Unexpectedness and Bayes' Rule
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Unexpectedness and Bayes' rule discusses how Simplicity Theory captures the concept of unexpectedness, which is a more general measure than Bayes' rule. It shows that Bayes' rule is a specific instantiation of ST's unexpectedness formula. ST's unexpectedness does not require explicitly stating a candidate cause and does not take a frequentist approach to encoding observations. Unexpectedness can be used to build an informational principle of framing, a model of derived likelihood, and explain the prosecutor's fallacy. It is argued that all priors are posteriors of some other prior from a complexity perspective, whereas this is not necessarily true from a probability perspective.
Unexpectedness and Bayes' Rule
- 1. Unexpectedness and Bayes’ rule 6 December 2021, CIFMA workshop Giovanni Sileno Jean-Louis Dessalles g.sileno@uva.nl jean-louis.dessalles@telecom-paris.fr University of Amsterdam Télécom Paris -- Institut Polytechnique de Paris
- 2. ● Human experience unfolds in patterns (tendencies, rules, laws, …) as much as in lack of determinism, even without taking into account quantum mechanics. We live in a “probabilistic” world /1
- 3. ● Started by investigating gambling, probability theory has grown to be the most important ingredient of formal accounts dealing with how rational agents (artificial or natural) reason in conditions of uncertainty. We live in a “probabilistic” world /2
- 4. ● Started by investigating gambling, probability theory has grown to be the most important ingredient of formal accounts dealing with how rational agents (artificial or natural) reason in conditions of uncertainty. ● Fundamental basis of Shannon’s theory of information. We live in a “probabilistic” world /2
- 5. ● The probabilistic formula named after Thomas Bayes (Bayes’ rule) has a special role in this success, as it is used for ○ Bayesian models (e.g. Bayesian networks), ○ Bayesian inference, ○ maximum a posteriori (MAP) estimation in statistics, ○ core component of machine learning methods (e.g. variational autoencoders) ○ … Bayes’ rule
- 6. ● Applications supporting or reproducing human decision-making, e.g. ○ medical diagnosis ○ evidential reasoning (eg. in criminal court settings) ○ … ● Cognitive models of ○ animal learning ○ visual perception ○ motor control ○ language processing ○ forms of social cognition ○ … Uses of Bayes’ rule
- 7. ● Applications supporting or reproducing human decision-making, e.g. ○ medical diagnosis ○ evidential reasoning (eg. in criminal court settings) ○ … ● Cognitive models of ○ animal learning ○ visual perception ○ motor control ○ language processing ○ forms of social cognition ○ … Uses of Bayes’ rule PRESCRIPTIVE accounts: how agents should reason DESCRIPTIVE accounts: how agents do produce inferences
- 8. ● clarity of the theoretical framework, ● proven practical value PROs of probability theory
- 9. ● clarity of the theoretical framework, ● proven practical value PROs of probability theory …and CONs as a FORMAL system probability theory relies on a series of axioms, e.g. a measurable space of events
- 10. ● clarity of the theoretical framework, ● proven practical value PROs of probability theory …and CONs as a FORMAL system probability theory relies on a series of axioms, e.g. a measurable space of events but our experience of the world defies this closure
- 11. ● clarity of the theoretical framework, ● proven practical value PROs of probability theory …and CONs as a FORMAL system probability theory relies on a series of axioms, e.g. a measurable space of events as a MODELLING framework several cognitive patterns (often called biases or fallacies) are not predicted by probability theory but our experience of the world defies this closure
- 12. ● clarity of the theoretical framework, ● proven practical value PROs of probability theory …and CONs as a FORMAL system probability theory relies on a series of axioms, e.g. a measurable space of events as a MODELLING framework several cognitive patterns (often called biases or fallacies) are not predicted by probability theory but our experience of the world defies this closure in particular, there is a mismatch in what humans perceive as informative w.r.t. Shannon’s notion of information
- 13. ● Simplicity Theory (ST) is a computational model of cognition whose investigation started by observing the “informativity” mismatch. Simplicity Theory NOISE SOURCE: maximally informative following Shannon’s theory of information
- 14. ● Simplicity Theory (ST) is a computational model of cognition whose investigation started by observing the “informativity” mismatch. ● ST predicts diverse human phenomena related to relevance: ○ unexpectedness ○ narrative interest ○ coincidences ○ near-miss experiences ○ emotional interest ○ responsibility ● ST has been used for experiments in artificial creativity. Simplicity Theory
- 15. ● Formally, ST builds upon Algorithmic Information Theory (AIT). Simplicity Theory: formal background
- 16. ● Formally, ST builds upon Algorithmic Information Theory (AIT). ● In AIT, the complexity of a string is the minimal length of a program that, given a certain optional input parameter, produces that string as an output (Kolmogorov complexity) Simplicity Theory: formal background underlying Turing machine target string additional input in support executable program
- 17. Simplicity Theory: formal background underlying Turing machine target string additional input in support executable program ≠ ● Formally, ST builds upon Algorithmic Information Theory (AIT). ● In AIT, the complexity of a string is the minimal length of a program that, given a certain optional input parameter, produces that string as an output (Kolmogorov complexity) how much information is needed for a program constructing the object how much time or space is needed for running it (algorithmic or time-complexity)
- 18. ● Formally, ST builds upon Algorithmic Information Theory (AIT). ● In AIT, the complexity of a string is the minimal length of a program that, given a certain optional input parameter, produces that string as an output (Kolmogorov complexity) Simplicity Theory: formal background underlying Turing machine target string additional input in support executable program ● Kolmogorov complexity is generally incomputable (due to the halting problem), but it is computable on bounded Turing machines.
