Bayesian Network By DengKe Dong. Key Points Today Intro to Graphical Model Conditional Independence Intro to Bayesian Network Reasoning BN: D-Separation

Bayesian Network

By DengKe Dong

Key Points Today

Intro to Graphical Model Conditional Independence Intro to Bayesian Network Reasoning BN: D-Separation Inference In Bayesian Network Belief Propagation Learning in Bayesian Network

Intro to Graphical Model

Two types of graphical models:

-- Directed graphs (aka Bayesian Networks)

-- Undirected graphs (aka Markov Random Fields)

Graphical Structure

Plus associated parameters define joint probability

distribution over of variables / nodes

Key Points Today


Conditional Independence

DefinitionX is conditionally independent of Y given Z, if the probability

distribution governing X is independent of the value of Y, given

the value of Z

we denote it as: P (X ⊥ Y | Z), if

Conditional Independence

Condition on its parentsA conditional probability distribution (CPD) is associated

with each node N, defining as:

P(N | Parents(N))

where the function Parents(N) returns the set of N’s immediate

parents

Key Points Today


Bayesian Network Definition

Definition:A directed acyclic graph defining a joint probability distribution

over a set of variables,

where each node denotes a random variable, and each edge

denotes the dependence between the connected nodes.

for example:


Conditional Independencies in Bayesian Network:

Each node is conditionally independent of its non-descendents,

given only its immediate parents,

So, the joint distribution over all variables in the network is

defined in terms of these CPD’s, plus the graph.

Bayesian Network Definition example: Chain rules for Probability:

P(S,L,R,T,W) = P(S)P(L|S)P(R|S,L)P(T|S,L,R)P(W|S,L,R,T)

CPD for each node Xi describing as P(Xi | Pa(Xi)):

P(S,L,R,T,W) = P(S)P(L|S)P(R|S)P(T|L)P(W|L,R)

So, in a Bayes net


Construction Choose an ordering over variables, e.g. X1, X2, …, Xn

For i=1 to n

Add Xi to the network

Select parents Pa(Xi) as minimal subset of X1…Xi-1, such that

Notice this choice of parents assures

Key Points Today


Reasoning BN: D-Separation

Conditional Independence, Revisited We said:

-- Each node is conditionally independent of its non-descendents,

given its immediate parents

Does this rule given us all of the conditional independence

relations implied by the Bayes Network ？a. NO

b. E.g., X1 and X4 are conditionally independent given {X2, X3}

c. But X1 and X4 not conditionally independent given X3

d. For this, we need to understand D-separation …


Three example to understand D-Separation Head to Tail

Tail to Tail

Head to Head


Head to TailP(a,b,c) = P(a) P(c|a) P(b|c)

Given C

P(a,b|c) = P(a,b,c) / P(c) = P(a|c) P(b|c)

Not Given C

P(a,b|c) = not equal to P(a|c) P(b|c)


Tail to TailP(a,b,c) = P(c) P(a|c) P(b|c)

Given C

P(a,b|c) = P(a,b,c) / P(c) = P(a|c) P(b|c)

Not Given C

P(a,b|c) = not equal to P(a|c) P(b|c)


Head to HeadP(a,b,c) = P(c|a,b) P(a) P(b)

Given C

P(a,b|c) = P(a,b,c) / P(c) not equal to P(a|c) P(b|c)

Not Given C

P(a,b|c) = = P(a) P(b)

X and Y are conditionally independent given Z, if and only if X and Y are D-separated by Z

Suppose we have three sets of random variables: X, Y and Z

X and Y are D-separated by Z (and therefore conditionally independence, given Z)

iff every path from any variable in X to any variable in Y is blocked

A path from variable A to variable B is blocked if it includes a node such that either arrows on the path meet either head-to-tail or tail-to-tail at the

node and this node is in Z

the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, is in Z

Key Points Today


Inference In Bayesian Network In general, intractable (NP-Complete)

For certain case, tractable

• Assigning probability to fully observed set of variables

• Or if just one variable unobserved

• Or for singly connected graphs (ie., no undirected loops)

• Variable elimination

• Belief propagation

For multiply connected graphs

• Junction tree

Sometimes use Monte Carlo method

• Generate many samples according to the Bayes Net distribution,

then count up the results

Variational methods for tractable approximate solutions

Inference In Bayesian Network Prob. of Joint assignment: easy

Suppose we are interested in joint assignment

<F=f,A=a,S=s,H=h,N=n>

What is P(f,a,s,h,n)?

