Tutorial on EM algorithm – Part 2

Tutorial on EM algorithm – Part 2
Prof. Dr. Loc Nguyen, PhD, PostDoc
Founder of Loc Nguyen’s Academic Network, Vietnam
Email: ng_phloc@yahoo.com
Homepage: www.locnguyen.net
EM Tutorial P2 - Loc Nguyen
30/05/2022 1

Abstract
This is the chapter 2 in my book “Tutorial on EM algorithm”, which
focuses on essential description of EM algorithm.
30/05/2022 EM Tutorial P2 - Loc Nguyen 2

Table of contents
This report focuses on EM description in detail.
1. Traditional EM algorithm
2. Practical EM algorithm
3
30/05/2022

Expectation maximization (EM) algorithm has many iterations and each iteration has two steps in which expectation step (E-
step) calculates sufficient statistic of hidden data based on observed data and current parameter whereas maximization step
(M-step) re-estimates parameter. When DLR proposed EM algorithm (Dempster, Laird, & Rubin, 1977), they firstly
concerned that the PDF f(X | Θ) of hidden space belongs to exponential family. E-step and M-step at the tth iteration are
described in table 2.1 (Dempster, Laird, & Rubin, 1977, p. 4), in which the current estimate is Θ(t), with note that f(X | Θ)
belongs to regular exponential family.
Table 2.1. E-step and M-step of EM algorithm given regular exponential PDF f(X|Θ)
EM algorithm stops if two successive estimates are equal, Θ* = Θ(t) = Θ(t+1), at some tth iteration. At that time we conclude that
Θ* is the optimal estimate of EM process. Please see table 1.2 to know how to calculate E(τ(X) | Θ(t)) and E(τ(X) | Y, Θ(t)). As a
convention, the estimate of parameter Θ resulted from EM process is denoted Θ* instead of Θ in order to emphasize that Θ* is
solution of optimization problem.
E-step: We calculate current value τ(t) of the sufficient statistic τ(X) from observed Y and current parameter Θ(t) according to
equation 2.6:
𝜏 𝑡 = 𝐸 𝜏 𝑋 𝑌, Θ 𝑡
M-step: Basing on τ(t), we determine the next parameter Θ(t+1) as solution of equation 2.3:
𝐸 𝜏 𝑋 Θ = 𝜏 𝑡
Note, Θ(t+1) will become current parameter at the next iteration ((t+1)th iteration).

It is necessary to explain E-step and M-step as well as convergence of EM algorithm. Essentially, the
two steps aim to maximize log-likelihood function of Θ, denoted L(Θ), with respect to observation Y.
Θ∗
= argmax
Θ
𝐿 Θ
Where, 𝐿 Θ = log 𝑔 𝑌 Θ . Note that log(.) denotes logarithm function. Therefore, EM algorithm
is an extension of maximum likelihood estimation (MLE) method. In fact, let l(Θ) be log-likelihood
function of Θ with respect to X.
𝑙 Θ = log 𝑓 𝑋 Θ = log 𝑏 𝑋 + Θ𝑇𝜏 𝑋 − log 𝑎 Θ (2.1)
By referring to table 1.2, the first-order derivative of l(Θ) is:
d𝑙 Θ
dΘ
=
dlog 𝑓 𝑌 Θ
dΘ
= 𝜏 𝑋
𝑇
− log′ 𝑎 Θ = 𝜏 𝑋
𝑇
− 𝐸 𝜏 𝑋 Θ
𝑇
(2.2)
We set the first-order derivative of l(Θ) to be zero with expectation that l(Θ) will be maximized.
Therefore, the optimal estimate Θ* is solution of the following equation which is specified in M-step.
𝐸 𝜏 𝑋 Θ = 𝜏 𝑋

The expression E(τ(X) | Θ) is function of Θ but τ(X) is still dependent on X. Let τ(t) be value of τ(X) at the tth
iteration of EM process, candidate for the best estimate of Θ is solution of equation 2.3 according to M-step.
𝐸 𝜏 𝑋 Θ = 𝜏 𝑡 (2.3)
Where 𝐸 𝜏 𝑋 Θ = 𝑋
𝑓 𝑋 Θ 𝜏 𝑋 d𝑋. Thus, we will calculate τ(t) by maximizing the log-likelihood function
L(Θ) given Y. Recall that maximizing L(Θ) is the ultimate purpose of EM algorithm.
Θ∗ = argmax
Θ
𝐿 Θ
Where,
𝐿 Θ = log 𝑔 𝑌 Θ = log
𝜑−1 𝑌
𝑓 𝑋 Θ d𝑋 (2.4)
Due to: 𝑘 𝑋 𝑌, Θ =
𝑓 𝑋 Θ
𝑔 𝑌 Θ
, it implies:
𝐿 Θ = log 𝑔 𝑌 Θ = log 𝑓 𝑋 Θ − log 𝑘 𝑋 𝑌, Θ
Because f(X | Θ) belongs to exponential family, we have:
𝑓 𝑋 Θ = 𝑏 𝑋 exp Θ𝑇𝜏 𝑋 𝑎 Θ
𝑘 𝑋 𝑌, Θ = 𝑏 𝑋 exp Θ𝑇
𝜏 𝑋 𝑎 Θ 𝑌
The log-likelihood function L(Θ) is reduced as follows:
𝐿 Θ = −log 𝑎 Θ + log 𝑎 Θ 𝑌

By referring to table 1.2, the first-order derivative of L(Θ) is:
d𝐿 Θ
dΘ
= −log′
𝑎 Θ + log′
𝑎 Θ 𝑌 = − 𝐸 𝜏 𝑋 Θ
𝑇
+ 𝐸 𝜏 𝑋 𝑌, Θ
𝑇
(2.5)
We set the first-order derivative of L(Θ) to be zero with expectation that L(Θ) will be maximized, as follows:
𝑇
+ 𝐸 𝜏 𝑋 𝑌, Θ
𝑇
= 0
It implies: 𝐸 𝜏 𝑋 Θ = 𝐸 𝜏 𝑋 𝑌, Θ . Let Θ(t) be the current estimate at some tth iteration of EM process. Derived
from the equality above, the value τ(t) is calculated as seen in equation 2.6.
𝜏 𝑡
= 𝐸 𝜏 𝑋 𝑌, Θ 𝑡
(2.6)
Where,
𝐸 𝜏 𝑋 𝑌, Θ 𝑡
=
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ 𝑡
𝜏 𝑋 d𝑋
Equation 2.6 specifies the E-step of EM process. After t iterations we will obtain Θ* = Θ(t+1) = Θ(t) such that E(τ(X) | Y,
Θ(t)) = E(τ(X) | Y, Θ*) = τ(t) = E(τ(X) | Θ*) = E(τ(X) | Θ(t+1)) when Θ(t+1) is solution of equation 2.3 (Dempster, Laird, &
Rubin, 1977, p. 5). This means that Θ* is the optimal estimate of EM process because Θ* is solution of the equation:
𝐸 𝜏 𝑋 Θ = 𝐸 𝜏 𝑋 𝑌, Θ
Thus, we conclude that Θ* is the optimal estimate of EM process.
Θ∗
= argmax
Θ
𝐿 Θ

