Bayesian Network

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

Conditional Independence • Definition X 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 parents A 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

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:

Bayesian Network Definition • 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 • 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

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？ • NO • E.g., X1 and X4 are conditionally independent given {X2, X3} • But X1 and X4 not conditionally independent given X3 • For this, we need to understand D-separation …

Reasoning BN: D-Separation • Three example to understand D-Separation • Head to Tail • Tail to Tail • Head to Head

Reasoning BN: D-Separation • Head to Tail P(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)

Reasoning BN: D-Separation • Tail to Tail P(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)

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

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 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 Compute the marginals

Inference On a Chain

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

Thanks

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