that an agent truly has one or more subgoals would come from observing
consistent deviations from the shortest path home: if a person regularly appears to go a
few blocks out of their way at the same spot and time of day, it suggests an intentional
action, such as stopping for groceries on the way home from work. Even greater evidence
for subgoals can come from observing consistent action sequences under different starting
conditions, particularly when they lead to large apparent detours from the shortest path to
the final goal. For instance, if a person occasionally heads home not from work but from
different points (a doctor’s office, an offsite meeting), and still passes through the same
apparent subgoal locations no matter how far away, it is a strong sign that they specifically
intend to visit those locations.
These patterns of evidence are related to the notion of equifinality (Heider, 1958), a
classic cue for intentional attribution from social psychology, based on identifying invari-
ant effects of actions across multiple situations. Our framework explains inferences about
whether an agent is pursuing a simple goal or a complex goal with subgoals through a ver-
sion of the Bayesian Occam’s razor (Jefferys & Berger, 1992). In general, Bayesian inference
naturally trades off simplicity and fit to the data in evaluating hypotheses. In the context
of action understanding with probabilistic planning models, Bayesian inference weighs the
greater likelihood that comes from explaining more detailed variation in an observed trajec-
tory against the lower prior probability that is assigned to more complex goals, with more
subgoals.
To test whether people’s judgments were consistent with these principles, Experiment
3 used a new task involving prediction of an agent’s future behavior, given examples of its
behavior in the past. This experiment used simple maze-world versions of the scenarios
described above. Our stimuli presented one or more trajectories in which the agent seemed
to change direction at a single midpoint location along the way to a final goal, suggesting
the possibility of a complex goal with a subgoal. We assessed whether subjects inferred
a subgoal by asking them to predict the agent’s hypothetical trajectory starting from a
different initial location; if they inferred a subgoal, they would predict a different, more
indirect path than if they inferred only a simple final goal. Different conditions varied the
amount of evidence for complex goals. In some cases, intended to suggest a simple goal
to observers, the change in direction could be naturally explained by environmental factors
(avoiding an obstacle) or as a correction from a small random path deviation. In other cases,
observers saw multiple trajectories starting from different initial positions, all with the same
intermediate switch point, which should strongly suggest a subgoal at that location. We
compared people’s judgments in these different conditions with those of our Bayesian inverse
planning models, using the models’ ability to predict future action sequences consistent with
the goals inferred from earlier observed action sequences (Equation 2).
Method
Participants.
Stimuli.
Experiment 3 used a maze-world stimulus paradigm similar to Experiments 1 and
2. Fig. 9 shows all stimuli from Experiment 3. Each stimulus displayed a complete action

sequence, ending with the agent achieving its goal. The environment had only one visible
goal, marked by a small orange triangle, and the size of the environment was 17 squares
wide by 8 squares high.
Design.
Experiment 3 used a 2 × 2 × 2 factorial design. Each row of Fig. 9 shows a different
condition of Experiment 3, and conditions are divided into groups A, B, C, and D. The first
factor varied the directness of the agent’s paths to the goal. Conditions A and C displayed
direct example paths to the goal, while conditions B and D displayed indirect example
paths. The second factor varied the presence or absence of an obstacle in the environment.
Conditions A and B displayed environments with a wall-like obstacle extending up from the
bottom edge, and conditions C and D displayed environments without the obstacle. The
third factor varied the location of the subgoal relative to the location of the marked goal,
which affected the length of the deviation from the direct path when the agent went through
the subgoal. In the “Far” subgoal conditions, the subgoal was farther from the marked goal,
given by the grid square directly above the obstacle, resulting in a larger deviation from the
most direct path. In the “Near” subgoal conditions, the subgoal was nearer to the marked
goal, corresponding to a point between the obstacle and the marked goal, resulting in a
smaller deviation from the most direct path.
Each condition presented 4 trials, which varied the number of paths subjects saw
before predicting the agent’s action in a new situation. Each trial had two phases. In
the example phase of each trial, a new, animated example path was presented to subjects,
following the order shown in Fig. 9. The example paths varied the agent’s starting point
in the environment, but always ended with the agent reaching the same final point. The
example paths in each condition had additional structure beyond the factorial design: con-
ditions A, B and D displayed example paths consistent with a complex goal, while condition
C displayed paths that were not consistent with any single complex goal (i.e. there was
no common subgoal that all example paths passed through). Example paths from corre-
sponding trials of conditions A, C and D all started from the same point, while paths from
corresponding trials of condition B started from these points reflected to the opposite side
of the obstacle.