The EM algorithm shown in table 2.1 is totally exact with assumption that f(X|Θ) belongs to regular
exponential family. If f(X|Θ) is not regular, the maximal point (maximizer) of the log-likelihood
function l(Θ) is not always the stationary point Θ* so that the first-order derivative of l(Θ) is zero,
l’(Θ*) = 0. However, if f(X|Θ) belongs to curved exponential family, the M-step of the EM algorithm
shown in table 2.1 is modified as follows (Dempster, Laird, & Rubin, 1977, p. 5):
Θ 𝑡+1 = argmax
Θ∈Ω0
𝑙 Θ = argmax
Θ∈Ω0
𝑙 Θ 𝜏 𝑡 = argmax
Θ∈Ω0
Θ𝑇𝜏 𝑡 − log 𝑎 Θ (2.7)
Where τ(t) is calculated by equation 2.6 in E-step. This means that, in more general manner, the
maximizer Θ(t+1) will be found by some way. Recall that if Θ lies in a curved sub-manifold Ω0 of Ω
where Ω is the domain of Θ then, f(X | Θ) belongs to curved exponential family.
In general, given exponential family, within simple EM algorithm, E-step aims to calculate the
current sufficient statistic τ(t) that the log-likelihood function L(Θ(t)) gets maximal with such τ(t) at
current Θ(t) given Y whereas M-step aims to maximize the log-likelihood function l(Θ) given τ(t), as
seen in table 2.2. Note, in table 2.2, f(X|Θ) belongs to curved exponential family but it is not
necessary to be regular. Next slide will show table 2.2.

E-step: Given observed Y and current Θ(t), current value τ(t) of the sufficient statistic τ(X) is the value that the log-likelihood function L(Θ(t))
gets maximal with such τ(t). Concretely, suppose Θ* is a maximizer of L(Θ) given Y where L(Θ) is specified by equation 2.4.
Θ∗
= argmax
Θ
𝐿 Θ = argmax
Θ
𝐿 Θ 𝑌
Suppose Θ* is formulated as function of τ(X), for instance, Θ* = h(τ(X)) with note that Θ* is not evaluated because τ(X) is not evaluated.
Thus, the equation Θ* = h(τ(X)) is only symbolic formula. Let τ(t) be a value of τ(X) such that Θ(t) = h(τ(X)). This means 𝜏 𝑡
∈ 𝜏 𝑋 : Θ 𝑡
=
Table 2.2. E-step and M-step of EM algorithm given exponential PDF f(X|Θ)

EM algorithm stops if two successive estimates are equal, Θ* = Θ(t) = Θ(t+1), at some tth iteration. At that time, Θ* is the optimal estimate of EM
process, which is an optimizer of L(Θ) as Θ∗
= argmax
Θ
𝐿 Θ . Going back example 1.1, given the tth iteration, sufficient statistics x1 and x2 are
estimated as x1
(t) and x2
(t) based on current parameter θ(t) in E-step according to equation 2.6.
𝑥1
𝑡
+ 𝑥2
𝑡
= 𝑦1
𝑡
= 𝐸 𝑦1 𝑌, 𝜃 𝑡
Given py1 = 1/2 + θ/4, which implies that: 𝑥1
𝑡
+ 𝑥2
𝑡
= 𝐸 𝑦1 𝑌, 𝜃 𝑡
= 𝑦1𝑝𝑦1
= 𝑦1
1
2
+
𝜃 𝑡
4
. Because the probability of y1 is 1/2 + θ/4 and
y1 is sum of x1 and x2, let 𝑝𝑥1 𝑦1
be conditional probability of x1 given y1 and let 𝑝𝑥2 𝑦1
be conditional probability of x2 given y1 such that
𝑝𝑥1 𝑦1
=
𝑃 𝑥1, 𝑦1
𝑝𝑦1
=
𝑃 𝑥1, 𝑦1
1 2 + 𝜃 4
𝑝𝑥2 𝑦1
=
𝑃 𝑥2, 𝑦1
𝑝𝑦1
=
𝑃 𝑥2, 𝑦1
1 2 + 𝜃 4
𝑝𝑥1 𝑦1
+ 𝑝𝑥2 𝑦1
= 1
Where P(x1, y1) and P(x2, y1) are joint probabilities of (x1, y1) and (x2, y1), respectively. We can select P(x1, y1) = 1/2 and P(x2, y1) = θ/4, which
implies:
𝑥1
𝑡
= 𝐸 𝑥1 𝑌, 𝜃 𝑡
= 𝑦1
𝑡
𝑝𝑥1 𝑦1
= 𝑦1
𝑡 1 2
1 2 + 𝜃 𝑡 4
𝑥2
𝑡
= 𝐸 𝑥2 𝑌, 𝜃 𝑡
= 𝑦1
𝑡
𝑝𝑥2 𝑦1
= 𝑦1
𝑡 𝜃 𝑡
4
1 2 + 𝜃 𝑡 4
Such that
𝑥1
𝑡
+ 𝑥2
𝑡
= 𝑦1
𝑡

Note, we can select alternately as P(x1, y1) = P(x2, y1) = (1/2 + θ/4) / 2, for example but fixing P(x1, y1) as 1/2 is better because the next estimate
θ(t+1) known later depends only on x2
(t). When y1 is evaluated as y1 = 125, we obtain:
𝑥1
𝑡
= 125
1 2
1 2 + 𝜃 𝑡 4
𝑥2
𝑡
= 125
𝜃 𝑡 4
1 2 + 𝜃 𝑡 4
The expectation y1
(t) = E(y1 | Y, θ(t)) gets value 125 when y1 is evaluated as y1 = 125 and the probability corresponding to y1 gets maximal as 1/2 +
θ(t)/4 = 1. Essentially, equation 2.3 specifying M-step is result of maximizing the log-likelihood function l(Θ). This log-likelihood function is:
𝑙 Θ = log 𝑓 𝑋 𝜃 = log
𝑖=1
5
𝑥𝑖 !
𝑖=1
5
𝑥𝑖!
− 𝑥1 + 2𝑥2 + 2𝑥3 + 2𝑥4 + 2𝑥5 log 2 + 𝑥2 + 𝑥5 log 𝜃 + 𝑥3 + 𝑥4 log 1 − 𝜃
The first-order derivative of log(f(X | θ) is
dlog 𝑓 𝑋 𝜃
d𝜃
=
𝑥2+𝑥5− 𝑥2+𝑥3+𝑥4+𝑥5 𝜃
𝜃 1−𝜃
. Because y2 = x3 = 18, y3 = x4 = 20, y4 = x5 = 34 and x2 is
approximated by x2
(t), we have
𝜕log 𝑓 𝑋 𝜃
𝜕𝜃
=
𝑥2
𝑡
+34− 𝑥2
𝑡
+72 𝜃
𝜃 1−𝜃
. As a maximizer of log(f(X | θ), the next estimate θ(t+1) is solution of the following
equation
𝜕log 𝑓 𝑋 𝜃
𝜕𝜃
= 0. So we have:
𝜃 𝑡+1 =
𝑥2
𝑡
+ 34
𝑥2
𝑡
+ 72
Where, 𝑥2
𝑡
= 125
𝜃 𝑡 4
1 2+𝜃 𝑡 4

For example, given the initial θ(1) = 0.5, at the first iteration, we
have:
𝑥2
1
= 125
𝜃 1
4
1 2 + 𝜃 1 4
=
125 ∗ 0.5/4
0.5 + 0.5/4
= 25
𝜃 2
=
𝑥2
1
+ 34
𝑥2
1
+ 72
=
25 + 34
25 + 72
= 0.6082
After five iterations we gets the optimal estimate θ*:
𝜃∗
= 𝜃 5
= 𝜃 6
= 0.6268
Table 1.3 (Dempster, Laird, & Rubin, 1977, p. 3) show resulted
estimation