Inference In Bayesian Network Prob. of Joint assignment: easy

Suppose we are interested in joint assignment

<F=f,A=a,S=s,H=h,N=n>

What is P(f,a,s,h,n)?

Inference In Bayesian Network Prob. of marginals: not so easy

How do we calculate P(N = n) ?

Inference In Bayesian Network Prob. of marginals: not so easy

How do we calculate P(N = n) ?

P(N = n) =

=

=

Inference In Bayesian Network Generating a sample from joint distribution:

easy How can we generate random samples

according to P(F, A, S, H, N)?

Randomly draw a value for F=f

draw r [0, 1] uniformly

f r < , then output f=1, else f = 0

Note we can estimate marginal like P(N=n) by generating many

samples from joint distribution, by summing up the probability

mass for which N = n

Similarly, for anything else we can care P(F=1 |H=1, N=0)

Inference On a Chain Converting Directed to Undirected Graphs

Inference On a Chain Converting Directed to Undirected Graphs

Inference On a ChainCompute the marginals




Inference On a Chain

Key Points Today


Belief Propagation

基于贝叶斯网络、 MRF’s 、因子图的置信传播算法已分别被开发出来，基于不同模型的置信传播算法在数学上是等价的。从叙述简洁的角度考虑，这里主要介绍基于 MRF’s 的标准置信传播算法。

Belief Propagation

基于贝叶斯网络、 MRF’s 、因子图的置信传播算法已分别被开发出来，基于不同模型的置信传播算法在数学上是等价的。从叙述简洁的角度考虑，这里主要介绍基于 MRF’s 的标准置信传播算法以马尔可夫随机场为例

给定某个观察得到的状态，推断其对应的潜在状态

Belief Propagation

基于贝叶斯网络、 MRF’s 、因子图的置信传播算法已分别被开发出来，基于不同模型的置信传播算法在数学上是等价的。从叙述简洁的角度考虑，这里主要介绍基于 MRF’s 的标准置信传播算法以马尔可夫随机场为例

给定某个观察得到的状态推断其对应的潜在状态

Belief Propagation 引入变量表示隐状态结点 i 传递给隐状态结点 j 的信息（ message ）表明了结点 j 应该处于何种状态的维度与相同，的每一维表明结点 i 认为结点 j 处于相应的状态的可能

Belief Propagation Belief （置信度）：

我们近似计算得到的边缘概率成为 belief ，并将结点 i 的置信度表示为 b()

那么结点 i 的 belief 为

eq(1)

其中 k 为归一化因子，保证所有置信度之和为 1 ， N(i) 表示结点 i 的所有相邻结点。置信传播的信息由公式 eq(2) 更新，该公式能够保证信息的一致性

eq(2 ）

公式 (2) 中除了由结点 j 传递给结点 i 的信息，其他所有传递给结点 i 的信息都被连乘起来；另外公式中的求和符号表示将结点 i 的所有可能状态累加起来。

Belief Propagation

在实际计算中， belief 从图边缘的结点开始计算，并且只更新所有必需信息都已知的信息。利用公式 1 和公式 2 ，依次计算每个结点的 belief 。

对于无环的 MRF’s ，通常情况下，每个信息只需计算一次，极大地提升了计算效率

Key Points Today


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Learning in Bayesian Network What you should know

Learning in Bayesian Network

Learning CPTs from Fully Observed Data

MLE estimate of from fully observed data




EM Algorithm

EM Algorithm

EM Algorithm

EM Algorithm

Using Unlabeled Data to Help Train Naïve Bayes Classifier




Summary


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Bayesian Network By DengKe Dong. Key Points Today Intro to Graphical Model Conditional Independence Intro to Bayesian Network Reasoning BN: D-Separation