Probability map Model (PGM) learning Notes (iii) pattern judgment and probability graph flow

We still use the "Student Network" as a sample, 1.

Figure 1

First, the intuitive interpretation of causal judgment (causal reasoning) is given.

To figure it out.

That is, the probability of a student getting a good recommendation is about 0.5.

But suppose we know that students have a lower IQ. Then the probability of getting a good letter of recommendation is reduced:

Further. Assuming that the exam is very difficult at the same time, the probability of getting a good recommendation has risen again. It can even exceed the initial probability:

the above process is causal judgment , you see it is the direction of the Arrow to judge.

Secondly, the intuition explanation of the reliability Judgment (evidential reasoning) is given. 2.

Figure 2

The odds were that the exam was difficult and the students were very smart, 0.4 and 0.3, respectively.

Now we suddenly know that this tragic classmate in the exam got C and so on.

Now the probability of a high test is rising, and the probability of a student being very clever is declining:

The above process is the reliability of judgment, you see it is against the direction of the Arrow to judge.

Again gives the intuitive interpretation of cross causal judgment (InterCausal reasoning), 3.

Figure 3

The reliability judgment points out. After the students have been known to test C and so on. The probability of his being very clever dropped to 0.08,

Suppose at this point we know that the exam is very difficult, then he is very clever probability will have a slight rise. To 0.11:

The characteristic of cross causal judgment is that the difficulty follows the arrows to the grade, and the arrows affect the intelligence.

why is that? Let's consider one of the simplest cases . 4.

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Figure 4

At the beginning, X1 and X2 are completely independent. At the same time if there is a known Y=1

P\left ({{x_1} = 1} \right) = P\left ({{x_2} = 1} \right) = \frac{2}{3} ">

However, after we have known x1=1. The probability of x2=1 has decreased:

P\left ({{x_1} = 1} \right) = \frac{1}{2} ">

Again to see this classmate Test B is what effect, 5.

Figure 5

The probability that the students would be very clever is 0.3. Then I knew he had a B, so he was very smart and dropped to 0.175.

Now I know that the exam is actually quite difficult.

Then the probability of his being very clever went up to 0.34. than the original 0.3.

now consider a situation in which the student got a in the SAT Test. 6.

Figure 6

Does this have any effect on the probability that the exam is very difficult and the students are very clever? I went back to the tragic classmate who had tested C.

This classmate tested C, so the probability of a very difficult exam is 0.63, the probability of students very smart to 0.08

Now, suddenly know this classmate pretty good, in the SAT exam a

Thus, the probability of a very difficult exam reached 0.76. The probability of a student being very clever is 0.58, and both of them greatly exceed their original probabilities .

This is due to. Students ' SAT scores for a changed our understanding of their IQ, thus affecting the difficulty of his examination when he was a C test.

Through the above visual analysis, we find that the nodes in the probability graph can affect each other . Detailed analysis is done below. 7.

Figure 7

In what circumstances can the random variables x and y be set to affect each other?

1.X is directly connected to Y when they can interact with each other.

For example, to tell you that the test is very easy, the probability of your high score rises naturally.

I'm telling you, C. Then the probability of a very easy exam will fall.

A w is spaced between 2.X and Y, and X and Y can interact with each other without changing the direction of the connection arrows.

To tell you that this classmate has received a good letter of recommendation, then the probability of a simple exam has risen. Tell you that the exam is very difficult, then the probability that he can get a good recommendation is down.

3.X and y are separated by a W, assuming that the arrows are pointing outward. X and y can affect each other.

For example, students ' SAT scores obviously interact with his grade. It's like a person can get a high score every time a mock exam, we naturally have reason to believe that he is very capable. Enough to get good grades in the college entrance examination.

4.X and y are separated by a W. Assuming that the arrows are pointing inward, then x and Y cannot affect each other.

For example, it is difficult to tell you the exam, but what does it have to do with the IQ of your classmates? Vice versa.

In short, suppose a chain of relationships

{X_1}-\ldots {x_m} "there is no structure in the shape of the >, then this chain can pass the influence down."

The above discussion is that we have no knowledge of the middle-link w situation.

Suppose we know the information about the intermediate W, will the interaction between x and y change? we use the Z-set to indicate that we know the meaning of the relevant information. 8.

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Figure 8

The left side of the column is what we discussed above: we know nothing about W.

The right column refers to the probability that we already know about W .

And then we'll take a second view of the effect between x and Y.

The wonderful thing arises: Suppose we know the probability of W, we break the previously unobstructed relationship chain, and we get through the chain of relationships that was previously blocked.

In detail: The student sat got a. But we know that this classmate IQ is actually super stupid, then he is the probability of high score in the exam will be due to his sat on the dog luck increase? Not at all. By definition, the test scores were only related to his IQ and the difficulty of the exam, and had nothing to do with the SAT that he happened to test. As we already knew he was actually very stupid, the SAT was just an accident.

The previously blocked links are now in flux.

For example, the exam is very difficult, and the IQ of the students no matter what relationship, but suppose I know that the exam is very difficult, the students test A, then we have good reason to believe. Students should be very clever.

In this picture. s-i-g-d this path in I do not know, G know the circumstances of the ability unobstructed.

Tips: In fact, this conclusion should be extended.

Known Test paper very difficult, do not know how many points, but we know this classmate use this score to get a very good letter of recommendation, we have reason to believe that he should test well, and then believe that he should be a very clever children's shoes.

Anyway.

Suppose a chain of relationships

{X_1}-\ldots {x_m} "> in every structure we know that Xi or at least knows the probability of one of his child nodes (as though we don't know grade.) But we know the probability of the letter, then the relationship chain will be able to transfer the impact down.

Independence

The definition of independence can be described in the following 3 descriptive narratives:

Similarly, conditional independence can be written in this way

\begin{array}{l}p\left ({x,y\left| Z \right.} \right) = P\left ({x\left| Z \right.} \right) P\left ({y\left| Z \right.} \right) \\P\left ({x\left| {y,z} \right.} \right) = P\left ({x\left| Z \right.} \right) \\P\left ({y\left| {x,z} \right.} \right) = P\left ({y\left| Z \right.} \right) \\P\left ({x, y, z} \right) \propto {\phi _1}\left ({x,z} \right) \phi \left ({y,z} \right) \end{array} ">

The following is a visual sense of conditional independence , 9

Figure 9

There are 2 coins, one just evenly. There is also an uneven and 90% probability of being able to face up. Of course, the two coins look exactly the same.

Now let you take out one. Ready to throw 2 times.

You throw it first, you find the face up, and you can believe it. The second or front-facing probability is definitely added. The result of the second coin was affected by the first coin cast.

And now I'm telling you, in fact, you just cast a uniform coin (or uneven). It doesn't matter), then the probability of your second coin toss and the result of your first shot loses contact.

This shows that the condition sometimes causes the correlation between variables to be lost .

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Probability map Model (PGM) learning Notes (iii) pattern judgment and probability graph flow