For further research, DLR gave a preeminent generality of EM algorithm (Dempster, Laird, & Rubin, 1977, pp. 6-
11) in which f(X | Θ) specifies arbitrary distribution. In other words, there is no requirement of exponential family.
They define the conditional expectation Q(Θ’ | Θ) according to equation 2.8 (Dempster, Laird, & Rubin, 1977, p. 6).
𝑄 Θ′
Θ = 𝐸 log 𝑓 𝑋 Θ′
𝑌, Θ =
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ log 𝑓 𝑋 Θ′
d𝑋 (2.8)
If X and Y are discrete, equation 2.8 can be re-written as follows:
𝑄 Θ′
𝑌, Θ =
𝑋∈𝜑−1 𝑌
The two steps of generalized EM (GEM) algorithm aim to maximize Q(Θ | Θ(t)) at some tth iteration as seen in table
2.3 (Dempster, Laird, & Rubin, 1977, p. 6).
Table 2.3. E-step and M-step of GEM algorithm
E-step: The expectation Q(Θ | Θ(t)) is determined based on current parameter Θ(t), according to equation 2.8.
Actually, Q(Θ | Θ(t)) is formulated as function of Θ.
M-step: The next parameter Θ(t+1) is a maximizer of Q(Θ | Θ(t)) with subject to Θ. Note that Θ(t+1) will become
current parameter at the next iteration (the (t+1)th iteration).

DLR proved that GEM algorithm converges at some tth iteration. At that time, Θ* = Θ(t+1) = Θ(t) is the optimal
estimate of EM process, which is an optimizer of L(Θ).
Θ∗ = argmax
Θ
𝐿 Θ
It is deduced from E-step and M-step that Q(Θ | Θ(t)) is increased after every iteration. How to maximize
Q(Θ|Θ(t)) is the optimization problem which is dependent on applications. For example, the estimate Θ(t+1) can
be solution of the equation created by setting the first-order derivative of Q(Θ|Θ(t)) regarding Θ to be zero,
DQ(Θ|Θ(t)) = 0T. If solving such equation is too complex or impossible, some popular methods to solve
optimization problem are Newton-Raphson (Burden & Faires, 2011, pp. 67-71), gradient descent (Ta, 2014),
and Lagrange duality (Wikipedia, Karush–Kuhn–Tucker conditions, 2014). Note, solving the equation
DQ(Θ|Θ(t)) = 0T may be incorrect in some case, for instance, in theory, Θ(t+1) such that DQ(Θ(t+1)|Θ(t)) = 0T
may be a saddle point (not a maximizer).
GEM algorithm still aims to maximize the log-likelihood function L(Θ) specified by equation 2.4, which is
explained here. Following is proof of equation 2.8. Suppose the current parameter is Θ after some iteration.
Next we must find out the new estimate Θ* that maximizes the next log-likelihood function L(Θ’).
Θ∗ = argmax
Θ′
𝐿 Θ′ = argmax
Θ′
log 𝑔 𝑌 Θ′

The next log-likelihood function L(Θ’) is re-written as follows:
𝐿 Θ′
= log
𝜑−1 𝑌
𝑓 𝑋 Θ′
d𝑋 = log
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ
𝑓 𝑋 Θ′
𝑘 𝑋 𝑌, Θ
d𝑋
Due to 𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ′
d𝑋 = 1, by applying Jensen’s inequality (Sean, 2009, pp. 3-4) with concavity of logarithm function
log
𝑥
𝑢 𝑥 𝑣 𝑥 d𝑥 ≥
𝑥
𝑢 𝑥 log 𝑣 𝑥 d𝑥 , where
𝑥
𝑢 𝑥 d𝑥 = 1
into L(Θ’), we have (Sean, 2009, p. 6):
𝐿 Θ′
≥
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ log
𝑓 𝑋 Θ′
𝑘 𝑋 𝑌, Θ
d𝑋 =
𝜑−1 𝑌
− log 𝑘 𝑋 𝑌, Θ d𝑋
=
𝜑−1 𝑌
d𝑋 −
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ log 𝑘 𝑋 𝑌, Θ d𝑋 = 𝑄 Θ′
Θ − 𝐻 Θ Θ
Where,
𝑄 Θ′
Θ =
𝜑−1 𝑌
d𝑋
𝐻 Θ′
Θ =
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ log 𝑘 𝑋 𝑌, Θ′
d𝑋

The lower-bound of L(Θ’) is defined as follows:
lb(Θ’ | Θ) = Q(Θ’ | Θ) – H(Θ | Θ)
Of course, we have:
L(Θ’) ≥ lb(Θ’ | Θ)
Suppose at some tth iteration, when the current parameter is Θ(t), the lower-bound of L(Θ) is re-written:
lb(Θ | Θ(t)) = Q(Θ | Θ(t)) – H(Θ(t) | Θ(t))
Of course, we have:
L(Θ) ≥ lb(Θ | Θ(t))
The lower bound lb(Θ | Θ(t)) has following property (Sean, 2009, p. 7):
lb(Θ(t) | Θ(t)) = Q(Θ(t) | Θ(t)) – H(Θ(t) | Θ(t)) = L(Θ(t))
Indeed, we have:
𝑙𝑏 Θ 𝑡 Θ 𝑡 = 𝑄 Θ 𝑡 Θ 𝑡 − 𝐻 Θ 𝑡 Θ 𝑡
=
𝜑−1 𝑌
log 𝑓 𝑋 Θ 𝑡
d𝑋 −
𝜑−1 𝑌
log 𝑘 𝑋 𝑌, Θ 𝑡
d𝑋
=
𝜑−1 𝑌
log
𝑓 𝑋 Θ 𝑡
d𝑋 =
𝜑−1 𝑌
log 𝑔 𝑌 Θ 𝑡
d𝑋 = log 𝑔 𝑌 Θ 𝑡
𝜑−1 𝑌
d𝑋
= log 𝑔 𝑌 Θ 𝑡 = 𝐿 Θ 𝑡

Recall that the main purpose of GEM algorithm is to maximize the log-likelihood L(Θ) =
log(g(Y|Θ)) with observed data Y. However, it is too difficult to maximize log(g(Y | Θ))
because g(Y | Θ) is not well-defined when g(Y | Θ) is integral of f(X | Θ) given a general
mapping function. DLR solved this problem by an iterative process which is an instance of
GEM algorithm. The lower-bound (Sean, 2009, pp. 7-8) of L(Θ) is maximized over many
iterations of the iterative process so that L(Θ) is maximized finally. Such lower-bound is
determined indirectly by the condition expectation Q(Θ | Θ(t)) so that maximizing Q(Θ | Θ(t))
is the same to maximizing the lower bound. Suppose Θ(t+1) is a maximizer of Q(Θ | Θ(t)) at
tth iteration, which is also a maximizer of the lower bound at tth iteration.
Θ 𝑡+1 = argmax
Θ
𝑙𝑏 Θ Θ 𝑡 = argmax
Θ
𝑄 Θ Θ 𝑡
Note, H(Θ(t) | Θ(t)) is constant with regard to Θ. The lower bound is increased after every
iteration. As a result, the maximizer Θ* of the final lower-bound after many iterations will
be expected as a maximizer of L(Θ) in final. Therefore, the two steps of GEM is interpreted
with regard to the lower bound lb(Θ | Θ(t)) as seen in table 2.4.