Next, in the response phase, two different hypothetical paths that the agent could take
if starting from a different location in the environment were displayed. All response paths
started and ended at the same point, and used the same environments and goal locations
as the example paths in their respective conditions. Each pair of response paths featured
one direct path to the goal and one indirect path to the goal, with the indirect path passing
through the subgoal from its condition. Conditions with the same subgoal location all used
the same response paths.
Procedure.
Before each condition, subjects were told verbally that they would be watching a
series of videos of intelligent aliens moving in their natural environment, and that each
video would show the same alien moving in the same environment, with walls through
which aliens could not move. They were told that after each video, they would see two
possible movements the alien might take when starting from a different location, and that

B
C
D
A
B
C
D
Example phase
1 2 1 2
3 4
Response phase
A
“Near” subgoal
“Far” subgoal
Figure 9. All stimuli from Experiment 3. The experiment followed a 2 × 2 × 2 design, which
varied the directness of the agent’s path, the presence or absence of an obstacle, and the location
of potential subgoals. Each condition had 4 trials, which each presented a different example path,
and then asked subjects to predict which of 2 response paths was more likely given all previously
displayed example paths from that condition.
they would rate how likely each movement would be given their observations of previous
movements.
The first trial of each condition presented an animation of the first example path.
Subjects then rated the relative likelihood of the two response paths on a 9-point scale from
1: “Definitely path 1”, to 5: “Paths 1 and 2 are equally likely”, to 9: “Definitely path 2”.
In subsequent trials of each condition, animations of the next example path were presented,

and all previous example paths remained onscreen to aid recall. Subjects were instructed to
base their judgments on the evidence from all the agent’s example paths in that condition.
Stimuli were shown from an overhead perspective, with an animated schematic trace
of the agent’s path as it moved through the environment. After each example path was
displayed, response paths were presented simultaneously, and the side of the screen on
which each response path was displayed was randomized in different conditions. Stimulus
conditions were presented to subjects in pseudo-random order. To minimize presentation
order effects, environments and paths were reflected and rotated in randomized increments
of 180 degrees between conditions to reduce the impression of repeated paths. To further
minimize order effects between conditions, after each condition, subjects were told verbally
that they would observe a new alien, of a different color than the previous alien(s). They
were told that this alien might be different than the previous alien(s), so they should not
use information about the previous alien(s) when making judgments about the new alien.
Modeling.
Model predictions take the form of probability distributions over agents’ future ac-
tions, conditioned on their previously observed actions, given by Equation 2. To compare
model predictions with subjects’ judgments, we computed the log posterior odds ratio of the
two response paths in each condition, conditioned on the previously observed example paths.
We then mapped the z-scores of the log posterior odds ratio through the sigmoidal standard
normal cumulative density function, and computed correlation coefficients between these
values and participants’ average ratings. We discuss this analysis further in the appendix.
M3 assumed that the agent had either a simple goal or a complex goal with one subgoal,
and that both goals and subgoals could be visible or invisible, and could correspond to any
grid square in the environment.
Results and Discussion
Our analysis of Experiment 3 compared subjects’ judgments with the predictions
of our models to test whether people’s goal-based generalization could be explained by
Bayesian inverse planning. We will focus on the predictions of M3, because it was the only
model that could represent and infer the kinds of complex goals presented in Experiment 3.
M1 performed poorly in this task, with a bootstrap cross-validated correlation coefficient of
hri = −0.03. M2 and H did not naturally allow goal-based generalization, and an extension
of them to do so performed poorly as well, with a bootstrap cross-validated correlation
coefficient of hri = 0.54 for both models. We analyze these models further in the appendix,
but it is easy to characterize their problems in intuitive terms. M1 is too inflexible, always
predicting that an agent will take the most direct path to its final goal. M2 and H, in
contrast, are too flexible. They explain apparent subgoals as arbitrary changes in the
agent’s final goal, which leads to predictions of future arbitrary goal changes that cause
noisy, unstable behavior.