- 19. ● Formally, ST builds upon Algorithmic Information Theory (AIT). ● In AIT, the complexity of a string is the minimal length of a program that, given a certain optional input parameter, produces that string as an output (Kolmogorov complexity) Simplicity Theory: formal background underlying Turing machine target string additional input in support executable program ● Kolmogorov complexity is generally incomputable (due to the halting problem), but it is computable on bounded Turing machines. We denote bounded complexities with
- 20. ● ST starts from the observation that humans are highly susceptible to complexity drops, ie. for them situations are relevant if they are simpler to describe than to explain Unexpectedness
- 21. ● ST starts from the observation that humans are highly susceptible to complexity drops, ie. for them situations are relevant if they are simpler to describe than to explain ● Formally, this is captured by the formula of unexpectedness, expressed as divergence of complexity computed on two distinct machines Unexpectedness causal complexity via world machine description complexity via description machine situation
- 22. ● ST starts from the observation that humans are highly susceptible to complexity drops, ie. for them situations are relevant if they are simpler to describe than to explain ● Formally, this is captured by the formula of unexpectedness, expressed as divergence of complexity computed on two distinct machines Unexpectedness causal complexity via world machine description complexity via description machine situation
- 23. Unexpectedness: examples ● remarkable lottery draws: 11111 is more unexpected than 64178, even if the lottery is fair ● coincidences: meeting by chance an old friend from yours abroad is more unexpected than meeting there any random unknown person. ● deterministic yet unexpected events: e.g. a lunar eclipse causal complexity via world machine description complexity via description machine situation
- 24. Aim of the paper Bayes’ rule is a specific instantiation of a more general template captured in ST by Unexpectedness ● Provide further arguments in support to non-probabilistic computational models in cognition, in particular focusing on the following: conjecture
- 25. Bayes’ rule ● From the definition of conditional probability: we can obtain the formula of Bayes simply: model observation often informally rewritten as:
- 26. Unexpectedness as posterior ● In previous works, it has been hypothesized that ST’s Unexpectedness offers as non-extensional measure of posterior subjective probability:
- 27. Unexpectedness as posterior ● In previous works, it has been hypothesized that ST’s Unexpectedness offers as non-extensional measure of posterior subjective probability: ● Starting from this hypothesis, we looked for a mapping from Unexpectedness to Bayes’ rules, and indeed we see that:
- 28. Unexpectedness as posterior ● In previous works, it has been hypothesized that ST’s Unexpectedness offers as non-extensional measure of posterior subjective probability: ● Starting from this hypothesis, we looked for a mapping from Unexpectedness to Bayes’ rules, and indeed we see that: problem: 1 parameter with unexpectednes, 2 with posterior
- 29. Unexpectedness as posterior ● In previous works, it has been hypothesized that ST’s Unexpectedness offers as non-extensional measure of posterior subjective probability: ● Starting from this hypothesis, we looked for a mapping from Unexpectedness to Bayes’ rules, and indeed we see that: problem: 1 parameter with unexpectednes, 2 with posterior let’s investigate these two terms…
- 30. Causal complexity ● The causal complexity is the length in bits of the shortest path that, according to the agent's world model, produces the situation.