E-step: The lower bound lb(Θ | Θ(t)) is re-calculated
based on Q(Θ | Θ(t)).
M-step: The next parameter Θ(t+1) is a maximizer of Q(Θ
| Θ(t)) which is also a maximizer of lb(Θ | Θ(t)) because
H(Θ(t) | Θ(t)) is constant.
Θ 𝑡+1 = argmax
Θ
𝑙𝑏 Θ Θ 𝑡 = argmax
Θ
𝑄 Θ Θ 𝑡
Note that Θ(t+1) will become current parameter at the
next iteration so that the lower bound is increased in the
next iteration.
Table 2.4. An interpretation of GEM with lower bound
Because Q(Θ | Θ(t)) is defined fixedly in E-step, most
variants of EM algorithm focus on how to maximize
Q(Θ’ | Θ) in M-step more effectively so that EM is faster
or more accurate. Figure 2.1 (Borman, 2004, p. 7) shows
relationship between the log-likelihood function
L(Θ) and its lower-bound lb(Θ | Θ(t)).
Figure 2.1. Relationship between the log-
likelihood function and its lower-bound
Now ideology of GEM is explained in detail.

The next section focuses on convergence of GEM algorithm proved by DLR (Dempster, Laird, & Rubin, 1977, pp. 7-10) but
firstly we should discuss some features of Q(Θ’ | Θ). In special case of exponential family, Q(Θ’ | Θ) is modified by equation
2.9.
𝑄 Θ′
Θ = 𝐸 log 𝑏 𝑋 𝑌, Θ + Θ′ 𝑇
𝜏Θ − log 𝑎 Θ′
(2.9)
Where,
𝐸 log 𝑏 𝑋 𝑌, Θ =
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ log 𝑏 𝑋 d𝑋
𝜏Θ = 𝐸 𝜏 𝑋 𝑌, Θ =
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ 𝜏 𝑋 d𝑋
Following is a proof of equation 2.9.
𝑄 Θ′
𝑌, Θ =
𝜑−1 𝑌
d𝑋 =
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ log 𝑏 𝑋 exp Θ′ 𝑇
𝜏 𝑋 𝑎 Θ′
d𝑋
=
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ log 𝑏 𝑋 + Θ′ 𝑇
𝜏 𝑋 − log 𝑎 Θ′
d𝑋 =
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ log 𝑏 𝑋 d𝑋 +
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ Θ′ 𝑇
𝜏 𝑋 d𝑋 −
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ log 𝑎 Θ′
d𝑋
= 𝐸 log 𝑏 𝑋 𝑌, Θ + Θ′ 𝑇
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ 𝜏 𝑋 d𝑋 − log 𝑎 Θ′
= 𝐸 log 𝑏 𝑋 𝑌, Θ + Θ′ 𝑇
𝐸 𝜏 𝑋 𝑌, Θ − log 𝑎 Θ′
Because k(X | Y, Θ) belongs exponential family, the expectation E(τ(X) | Y, Θ) is function of Θ, denoted τΘ. It implies:
𝑄 Θ′
Θ = 𝐸 log 𝑏 𝑋 𝑌, Θ + Θ′ 𝑇
𝜏Θ − log 𝑎 Θ′
∎

If f(X|Θ) belongs to regular exponential family, Q(Θ’ | Θ) gets maximal at the stationary point Θ* so that the first-order
derivative of Q(Θ’ | Θ) is zero. By referring to table 1.2, the first-order derivative of Q(Θ’ | Θ) with regard to Θ’ is:
d𝑄 Θ′ Θ
dΘ′ = 𝜏Θ
𝑇
− log′
𝑎 Θ = 𝜏Θ
𝑇
𝑇
Let τ(t) be the value of τΘ at the tth iteration.
𝜏 𝑡 = 𝐸 𝜏 𝑋 𝑌, Θ 𝑡 =
𝜑−1 𝑌
𝑘 𝑋 𝑌, Θ 𝑡 𝜏 𝑋 d𝑋
The equation above is indeed equation 2.6. The next parameter Θ(t+1) is determined at M-step as solution of the following
equation.
d𝑄 Θ′
Θ
dΘ′ = 𝜏 𝑡 𝑇
𝑇
= 𝟎𝑇
This implies 𝐸 𝜏 𝑋 Θ = 𝜏 𝑡
. The equation above is indeed equation 2.3. If f(X|Θ) belongs to curved exponential family,
Θ(t+1) is determined as follows:
Θ 𝑡+1
= argmax
Θ′
𝑄 Θ′
Θ = argmax
Θ′
Θ′ 𝑇
𝜏 𝑡
− log 𝑎 Θ′
The equation above is indeed equation 2.7. Therefore, GEM shown in table 2.3 degrades into EM shown in table 2.1 and table
2.2 if f(X|Θ) belongs to exponential family. Of course, this recognition is trivial. Example 1.1 is also a good example for GEM
when multinomial distribution belongs to exponential family and then we apply equation 2.7 into maximizing Q(Θ’ | Θ).

In practice, if Y is observed as particular N observations Y1, Y2,…, YN. Let 𝒴 = {Y1, Y2,…, YN} be the observed sample of size
N with note that all Yi (s) are mutually independent and identically distributed (iid). Given an observation Yi, there is an
associated random variable Xi. All Xi (s) are iid and they are not existent in fact. Each 𝑋𝑖 ∈ 𝑿 is a random variable like X. Of
course, the domain of each Xi is X. Let 𝒳 = {X1, X2,…, XN} be the set of associated random variables. Because all Xi (s) are
iid, the joint PDF of 𝒳 is determined as follows:
𝑓 𝒳 Θ = 𝑓 𝑋1, 𝑋2, … , 𝑋𝑁 Θ =
𝑖=1
𝑁
𝑓 𝑋𝑖 Θ
Because all Xi (s) are iid and each Yi is associated with Xi, the conditional joint PDF of 𝒳 given 𝒴 is determined as follows:
𝑘 𝒳 𝒴, Θ = 𝑘 𝑋1, 𝑋2, … , 𝑋𝑁 𝑌1, 𝑌2, … , 𝑌𝑁, Θ =
𝑖=1
𝑁
𝑘 𝑋𝑖 𝑌1, 𝑌2, … , 𝑌𝑁, Θ =
𝑖=1
𝑁
𝑘 𝑋𝑖 𝑌𝑖, Θ
The conditional expectation Q(Θ’ | Θ) given samples X and Y is determined as follows:
𝑄 Θ′
Θ =
𝜑−1 𝒴
𝑘 𝒳 𝒴, Θ log 𝑓 𝒳 Θ′
d𝒳
=
𝜑−1 𝑌1 𝜑−1 𝑌2
…
𝜑−1 𝑌𝑁
𝑗=1
𝑁
𝑘 𝑋𝑗 𝑌
𝑗, Θ ∗ log
𝑖=1
𝑁
𝑓 𝑋𝑖 Θ′ d𝑋𝑁 … d𝑋2 d𝑋1