Fig. 10 compares participants’ average ratings from all conditions of Experiment 3
with the predictions of M3 using the best-fitting parameter values. M3 predicted people’s
judgments very accurately overall, and also predicted the qualitative effects of varying each
stimulus factor. M3 correctly predicted that subjects would infer a subgoal in conditions
B and D, but not in conditions A and C. M3 also captured the rate at which evidence

accumulated for or against the subgoal in each condition, fitting each of the learning curves
across trials very closely.
1 2 3 4
0
0.25
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0.75
1
1 2 3 4
0
0.25
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0.75
1
r=0.97
M3 (β = 5, κ = 0.6)
“Far”
subgoal
“Near”
subgoal
Trial
Condition
1 2 3 4
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A P(Path
2|Actions)
P(Path
2|Actions)
P(Path
2|Actions)
P(Path
2|Actions)
D
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C
1 2 3 4
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People
Model
Figure 10. Subjects versus M3 with the best-fitting parameters for all stimuli from Experiment 3.
Ratings correspond directly to conditions from Fig. 9.
Two intuitive principles underlie these predictions. First, a sequence of actions pro-
vides more evidence for a subgoal interpretation to the extent that it embodies a larger
deviation from the shortest path linking the agent’s starting point to the final goal. The
larger the deviation, the more of the agent’s actions would have to be attributed arbitrarily
to chance under a simple goal interpretation – lowering the probability of that interpretation
and raising the probability of a subgoal. This explains why the paths in conditions B and
D suggested a subgoal while the condition A paths did not. It also explains why condition
B provided stronger evidence for a subgoal than condition D, and why the “Far” subgoal

conditions provided stronger evidence than the “Near” subgoal conditions. Second, because
the agent’s goal was assumed to be constant across trajectories within a single condition,
evidence for a subgoal accumulated when the same midpoint location appeared to be a
subgoal in multiple trajectories starting from different initial conditions. This explains why
subgoal inferences became stronger with more evidence in conditions B and D, and why they
were even weaker in condition C than condition A – because A paths were consistent with
a single candidate subgoal while C paths did not even pass through a common midpoint
location.
The quantitative fit of M3 to subjects’ data was also very accurate. The bootstrap
cross-validated correlation coefficient of M3 with subjects’ judgments was hri = 0.96, which
was significantly higher than the correlation of the other models (pBSCV < 0.0001). To
assess the dependence of the model predictions on the parameter values, we compared
subjects’ judgments to model predictions under a range of different parameter settings.
Plots of the correlation of our models with participants’ data across all tested parameter
values are shown in the appendix. M3 correlated very highly with subjects’ judgments for a
wide range of parameter values, with β ≥ 2.0 and κ ≤ 0.95. The best-fitting correlation for
M3 occurred with β = 5.0 and κ = 0.6, with a correlation coefficient of r = 0.97. The best-
fitting β value was higher than in previous experiments because the paths in Experiment 3
were less noisy, and M3 could assume a more deterministic agent.
In sum, Experiment 3 showed how inverse planning models that represent subgoals
can capture human action understanding in the maze-world domain, and can generalize from
previous observations to predict novel action sequences by inferring invariant goals. Our
subgoal model M3 predicted people’s judgments with high accuracy, explaining the strength
of subgoal inferences based on two intuitive stimulus factors: the number of independent
action sequences consistent with a putative subgoal, and the length of the deviation in these
paths relative to the shortest paths from each initial state to the final goal.