- 31. Causal complexity ● The causal complexity is the length in bits of the shortest path that, according to the agent's world model, produces the situation. ● The causal path is temporally unfolded. The chain rule has the form: sequential composition causal link (implicit: from the current situation)
- 32. Causal complexity ● The causal complexity is the length in bits of the shortest path that, according to the agent's world model, produces the situation. ● The causal path is temporally unfolded. The chain rule has the form: ● Being a Kolmogorov complexity, the cause can be omitted if it lies on the shortest path sequential composition causal link (implicit: from the current situation)
- 33. Causal complexity ● The causal complexity is the length in bits of the shortest path that, according to the agent's world model, produces the situation. ● The causal path is temporally unfolded. The chain rule has the form: ● Being a Kolmogorov complexity, the cause can be omitted if it lies on the shortest path sequential composition causal link (implicit: from the current situation) the Unexpectedness formula abstracts the causally explanatory factor
- 34. Description complexity ● The description complexity is the length in bits of the shortest program that, leveraging mental resources, determines the situation
- 35. Description complexity ● The description complexity is the length in bits of the shortest program that, leveraging mental resources, determines the situation ○ e.g. determination could correspond to retrieve the situation from memory, so informationally, we need to specify the address where to look at (an encoding)
- 36. Description complexity ● The description complexity is the length in bits of the shortest program that, leveraging mental resources, determines the situation ○ e.g. determination could correspond to retrieve the situation from memory, so informationally, we need to specify the address where to look at (an encoding) ● In the proposed mapping, corresponds to , the probability of observing that situation. a theoretical link can be then established through optimal encoding in Shannon’s terms, where probability is assessed through frequency.
- 37. Description complexity ● The description complexity is the length in bits of the shortest program that, leveraging mental resources, determines the situation ○ e.g. determination could correspond to retrieve the situation from memory, so informationally, we need to specify the address where to look at (an encoding) ● In the proposed mapping, corresponds to , the probability of observing that situation. a theoretical link can be then established through optimal encoding in Shannon’s terms, where probability is assessed through frequency. ● Complexity is however a more general measure, as it allows us to consider compositional effects (eg. à la Gestalt) via adequate mental operations
- 38. Bayes’ rule vs Unexpectedness ● Bayes’ rule is a specific instantiation of ST’s Unexpectedness that: ○ makes a candidate “cause” explicit and does not select automatically the best one ○ takes a frequentist-like approach for encoding observables.
- 39. Why is this relevant? ● Unexpectedness is a more generally applicable measure. ● In the paper we show that it can be used to build: ○ an informational principle of framing ○ a model of derived likelihood ○ an explanation of the prosecutor’s fallacy
- 40. All prior is posterior of some other prior ● Let us consider an additional prior in Bayes’ formula, a sort of ‘environmental context’. Following probability theory we have two equivalent formulations for the posterior:
- 41. ● Let us consider an additional prior in Bayes’ formula, a sort of ‘environmental context’. Following probability theory we have two equivalent formulations for the posterior: ● These formulations are not equivalent when expressed in complexity terms! All prior is posterior of some other prior
- 42. ● Let us consider an additional prior in Bayes’ formula, a sort of ‘environmental context’. Following probability theory we have two equivalent formulations for the posterior: ● These formulations are not equivalent when expressed in complexity terms! All prior is posterior of some other prior abstracting c as before
- 43. ● Let us compute the difference between the two formulations: All prior is posterior of some other prior
- 44. ● Let us compute the difference between the two formulations: ● Two distinct chain rules apply on the world and description machines: All prior is posterior of some other prior
- 45. ● Let us compute the difference between the two formulations: ● Two distinct chain rules apply on the world and description machines: All prior is posterior of some other prior describing e and s together may be simpler than fully determining one term before the other (cf. informed search) the temporal constraint is dropped
- 46. ● Let us compute the difference between the two formulations: All prior is posterior of some other prior applying the chain rules…
- 47. ● Let us compute the difference between the two formulations: a necessary condition for which the two formulations may be equivalent is that the contextual prior is not unexpected. All prior is posterior of some other prior applying the chain rules…
- 48. ● Let us compute the difference between the two formulations: a necessary condition for which the two formulations may be equivalent is that the contextual prior is not unexpected. All prior is posterior of some other prior applying the chain rules… shared facts, defaults, and also improbable but descriptively complex situations
- 49. ● Let us compute the difference between the two formulations: a necessary condition for which the two formulations may be equivalent is that the contextual prior is not unexpected. All prior is posterior of some other prior applying the chain rules… informational principle of framing all contextual situations which are not unexpected provide grounds to be neglected; the remaining situations provide the “relevant” context for the situation in focus. shared facts, defaults, and also improbable but descriptively complex situations
- 50. ● Following ST, we do not have direct access to the causal complexity, as we need always to pass through a descriptive step to identify what to compute. Derived likelihood