𝑄 Θ′ Θ = ⋯ =
𝜑−1 𝑌1 𝜑−1 𝑌2
…
𝜑−1 𝑌𝑁
𝑗=1
𝑁
𝑘 𝑋𝑗 𝑌
𝑗, Θ ∗
𝑖=1
𝑁
log 𝑓 𝑋𝑖 Θ′
d𝑋𝑁 … d𝑋2 d𝑋1
=
𝜑−1 𝑌1 𝜑−1 𝑌2
…
𝜑−1 𝑌𝑁
𝑖=1
𝑁
𝑗=1
𝑁
𝑘 𝑋𝑗 𝑌
𝑗, Θ ∗ log 𝑓 𝑋𝑖 Θ′
d𝑋𝑁 … d𝑋2 d𝑋1
=
𝑖=1
𝑁
𝜑−1 𝑌1 𝜑−1 𝑌2
…
𝜑−1 𝑌𝑁
log 𝑓 𝑋𝑖 Θ′
∗
𝑗=1
𝑁
𝑘 𝑋𝑗 𝑌
𝑗, Θ d𝑋𝑁 … d𝑋2 d𝑋1
(Suppose f(Xi | Θ) and k(Xj | Yj, Θ) are analytic functions)
=
𝑖=1
𝑁
𝜑−1 𝑌1 𝜑−1 𝑌2
…
𝜑−1 𝑌𝑁 𝑋
𝛿 𝑋, 𝑋𝑖 log 𝑓 𝑋 Θ′
∗
𝑗=1
𝑁
𝑘 𝑋𝑗 𝑌
𝑗, Θ d𝑋 d𝑋𝑁 … d𝑋2 d𝑋1
𝛿 𝑋, 𝑋𝑖 =
1 if 𝑋 = 𝑋𝑖
0 if 𝑋 ≠ 𝑋𝑖
⇒
𝑋
𝛿 𝑋, 𝑋𝑖 𝑢 𝑋 d𝑋 = 𝑢 𝑋𝑖
like Riemann integral
with note that the domain of 𝑋 and 𝑋𝑖 is 𝑿

𝑄 Θ′
Θ = ⋯ =
𝑖=1
𝑁
𝑋 𝜑−1 𝑌1 𝜑−1 𝑌2
…
𝜑−1 𝑌𝑁
𝛿 𝑋, 𝑋𝑖 log 𝑓 𝑋 Θ′
∗
𝑗=1
𝑁
𝑘 𝑋𝑗 𝑌
𝑗, Θ d𝑋𝑁 … d𝑋2 d𝑋1 𝑑𝑋
=
𝑖=1
𝑁
𝑋 𝜑−1 𝑌1 ,𝜑−1 𝑌2 ,…,𝜑−1 𝑌𝑁
𝛿 𝑋, 𝑋𝑖 log 𝑓 𝑋 Θ′ ∗
𝑗=1
𝑁
𝑘 𝑋𝑗 𝑌
𝑗, Θ d𝑋𝑁d𝑋2d𝑋1 … 𝑑𝑋
=
𝑖=1
𝑁
𝑋
log 𝑓 𝑋 Θ′ ∗
𝜑−1 𝑌1 ,𝜑−1 𝑌2 ,…,𝜑−1 𝑌𝑁
𝛿 𝑋, 𝑋𝑖 ∗
𝑗=1
𝑁
𝑘 𝑋𝑗 𝑌
=
𝑖=1
𝑁
𝑋
log 𝑓 𝑋 Θ′ ∗
𝜑−1 𝑌1 ,𝜑−1 𝑌2 ,…,𝜑−1 𝑌𝑁
𝛿 𝑋, 𝑋𝑖 𝑘 𝑋𝑖 𝑌𝑖, Θ ∗
𝑗=1,𝑗≠𝑖
𝑁
𝑘 𝑋𝑗 𝑌
=
𝑖=1
𝑁
𝑋
log 𝑓 𝑋 Θ′
∗
𝜑−1 𝑌1 ,𝜑−1 𝑌2 ,…,𝜑−1 𝑌𝑖−1 ,
𝜑−1 𝑌𝑖 ,𝜑−1 𝑌𝑖+1 ,…,𝜑−1 𝑌𝑁
𝑗=1,𝑗≠𝑖
𝑁
𝑘 𝑋𝑗 𝑌
𝑗, Θ d𝑋𝑁 … d𝑋𝑖+1d𝑋𝑖d𝑋𝑖−1 … d𝑋2d𝑋1 … d𝑋
=
𝑖=1
𝑁
𝑋
log 𝑓 𝑋 Θ′
∗
𝜑−1 𝑌1 ,𝜑−1 𝑌2 ,…𝜑−1 𝑌𝑖−1 𝜑−1 𝑌𝑖
𝜑−1 𝑌𝑖+1 ,…,𝜑−1 𝑌𝑁
𝑗=1,𝑗≠𝑖
𝑁
𝑘 𝑋𝑗 𝑌
𝑗, Θ d𝑋𝑁 … d𝑋𝑖+1 d𝑋𝑖 d𝑋𝑖−1 … d𝑋2d𝑋1 d𝑋

𝑄 Θ′
Θ = ⋯ =
𝑖=1
𝑁
𝑋
log 𝑓 𝑋 Θ′
∗
𝜑−1 𝑌𝑖
𝛿 𝑋, 𝑋𝑖 𝑘 𝑋𝑖 𝑌𝑖, Θ d𝑋𝑖 ∗
𝜑−1 𝑌1 ,𝜑−1 𝑌2 ,…,
𝜑−1 𝑌𝑖−1 ,𝜑−1 𝑌𝑖+1 ,…,𝜑−1 𝑌𝑁
𝑗=1,𝑗≠𝑖
𝑁
𝑘 𝑋𝑗 𝑌
𝑗, Θ d𝑋𝑁 … d𝑋𝑖+1d𝑋𝑖−1 … d𝑋2d𝑋1 d𝑋
=
𝑖=1
𝑁
𝑋
log 𝑓 𝑋 Θ′
∗
𝜑−1 𝑌𝑖
𝛿 𝑋, 𝑋𝑖 𝑘 𝑋𝑖 𝑌𝑖, Θ d𝑋𝑖 ∗
𝑗=1,𝑗≠𝑖
𝑁
𝜑−1 𝑌𝑗
𝑘 𝑋𝑗 𝑌
𝑗, Θ d𝑋𝑗 d𝑋
=
𝑖=1
𝑁
𝑋
log 𝑓 𝑋 Θ′
∗
𝜑−1 𝑌𝑖
𝛿 𝑋, 𝑋𝑖 𝑘 𝑋𝑖 𝑌𝑖, Θ d𝑋𝑖 d𝑋
Due to
𝜑−1 𝑌𝑗
𝑘 𝑋𝑗 𝑌
𝑗, Θ d𝑋𝑗 = 1
=
𝑖=1
𝑁
𝜑−1 𝑌𝑖 𝑋
𝛿 𝑋, 𝑋𝑖 𝑘 𝑋𝑖 𝑌𝑖, Θ log 𝑓 𝑋 Θ′
d𝑋 d𝑋𝑖
(Suppose f(Xi | Θ) and k(Xj | Yj, Θ) are analytic functions)

Like taking Riemann integral on 𝑋
𝛿 𝑋, 𝑋𝑖 𝑘 𝑋𝑖 𝑌𝑖, Θ log 𝑓 𝑋 Θ′
d𝑋, we have:
𝜑−1 𝑌𝑖 𝑋
𝛿 𝑋, 𝑋𝑖 𝑘 𝑋𝑖 𝑌𝑖, Θ log 𝑓 𝑋 Θ′ d𝑋 d𝑋𝑖 =
𝜑−1 𝑌𝑖
𝑘 𝑋𝑖 𝑌𝑖, Θ log 𝑓 𝑋𝑖 Θ′ d𝑋𝑖
As a result, the conditional expectation Q(Θ’ | Θ) given an observed sample 𝒴 = {Y1, Y2,…, YN} and a set of associated random
variables 𝒳 = {X1, X2,…, XN} is specified as follows:
𝑄 Θ′ Θ =
𝑖=1
𝑁
𝜑−1 𝑌𝑖
𝑘 𝑋𝑖 𝑌𝑖, Θ log 𝑓 𝑋𝑖 Θ′ d𝑋𝑖
Note, all Xi (s) are iid and they are not existent in fact. Because all Xi are iid, let X be the random variable representing every Xi and
the equation of Q(Θ’ | Θ) is re-written according to equation 2.10.
𝑄 Θ′
Θ =
𝑖=1
𝑁
𝜑−1 𝑌𝑖
𝑘 𝑋 𝑌𝑖, Θ log 𝑓 𝑋 Θ′
d𝑋 (2.10)
The similar proof of equation 2.10 in case that Xi (s) are discrete is found in (Bilmes, 1998, p. 4). If X and all Yi (s) are discrete,
equation 2.10 can be re-written as follows:
𝑄 Θ′
Θ =
𝑖=1
𝑁
𝑋∈𝜑−1 𝑌𝑖
𝑘 𝑋 𝑌𝑖, Θ log 𝑓 𝑋 Θ′