General Discussion
We presented a computational framework for modeling human action understanding,
and some of the first combined experimental and computational studies of adult goal infer-
ence and action prediction. Our studies made three main contributions. First, we presented
strong evidence that human action understanding can be formalized as Bayesian inverse
planning in Markov decision problems. In quantitative terms, our models correlated highly
with people’s judgments across multiple conditions and experiments. They also provided
insights into several qualitative phenomena of goal inference, such as the effect of alternative
available routes in Experiments 1 and 2, or the dependence of subgoal inferences on the
length of deviations from shortest paths in Experiment 3. Our experiments examined only
one domain of spatial navigation in simple mazes, but this task shares deep similarities with
more complex, naturalistic tasks. Likewise, our framework of inverse planning in Markov
decision problems extends to much richer settings. MDPs can be generalized to partially
observable environments (Kaelbling et al., 1998) and multi-agent situations (Littman, 1994;
Filar & Vrieze, 1997), and these generalizations can model domains and tasks that go well
beyond a single agent moving in a simple two-dimensional maze, such as motor action
(Todorov, 2004; Körding, 1997), games and other strategic interactions (Littman, 1994;
Filar & Vrieze, 1997), and a range of cooperative or communicative activities (Littman,

1994). In recent work, we have used these ideas to build inverse planning models of how
people infer “social goals” in multi-agent interactions, such as whether one agent is chasing
or fleeing from another (Baker et al., 2008); see Yoshida et al. (2008) for a similar approach.
Second, our experiments provided evidence that inverse planning models can predict
people’s judgments about an agent’s goals more accurately than a simple heuristic alterna-
tive that looks only at the agent’s current heading. Similar heuristics have been proposed to
explain how people categorize movement as intentional or animate in stimuli similar to our
own (Blythe et al., 1999; Zacks, 2004; Barrett et al., 2005). We showed how to formulate
this heuristic as a limiting case of inverse planning, providing insight into the situations
under which it approximates our ideal observer models of inverse planning. In these cases,
in particular where an agent appears to follow a single shortest path to a fixed goal, the
heuristic indeed predicted people’s judgments accurately. However, in cases where the ap-
proximation breaks down and the heuristic’s predictions diverge from our more general
inverse planning models, the latter gave a much better fit to people’s goal inferences.
Third, we showed how our framework can be used to distinguish different hypothe-
ses about the contents of people’s mental representations of agents’ goals, and provided
evidence for the importance of different kinds of goal representations in different contexts.
Experiment 1 showed that in online goal inference, goal representations must be more flex-
ible than just single fixed locations (M1); representations based on changing goals (M2) or
subgoals (M3) fit people’s judgments significantly better, with a small advantage for the
former. Experiment 2 provided definitive evidence for the use of representations of chang-
ing goals (M2) in a retrospective judgment task. Experiment 3 showed that given sufficient
evidence – multiple trajectories starting from different positions but consistent with the
same final goal and subgoal locations – people would engage more complex subgoal-based
representations.
Our models rest on a number of general assumptions about the background knowledge
underlying human action understanding, as well as specific assumptions about how subjects
approach our laboratory tasks. The rest of this discussion attempts to lay out these as-
sumptions and sketches some of our ongoing work aimed at relaxing them or explaining
them in a more satisfying way.
On each individual trial of the experiments, our subjects and our models observed
the same data, but the models were also given certain basic aspects of the MDP setup,
such as the agent’s cost function and the probabilistic dynamics by which the agent moves
through the environment. Since subjects were not explicitly told this information, how is it
reasonable to model their judgments as if they knew it? People certainly come to our tasks
– and any action understanding task – with rich background knowledge about intentional
action that goes far beyond what we have assumed here. This background knowledge
plausibly includes the assumptions made by our models: (1) that the agent’s cost function
depends on distance traveled to the ultimate goal, (2) that the agent probabilistically selects
actions as a function of their expected value, and (3) that the agent’s actions yield the
intended state transitions. Indeed, these assumptions are at the heart of previous qualitative
descriptions of human action understanding, like the “teleological stance” (Gergely et al.,
1995) and simple versions of the “intentional stance” (Dennett, 1987). Nevertheless, future
research should explicitly test these assumptions.
Another crucial piece of background knowledge assumed by our models, and conveyed

to our participants in the instructions, is that the entity under observation should be treated
as a intentional agent. In some real world contexts, by contrast, people must infer whether
an observed entity is in fact a rational or intentional agent. These inferences are likely
a function of both the entity’s observed actions in the context, and the observer’s prior
expectations based on how the entity appears or is described. Even young infants can
apply a teleological stance to a novel entity, without being explicitly told it is appropriate,
based either on the agents’ actions (Csibra et al., 1999) or its appearance (Johnson et al.,
1998; Guajardo & Woodward, 2004; Saxe et al., 2005).