- 51. ● Following ST, we do not have direct access to the causal complexity, as we need always to pass through a descriptive step to identify what to compute. ● So, how can we estimate likelihood? Counting back the description complexity! Derived likelihood
- 52. ● Consider the estimation of the likelihood that the wall changes colour if I close the door: Derived likelihood: examples
- 53. ● Consider the estimation of the likelihood that the wall changes colour if I close the door: Derived likelihood: examples because these elements are just in front of me because this never occurred
- 54. ● Consider the estimation of the likelihood that the wall changes colour if I close the door: Derived likelihood: examples because these elements are just in front of me because this never occurred it is implausible (if it occurred) it is improbable (to occur)
- 55. ● Consider the estimation of the likelihood that the wall changes colour if I close the door: ● The likelihood that a stone in the world moves if I close the door: Derived likelihood: examples because these elements are just in front of me because this never occurred it is implausible (if it occurred) it is improbable (to occur)
- 56. ● Consider the estimation of the likelihood that the wall changes colour if I close the door: ● The likelihood that a stone in the world moves if I close the door: Derived likelihood: examples because these elements are just in front of me because this never occurred it is implausible (if it occurred) it is improbable (to occur) because I need to specify of which stone I am talking because this never occurred
- 57. ● Consider the estimation of the likelihood that the wall changes colour if I close the door: ● The likelihood that a stone in the world moves if I close the door: Derived likelihood: examples because these elements are just in front of me because this never occurred it is implausible (if it occurred) it is improbable (to occur) because I need to specify of which stone I am talking because this never occurred it is plausible (if it occurred) it is improbable (to occur)
- 58. ● Consider the estimation of the likelihood that the wall changes colour if I close the door: ● The likelihood that a stone in the world moves if I close the door: Derived likelihood: examples because these elements are just in front of me because this never occurred it is implausible (if it occurred) it is improbable (to occur) because I need to specify of which stone I am talking because this never occurred it is plausible (if it occurred) it is improbable (to occur) NOTE: If the stone e.g. is in the room or was already described, we return to the first case!
- 59. ● Suppose that, following forensic studies, the probability that a certain DNA evidence appears if the defendant is guilty is deemed very high. Prosecutor’s fallacy
- 60. ● Suppose that, following forensic studies, the probability that a certain DNA evidence appears if the defendant is guilty is deemed very high. ● The prosecutor’s fallacy occurs when the probability that the defendant is guilty (given that there is DNA evidence) is also concluded to be comparatively high. Prosecutor’s fallacy
- 61. ● Suppose that, following forensic studies, the probability that a certain DNA evidence appears if the defendant is guilty is deemed very high. ● The prosecutor’s fallacy occurs when the probability that the defendant is guilty (given that there is DNA evidence) is also concluded to be comparatively high. Prosecutor’s fallacy this is a fallacy as it neglects the base rates
- 62. ● Let us reframe the problem in terms of complexity, introducing the definition of causally constrained unexpectedness, computed before the selection of the best cause in unexpectedness: Prosecutor’s fallacy: an explanation maps to posterior
- 63. ● Let us reframe the problem in terms of complexity, introducing the definition of causally constrained unexpectedness, computed before the selection of the best cause in unexpectedness: ● Applying the chain rule: Prosecutor’s fallacy: an explanation maps to likelihood maps to posterior
- 64. ● Let us reframe the problem in terms of complexity, introducing the definition of causally constrained unexpectedness, computed before the selection of the best cause in unexpectedness: ● Applying the chain rule: Prosecutor’s fallacy: an explanation because
- 65. ● Let us reframe the problem in terms of complexity, introducing the definition of causally constrained unexpectedness, computed before the selection of the best cause in unexpectedness: ● Applying the chain rule: Prosecutor’s fallacy: an explanation If the procurator finds plausible that the suspect is guilty:
- 66. ● Let us reframe the problem in terms of complexity, introducing the definition of causally constrained unexpectedness, computed before the selection of the best cause in unexpectedness: ● Applying the chain rule: Considering the limited number of suspects and proximity to the victim: Prosecutor’s fallacy: an explanation
- 67. ● Let us reframe the problem in terms of complexity, introducing the definition of causally constrained unexpectedness, computed before the selection of the best cause in unexpectedness: ● Applying the chain rule: Considering the limited number of suspects and proximity to the victim: Prosecutor’s fallacy: an explanation
- 68. ● Let us reframe the problem in terms of complexity, introducing the definition of causally constrained unexpectedness, computed before the selection of the best cause in unexpectedness: ● Applying the chain rule: Prosecutor’s fallacy: an explanation
- 69. ● The proposed conjecture provides further arguments in support to non-probabilistic computational models of cognition. ● A complexity-based account allows distinguishing between relevant and irrelevant contextual elements, while the probabilistic account treats them equally. ● Remaining open questions is how the underlying machines should be defined. ● Yet, the abstraction level of algorithmic information theory is already relevant to draw insights on cognitive processes, as we have shown here eg. with the analysis of the prosecutor’s fallacy. Conclusions
- 70. Unexpectedness and Bayes’ rule 6 December 2021, CIFMA workshop Giovanni Sileno Jean-Louis Dessalles g.sileno@uva.nl jean-louis.dessalles@telecom-paris.fr University of Amsterdam Télécom Paris -- Institut Polytechnique de Paris