In case that f(X | Θ) and k(X | Yi, Θ) belong to exponential family, equation 2.10
becomes equation 2.11 with an observed sample 𝒴 = {Y1, Y2,…, YN}.
𝑄 Θ′
Θ =
𝑖=1
𝑁
𝐸 log 𝑏 𝑋 𝑌𝑖, Θ + Θ′ 𝑇
𝑖=1
𝑁
𝜏Θ,𝑌𝑖
− 𝑁log 𝑎 Θ′
(2.11)
Where,
𝐸 log 𝑏 𝑋 𝑌𝑖, Θ =
𝜑−1 𝑌𝑖
𝑘 𝑋 𝑌𝑖, Θ log 𝑏 𝑋 d𝑋
𝜏Θ,𝑌𝑖
= 𝐸 𝜏 𝑋 𝑌𝑖, Θ =
𝜑−1 𝑌𝑖
𝑘 𝑋 𝑌𝑖, Θ 𝜏 𝑋 d𝑋
Please combine equation 2.9 and equation 2.10 to comprehend how to derive equation
2.11. Note, 𝜏Θ,𝑌𝑖
is dependent on both Θ and Yi.

DLR (Dempster, Laird, & Rubin, 1977, p. 1) called X as complete data because the mapping φ:
X → Y is many-one function. There is another case that the complete space Z consists of hidden
space X and observed space Y with note that X and Y are separated. There is no explicit
mapping φ from X and Y but there exists a PDF of 𝑍 ∈ 𝒁 as the joint PDF of 𝑋 ∈ 𝑿 and 𝑌 ∈ 𝒀.
𝑓 𝑍 Θ = 𝑓 𝑋, 𝑌 Θ
In this case, the equation 2.8 is modified with the joint PDF f(X, Y | Θ). The PDF of Y becomes:
𝑓 𝑌 Θ =
𝑋
𝑓 𝑋, 𝑌 Θ d𝑋
The PDF f(Y|Θ) is equivalent to the PDF g(Y|Θ) mentioned in equation 1.34. Although there is
no explicit mapping from X to Y, the PDF of Y above implies an implicit mapping from Z to Y.
The conditional PDF of X given Z is specified according to Bayes’ rule as follows:
𝑓 𝑍 𝑌, Θ = 𝑓 𝑋, 𝑌 𝑌, Θ = 𝑓 𝑋 𝑌 𝑓 𝑌 𝑌 = 𝑓 𝑋 𝑌, Θ =
𝑓 𝑋, 𝑌 Θ
𝑓 𝑌 Θ
=
𝑓 𝑋, 𝑌 Θ
𝑋
The conditional PDF f(X|Y, Θ) is equivalent to the conditional PDF k(X|Y, Θ) mentioned in
equation 1.35.

Equation 2.12 specifies the conditional expectation Q(Θ’ | Θ) in case that there is no explicit mapping from X to
Y but there exists the joint PDF of X and Y.
𝑄 Θ′ Θ =
𝑋
𝑓 𝑍 𝑌, Θ log 𝑓 𝑍 Θ′ d𝑋 =
𝑋
𝑓 𝑋 𝑌, Θ log 𝑓 𝑋, 𝑌 Θ′ d𝑋 (2.12)
Note, X is separated from Y and the complete data Z = (X, Y) is composed of X and Y. For equation 2.12, the
existence of the joint PDF f(X, Y | Θ) can be replaced by the existence of the conditional PDF f(Y|X, Θ) and the
prior PDF f(X|Θ) due to:
𝑓 𝑋, 𝑌 Θ = 𝑓 𝑌 𝑋, Θ 𝑓 𝑋 Θ
In applied statistics, equation 2.8 is often replaced by equation 2.12 because specifying the joint PDF f(X, Y | Θ)
is more practical than specifying the mapping φ: X → Y. However, equation 2.8 is more general equation 2.12
because the requirement of the joint PDF for equation 2.12 is stricter than the requirement of the explicit
mapping for equation 2.8. In case that X and Y are discrete, equation 2.12 becomes:
𝑄 Θ′ Θ =
𝑋
𝑃 𝑋 𝑌, Θ log 𝑃 𝑋, 𝑌 Θ′
In case that X and Y are discrete, P(X, Y | Θ) is the joint probability of X and Y whereas P(X | Y, Θ) is the
conditional probability of X given Y.

Equation 2.12 can be proved alternately without knowledge related to complete data (Sean, 2009). This proof
is like the proof of equation 2.8. In fact, given hidden space X, observed space Y, and a joint PDF f(X, Y | Θ),
the likelihood function L(Θ’) is re-defined here as log(f(Y | Θ’)). The maximizer is:
Θ∗ = argmax
Θ′
𝐿 Θ′ = argmax
Θ′
log 𝑓 𝑌 Θ′
Suppose the current parameter is Θ after some iteration. Next we must find out the new estimate Θ* that
maximizes the next log-likelihood function L(Θ’). Suppose the total probability of observed data can be
determined by marginalizing over hidden data:
𝑓 𝑌 Θ′
=
𝑋
𝑓 𝑋, 𝑌 Θ′
d𝑋
The expansion of f(Y | Θ’) is total probability rule. The next log-likelihood function L(Θ’) is re-written:
𝐿 Θ′ = log 𝑓 𝑌 Θ′ = log
𝑋
𝑓 𝑋, 𝑌 Θ′ d𝑋 = log
𝑋
𝑓 𝑋 𝑌, Θ
𝑓 𝑋 𝑌, Θ
d𝑋
Because hidden X is the complete set of mutually exclusive variables, the sum of conditional probabilities of
X is equal to 1 given Y and Θ as 𝑋
𝑓 𝑋 𝑌, Θ d𝑋 = 1, where 𝑓 𝑋 𝑌, Θ =
𝑓 𝑋, 𝑌 Θ
𝑋

Applying Jensen’s inequality (Sean, 2009, pp. 3-4) with concavity of logarithm function log 𝑥
𝑢 𝑥 𝑣 𝑥 d𝑥 ≥
𝑥
𝑢 𝑥 log 𝑣 𝑥 d𝑥 where 𝑥
𝑢 𝑥 d𝑥 = 1 into L(Θ’), we have (Sean, 2009, p. 6):
𝐿 Θ′
≥
𝑋
𝑓 𝑋 𝑌, Θ log
𝑓 𝑋 𝑌, Θ
=
𝑋
𝑓 𝑋 𝑌, Θ log 𝑓 𝑋, 𝑌 Θ′
− log 𝑓 𝑋 𝑌, Θ d𝑋 =
𝑋
d𝑋 −
𝑋
𝑓 𝑋 𝑌, Θ log 𝑓 𝑋 𝑌, Θ d𝑋
= 𝑄 Θ′
Θ − 𝐻 Θ Θ
Where,
𝑄 Θ′
Θ =
𝑋
d𝑋
𝐻 Θ′
Θ =
𝑋
𝑓 𝑋 𝑌, Θ log 𝑓 𝑋 𝑌, Θ′
d𝑋
Obviously, the lower-bound of L(Θ’) is 𝑙𝑏 Θ′
Θ = 𝑄 Θ′
Θ − 𝐻 Θ Θ . As aforementioned, the lower-bound lb(Θ’|Θ) (Sean, 2009, pp. 7-8)
is maximized over many iterations of the iterative process so that L(Θ’) is maximized finally. Because H(Θ|Θ) is constant with regard to Θ’, it
is possible to eliminate H(Θ|Θ) so that maximizing Q(Θ’|Θ) is the same to maximizing the lower bound. In final, when GEM converges Θ(t) =
Θ(t+1) = Θ*, we have:
Θ∗
= argmax
Θ′
𝑙𝑏 Θ′
Θ = argmax
Θ′
𝑄 Θ′
Θ
We have the proof .