In ongoing work, we are modeling the inference that an entity observed in motion is an
intentional agent, using a hierarchical Bayesian model (HBM) (Good, 1980; A. Gelman et
al., 2003; Tenenbaum et al., 2006; Kemp et al., 2007). An HBM represents data at multiple
levels of abstraction. It allows us to consider probabilistic models of both intentional action
and other kinds of motion, due to inanimate objects or artifacts, or animate but non-
intentional beings. Classifying an agent as an intentional actor can be captured by an HBM
with two levels, with the higher level selecting among alternative models that could explain
some observed motion (intentional, inanimate, animate), and the lower level selecting among
specific explanations generated by these models (e.g., a specific goal interpretation that
explains the particular trajectory that a putatively intentional agent appears to follow).
Given observations of how an entity moves in some context, these models can simultaneously
make inferences over a hypothesis space of both intentional and non-intentional models,
evaluating each model, as well as the specific hypothesis within the classes generated by
these models, based on how well they explain the observed motion.
Hierarchical Bayesian models may also provide the solution to a puzzle in our current
data. In our experiments, two different models of goal-directed action provided the best fit
to participants’ judgments: in Experiments 1 and 2, M2 (which allowed for agents to change
their goals) provided the best fit, while in Experiment 3, the best fit was with M3 (allowing
subgoals within a single action sequence). Did our participants choose different kinds of
goal representations for the actions presented in these experiments, and if so, how did they
make this choice? We hypothesized that participants actually made a rational inference
based on the stimuli they observed, selecting not only the specific goal that best explained
the agent’s action on any one trial, but also selecting the class of goal representations that
best explained all of the agents’ actions observed across multiple trials. Selecting which
class of goal representations to use in a particular context can also be captured by an HBM
with two levels, with the higher level selecting a hypothesis space of goals (M1, M2, M3
or H), and the lower level selecting a specific goal from the hypothesis space given by the
higher level. This goal then generates a sequence of actions, conditioned on the agent’s
environment.
We make this HBM analysis precise in the appendix. For each model (M1, M2, M3
or H) and each experiment (1, 2 or 3), we compute log P(Stimuli|Model), a measure of
how well that model explains all the stimuli observed in that experiment as a whole. The
results of this analysis are shown in Table 3. In each case, the model with the highest
marginal likelihood (shown in bold) is the model that correlated most highly with people’s
judgments in that experiment. Thus, the goal representation that people appear to use in
each experiment is also the one that an ideal learner would see as the best explanation for
agents’ observed behavior in that experiment.

log P(Stimuli|Model)
M1 M2 M3 H
Experiment 1 -773.6 -641.9 -651.6 -778.5
Experiment 2 -1118.5 -832.4 -860.1 -1068.1
Experiment 3 -457.2 -294.3 -236.5 -391.5
Table 3: Log marginal likelihood of models and heuristic given all Experimental stimuli. Higher
log-likelihood indicates a better fit, and the highest value for each experiment is shown in bold.
The content of people’s representations of agents’ mental states is surely much more
complex than the models we consider here. Important directions for future research are
to apply our models in more complex environments, and to continue exploring the space
of complex goal structures. A potential framework for extending our models to struc-
tured, probabilistic environments is given by factored MDPs (Guestrin et al., 2003). This
framework represents the environment as a dynamic Bayesian network, and allows efficient
algorithms for planning over much larger and more complex state spaces than those feasible
in standard MDPs. A possible representation language for hierarchical goal structures is
provided by hierarchical MDPs (Dietterich, 2000; Parr & Russell, 1998), which provide a
natural hypothesis space for more complex goals, and can represent the space of goals we
consider here. With an appropriate prior on the space of models of goals, it should be pos-
sible to learn new goal structures from data using hierarchical Bayesian models like those
we sketched above.
Another important direction for future research will be to extend our framework
to modeling people’s reasoning about agent’s beliefs. This can be formulated as inverse
planning using partially observable MDPs (POMDPs) (Kaelbling et al., 1998) to handle
joint inferences about agents’ beliefs and goals. A sketch of this extension was described in
Fig. 1(c). Here we focused on the special case of Fig. 1(b) because it has been the target
of extensive empirical work with infants and adults, and also because it is computationally
much simpler to implement. Optimal planning in POMDPs is computationally intractable,
so we would require some approximate algorithm to generate expectations about agents’
beliefs and goal-dependent behaviors. In ongoing work we are beginning to explore the
more general POMDP case, along with various approximate inference schemes based on
cognitively plausible approximations to ideal optimal planning.
