title: 斯坦福-隨機圖模型-week1.3
tags: note
notebook: 6- 英文課程-9-Probabilistic Graphical Models 1: Representation
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斯坦福-隨機圖模型-week1.3

1。第 1 個問題

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 個問題1
point

2。第 2 個問題

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 個問題

*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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