Mixture model mentioned in subsection 5.1 is a good example for GEM without explicit mapping from
X to Y. Another well-known example is three-coin toss example (Collins & Barzilay, 2005) which
applies GEM into estimating parameters of binomial distributions without explicit mapping.
Example 2.1. There are three coins named coin 1, coin 2 and coin 3. Each coin has two sides such as
head (H) side and tail (T) side. Let hidden random variable X represent coin 1 where X is binary (X =
{H, T}). Let θ1 be probability of coin 1 receiving head side.
θ1 = P(X=H)
Of course, we have:
P(X=T) = 1 – θ1
Let observed random variable Y represent a sequence of tossing coin 2 or coin 3 three times. Such
sequence depends on first tossing coin 1. For instance, if coin 1 shows head side (X=H), the sequence is
result of tossing coin 2 three times. Otherwise, if coin 1 shows tail side (X=T), the sequence is result of
tossing coin 3 three times. For example, suppose first tossing coin 1 results X=H then, a possible result Y
= HHT means that we toss coin 2 three times resulting head, head, and tail from coin 2. Obviously, X is
hidden and Y is observed. In this example, we observe that
Y=HHT

Suppose Y conforms binomial distribution as follows:
𝑃 𝑌 𝑋 =
𝜃2
ℎ
1 − 𝜃2
𝑡 if 𝑋 = 𝐻
𝜃3
ℎ
1 − 𝜃3
𝑡
if 𝑋 = 𝑇
Where θ2 and θ3 are probabilities of coin 2 and coin 3 receiving head side, respectively. Note, h is the number of head side
from trials of tossing coin 2 (if X=H) or coin 3 (if X=T). Similarly, t is the number of tail side from trials of tossing coin 2 (if
X=H) or coin 3 (if X=T). The joint probability P(X, Y) is:
𝑃 𝑋, 𝑌 = 𝑃 𝑋 𝑃 𝑌 𝑋 =
𝜃1𝜃2
ℎ
1 − 𝜃2
𝑡
if 𝑋 = 𝐻
1 − 𝜃1 𝜃3
ℎ
1 − 𝜃3
𝑡
if 𝑋 = 𝑇
In short, we need to estimate Θ = (θ1, θ2, θ3)T from the observation Y=HHT by discrete version of Q(Θ’ | Θ). Given Y=HHT,
we have h=2 and t=1. Thus, the probability P(Y|X) becomes:
𝑃 𝑌 𝑋 = 𝑃 𝑌 = 𝐻𝐻𝑇 𝑋 =
𝜃2
2
1 − 𝜃2 if 𝑋 = 𝐻
𝜃3
2
1 − 𝜃3 if 𝑋 = 𝑇
The joint probability P(X, Y) becomes:
𝑃 𝑋, 𝑌 =
𝜃1𝜃2
2
1 − 𝜃2 if 𝑋 = 𝐻
1 − 𝜃1 𝜃3
2
1 − 𝜃3 if 𝑋 = 𝑇
The probability of Y is calculated as follows:
𝑃 𝑌 = 𝑃 𝑌 𝑋 = 𝐻 + 𝑃 𝑌 𝑋 = 𝑇 = 𝜃2
2
1 − 𝜃2 + 𝜃3
2
1 − 𝜃3

The conditional probability of X given Y is determined as follows:
𝑃 𝑋 𝑌 =
𝑃 𝑋, 𝑌
𝑃 𝑌
=
𝜃1𝜃2
2
1 − 𝜃2
𝜃2
2
1 − 𝜃2 + 𝜃3
2
1 − 𝜃3
if 𝑋 = 𝐻
1 − 𝜃1 𝜃3
2
1 − 𝜃3
𝜃2
2
1 − 𝜃2 + 𝜃3
2
1 − 𝜃3
if 𝑋 = 𝑇
The discrete version of Q(Θ’ | Θ) is determined as follows:
𝑄 Θ′
Θ =
𝑋
𝑃 𝑋 𝑌, Θ log 𝑃 𝑋, 𝑌 Θ′
= 𝑃 𝑋 = 𝐻 𝑌, Θ log 𝑃 𝑋 = 𝐻, 𝑌 Θ′
+ 𝑃 𝑋 = 𝑇 𝑌, Θ log 𝑃 𝑋 = 𝑇, 𝑌 Θ′
=
𝜃1𝜃2
2
1 − 𝜃2
𝜃2
2
1 − 𝜃2 + 𝜃3
2
1 − 𝜃3
log 𝜃1
′
+ 2log 𝜃2
′
+ log 1 − 𝜃2
′
+
1 − 𝜃1 𝜃3
2
1 − 𝜃3
𝜃2
2
1 − 𝜃2 + 𝜃3
2
1 − 𝜃3
log 1 − 𝜃1
′
+ 2log 𝜃3
′
+ log 1 − 𝜃3
′
Note, Q(Θ’|Θ) is function of Θ’ = (θ1’, θ2’, θ3’)T. The next parameter Θ(t+1) = (θ1
(t+1), θ2
(t+1), θ3
(t+1))T is a maximizer of
Q(Θ’|Θ) with regard to Θ’, which is solution of the equation created by setting the first-order derivative of Q(Θ’|Θ) to
be zero with note that the current parameter is Θ(t) = Θ.

The first-order partial derivative of Q(Θ’|Θ) with regard to θ1’ is:
𝜕𝑄 Θ′
Θ
𝜕𝜃1
′ =
𝜃1𝜃2
2
1 − 𝜃2 − 𝜃1
′
𝜃1𝜃2
2
1 − 𝜃2 + 1 − 𝜃1 𝜃3
2
1 − 𝜃3
𝜃1
′
1 − 𝜃1
′
𝜃2
2
1 − 𝜃2 + 𝜃3
2
1 − 𝜃3
Setting this partial derivative
𝜕𝑄 Θ′
Θ
𝜕𝜃1
′ to be zero, we obtain:
𝜃1
′
=
𝜃1𝜃2
2
1 − 𝜃2
𝜃1𝜃2
2
1 − 𝜃2 + 1 − 𝜃1 𝜃3
2
1 − 𝜃3
Therefore, in M-step, given current parameter Θ(t) = (θ1
(t), θ2
(t), θ3
(t))T, the next partial parameter θ1
(t+1) is calculated as follows:
𝜃1
𝑡+1
=
𝜃1
𝑡
𝜃2
𝑡
2
1 − 𝜃2
𝑡
𝜃1
𝑡
𝜃2
𝑡
2
1 − 𝜃2
𝑡
+ 1 − 𝜃1
𝑡
𝜃3
𝑡
2
1 − 𝜃3
𝑡
The first-order partial derivative of Q(Θ’|Θ) with regard to θ2’ is:
𝜕𝑄 Θ′
Θ
𝜕𝜃2
′ =
𝜃1𝜃2
2
1 − 𝜃2
𝜃2
2
1 − 𝜃2 + 𝜃3
2
1 − 𝜃3
2 − 3𝜃2
′
𝜃2
′
1 − 𝜃2
′
𝜕𝑄 Θ′ Θ
𝜕𝜃2
′ to be zero, we obtain:
𝜃2
′
=
2
3