Do people really interpret others’ actions using a causal model structured around an
explicit assumption of rationality, as our models do, or do they instead simulate others’
planning processes using their own planning mechanisms – in effect building in an implicit
assumption of rationality to the extent that their own planning mechanisms are rational?
The debate between “theory-based” and “simulation-based” accounts of action understand-
ing has generated much attention in the recent literature (Gopnik & Meltzoff, 1997; Gold-
man, 2006) and on first glance our work appears most consistent with the “theory-based”
approach. Formalizing an intuitive theory of mind was in fact one of our original moti-
vations, and elsewhere one of us has argued for theory-based accounts against simulation
accounts (Saxe, 2005). However, the models we propose here could be sensibly interpreted
under either account. On a theory-based interpretation, inverse planning consists of invert-
ing a causal theory of rational action to arrive at a set of goals that could have generated

the observed behavior, and inferring individual goals based on prior knowledge of the kinds
of goals the observed agent prefers. On a simulation account, goal inference is performed
by inverting one’s own planning process – the planning mechanism used in model-based
reinforcement learning – to infer the goals most likely to have generated another agent’s
observed behavior. In future work, we hope to be able to distinguish these accounts by
testing whether observers’ interpretations of other agents’ behavior in a particular context
can be predicted by inverting policies measured from how the observers themselves plan
actions in the same context.
Conclusion
Formal models of “ideal observers” or “ideal inference agents” have long played an
important role in the study of core cognitive capacities, such as visual perception, memory
retrieval, or language processing and acquisition (Liu et al., 1995; Weiss et al., 2002; Shiffrin
& Steyvers, 1997; Anderson, 1990; Hale, 2001; Gold, 1967). These models allow us to assess
how well and in what ways people’s mental representations of the world correspond to
reality, by seeing how close people’s inferences come to the best possible inferences under
different assumptions about how the relevant stimuli might be represented and processed.
Human percepts or inferences in some domains, particularly lower-level functions such as
visual motion or surface perception, often come remarkably close to the ideal limit.
Here we have taken a first step toward extending the ideal-observer approach to a
higher-level perceptual task of action understanding, specifically goal inference. Human
action is notoriously complex and difficult to predict, perhaps even irrational in many
situations (Kahneman et al., 1982; Hernstein & Prelec, 1991); it is certainly more complex
than the moving-dot or slanted-surface stimuli for which ideal observer analyses are better
known in vision. It is perhaps surprising then that people’s judgments about agents’ goals
appear so consistent with an ideal observer based on inverting a simple rational model of
action. However, in some domains, human action planning may be quite rational, and
our inferences may be based on inverting near-ideal models of behavior. Two-dimensional
navigation tasks, such as our maze-world stimuli, are a good candidate for a domain in
which humans have evolved to act and interpret others’ actions near-optimally (Barrett
et al., 2005). Or maybe in some cases, as in intuitive physics (McCloskey et al., 1980),
our mental models of the world are simpler than the world itself; if so, a probabilistic
rational-actor model seems to be a reasonable first approximation for the mind to adopt in
interpreting other people’s behavior. Future work should further explore the correspondence
between people’s intuitive models of goal-directed action and the actual mechanisms by
which humans produce action.
By applying our framework across different tasks and contexts, and exploring different
models of people’s goal representations, we found evidence for flexible use of different goal
representations depending on what the task and context suggests. It is not yet clear whether
the mental representations uncovered by our inverse planning models will generalize to
explain human action understanding outside of these laboratory tasks. More work here is
certainly needed. Our expectation, however, is that many areas of intuitive psychology will
be usefully illuminated by ideal inference models that combine a probabilistic principle of
rationality with increasingly powerful representations of intentional mental states.

Acknowledgments
This work was supported by AFOSR MURI contract FA9550-05-1-0321, the James S.
McDonnell Foundation Causal Learning Collaborative Initiative and the National Science
Foundation Graduate Fellowship (CLB). We thank Noah Goodman, Konrad Körding and
Pat Shafto for helpful comments and discussions.
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