斯坦福-随机图模型-week1.3_

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title: 斯坦福-随机图模型-week1.3
tags: note
notebook: 6- 英文课程-9-Probabilistic Graphical Models 1: Representation

斯坦福-随机图模型-week1.3

1。第 1 个问题

Factorization.

Given the same model as above, which of these is an appropriate decomposition of the joint distribution P(A,B,C,D)?

P(A,B,C,D)=P(A)P(B)P(C)P(D)

P(A,B,C,D)=P(A)P(B)P(A,B|C)P(B|D)

P(A,B,C,D)=P(A)P(B)P(C|A)P(C|B)P(D|B)

P(A,B,C,D)=P(A)P(B)P(C|A,B)P(D|B)

第 2 个问题1
point

2。第 2 个问题

Independent parameters.

How many independent parameters are required to uniquely define the CPD of C (the conditional probability distribution associated with the variable C) in the same graphical model as above, if A, B, and D are binary, and C and E have three values each?

If you haven't come across the term before, here's a brief explanation: A multinomial distribution over m possibilities x1,…,xm has m parameters, but m−1 independent parameters, because we have the constraint that all parameters must sum to 1, so that if you specify m−1 of the parameters, the final one is fixed. In a CPD P(X|Y), if X has m values and Y has k values, then we have k distinct multinomial distributions, one for each value of Y, and we have m−1 independent parameters in each of them, for a total of k(m−1). More generally, in a CPD P(X|Y1,…,Yr), if each Yi has ki values, we have a total of k1×…×kr×(m−1) independent parameters.

Example: Let's say we have a graphical model that just had X→Y, where both variables are binary. In this scenario, we need 1 parameter to define the CPD of X. The CPD of X contains two entries P(X=0) and P(X=1). Since the sum of these two entries has to be equal to 1, we only need one parameter to define the CPD.

Now we look at Y. The CPD for Y contains 4 entries which correspond to: P(Y=0|X=0),P(Y=1|X=0),P(Y=0|X=1),P(Y=1|X=1). Note that P(Y=0|X=0) and P(Y=1|X=0) should sum to one, so we need 1 independent parameter to describe those two entries; likewise, P(Y=0|X=1) and P(Y=1|X=1) should also sum to 1, so we need 1 independent parameter for those two entries.

Therefore, we need 1 independent parameter to define the CPD of X and 2 independent parameters to define the CPD of Y.

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3。第 3 个问题

*Inter-causal reasoning.

Consider the following model for traffic jams in a small town, which we assume can be caused by a car accident, or by a visit from the president (and the accompanying security motorcade).

Calculate P(Accident = 1 | Traffic = 1) and P(Accident = 1 | Traffic = 1, President = 1). Separate your answers with a space, e.g., an answer of

0.15 0.25

means that P(Accident = 1 | Traffic = 1) = 0.15 and P(Accident = 1 | Traffic = 1, President = 1) = 0.25. Round your answers to two decimal places and write a leading zero, like in the example above.

__ 0.35 0.14 __
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