Therefore, in M-step, given current parameter Θ(t) = (θ1
(t), θ2
(t), θ3
(t+1) is fixed 𝜃2
𝑡+1
=
2
3
.
The first-order partial derivative of Q(Θ’|Θ) with regard to θ3’ is
𝜕𝑄 Θ′
Θ
𝜕𝜃3
′ =
1 − 𝜃1 𝜃3
2
1 − 𝜃3
𝜃2
2
1 − 𝜃2 + 𝜃3
2
1 − 𝜃3
2 − 3𝜃3
′
𝜃3
′
1 − 𝜃3
′
𝜕𝑄 Θ′
Θ
𝜕𝜃3
′ to be zero, we obtain 𝜃3
′
=
2
3
. Therefore, in M-step, given current parameter Θ(t) =
(θ1
(t), θ2
(t), θ3
(t+1) is fixed 𝜃3
𝑡+1
=
2
3
. In short, in M-step of some tth iteration, given current
parameter Θ(t) = (θ1
(t), θ2
(t), θ3
(t))T, only θ1
(t+1) is updated whereas both θ2
(t+1) and θ3
(t+1) are fixed with observation Y=HHT.
𝜃1
𝑡+1
=
𝜃1
𝑡
𝜃2
𝑡
2
1 − 𝜃2
𝑡
𝜃1
𝑡
𝜃2
𝑡
2
1 − 𝜃2
𝑡
+ 1 − 𝜃1
𝑡
𝜃3
𝑡
2
1 − 𝜃3
𝑡
𝜃2
𝑡+1
= 𝜃3
𝑡+1
=
2
3

For instance, let Θ(1) = (θ1
(1), θ2
(1), θ3
(1))T be initialized arbitrarily as θ1
(1) = θ2
(1) = θ3
(1) = 0.5, at the
first iteration, we obtain:
𝜃1
2
=
0.5 ∗ 0.5 2
∗ 1 − 0.5
0.5 ∗ 0.5 2 ∗ 1 − 0.5 + 1 − 0.5 ∗ 0.5 2 ∗ 1 − 0.5
= 0.5
𝜃2
2
= 𝜃3
2
=
2
3
At the second iteration with current parameter Θ(2) = (θ1
(2)=0.5, θ2
(2)=2/3, θ3
(2)=2/3)T, we obtain:
𝜃1
3
=
0.5 ∗
2
3
2
∗ 1 −
2
3
0.5 ∗
2
3
2
∗ 1 −
2
3
+ 1 − 0.5 ∗
2
3
2
∗ 1 −
2
3
= 0.5
𝜃2
3
= 𝜃3
3
=
2
3
As a result, GEM inside this example converges at the second iteration with final estimate Θ(2) =
Θ(3) = Θ* = (θ1
*=0.5, θ2
*=2/3, θ3
*=2/3)T .

In practice, suppose Y is observed as a sample 𝒴 = {Y1, Y2,…, YN} of size N with note that
all Yi (s) are mutually independent and identically distributed (iid). The observed sample 𝒴
is associated with a a hidden set (latent set) 𝒳 = {X1, X2,…, XN} of size N. All Xi (s) are iid
and they are not existent in fact. Let 𝑋 ∈ 𝑿 be the random variable representing every Xi. Of
course, the domain of X is X. Equation 2.13 specifies the conditional expectation Q(Θ’ | Θ)
given such 𝒴.
𝑄 Θ′ Θ =
𝑖=1
𝑁
𝑋
𝑓 𝑋 𝑌𝑖, Θ log 𝑓 𝑋, 𝑌𝑖 Θ′ d𝑋 (2.13)
Equation 2.13 is a variant of equation 2.10 in case that there is no explicit mapping between
Xi and Yi but there exists the same joint PDF between Xi and Yi. Please see the proof of
equation 2.10 to comprehend how to derive equation 2.13. If both X and Y are discrete,
equation 2.13 becomes:
𝑄 Θ′
Θ =
𝑖=1
𝑁
𝑋
𝑃 𝑋 𝑌𝑖, Θ log 𝑃 𝑋, 𝑌𝑖 Θ′
(2.14)

If X is discrete and Y is continuous such that f(X, Y | Θ) = P(X|Θ)f(Y | X, Θ) then, according
to the total probability rule, we have:
𝑓 𝑌 Θ =
𝑋
𝑃 𝑋 Θ 𝑓 𝑌 𝑋, Θ
Note, when only X is discrete, its PDF f(X|Θ) becomes the probability P(X|Θ). Therefore,
equation 2.15 is a variant of equation 2.13, as follows:
𝑄 Θ′ Θ =
𝑖=1
𝑁
𝑋
𝑃 𝑋 𝑌𝑖, Θ log 𝑃 𝑋 Θ′ 𝑓 𝑌𝑖 𝑋, Θ′ (2.15)
Where P(X | Yi, Θ) is determined by Bayes’ rule, as follows:
𝑃 𝑋 𝑌𝑖, Θ =
𝑃 𝑋 Θ 𝑓 𝑌𝑖 𝑋, Θ
𝑋 𝑃 𝑋 Θ 𝑓 𝑌𝑖 𝑋, Θ
Equation 2.15 is the base for estimating the probabilistic mixture model by EM algorithm,
which will be described later in detail. Some other properties of GEM will be mentioned in
next section.

References
1. Bilmes, J. A. (1998). A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture
and Hidden Markov Models. International Computer Science Institute, Department of Electrical Engineering and Computer
Science. Berkeley: University of Washington. Retrieved from
http://melodi.ee.washington.edu/people/bilmes/mypubs/bilmes1997-em.pdf
2. Borman, S. (2004). The Expectation Maximization Algorithm - A short tutorial. University of Notre Dame, Department of
Electrical Engineering. South Bend, Indiana: Sean Borman's Home Page.
3. Burden, R. L., & Faires, D. J. (2011). Numerical Analysis (9th Edition ed.). (M. Julet, Ed.) Brooks/Cole Cengage Learning.
4. Collins, M., & Barzilay, R. (2005). Advanced Natural Language Processing - The EM Algorithm. Massachusetts Institute of
Technology, Electrical Engineering and Computer Science. MIT OpenCourseWare. Retrieved October 9, 2020, from
https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-864-advanced-natural-language-processing-fall-
2005/lecture-notes/lec5.pdf
5. Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum Likelihood from Incomplete Data via the EM Algorithm. (M.
Stone, Ed.) Journal of the Royal Statistical Society, Series B (Methodological), 39(1), 1-38.
6. Sean, B. (2009). The Expectation Maximization Algorithm - A short tutorial. University of Notre Dame, Indiana, Department of
Electrical Engineering. Sean Borman's Homepage.
7. Ta, P. D. (2014). Numerical Analysis Lecture Notes. Vietnam Institute of Mathematics, Numerical Analysis and Scientific
Computing. Hanoi: Vietnam Institute of Mathematics. Retrieved 2014
8. Wikipedia. (2014, August 4). Karush–Kuhn–Tucker conditions. (Wikimedia Foundation) Retrieved November 16, 2014, from
Wikipedia website: http://en.wikipedia.org/wiki/Karush–Kuhn–Tucker_conditions

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