CSE 571 LECTURE NOTES FOR NOVEMBER 17TH, 2003

Aleks Yasnogorodskiy

CSE 571 Lecture Notes

November 2003 Bayes nets have severe limitations. Statistics and conditional probability by themselfs only give diagnositcs. This is not good enough for an actual action. The trend in AI is to move away from Bayes nets. Consider the following table:

Simpson’s Paradox:

Males Recover ¬Recover # of People Recovery Rate
drug  18  12  30  60%
¬drug  7  3  10  70%

Females

drug  2  8  10  20%
¬drug  9  21  30  30%

Overall

drug  20  20  40  50%
¬drug  16  24  40  40%
Aleks Yasnogorodskiy  1

Notice that if you know the gender of the patient, the recovery rate suggests that the drug should not be taken, however, if you don’t know the gender of the patient, the recovery rate for patients suggests using the drug is a better option. . Consider another example: X=1 means there was treatment done. X=0 - no treatment. Y=1 the patient is dead. Y=0 - not dead. 50% of the patients recovered and 50% died in both treatment and non-treatment groups. The table representing this is given below:

X Y 0 0 25 0.25 0 1 25 0.25 1 0 25 0.25 1 1 25 0.25

Joe took the treatment and died.

Did Joe’s death occur due to treatment? What is the probability that Joe, who died under treatment (X=1, Y=1) would have recovered had he not been treated (X=0, Y=0)?

It is not enough to calculate conditional probabilities to answer this question. We now introduce new notations. Exoginous variables: U_{1}, U_{2}, ..., U_{n} Endoginous variables: V_{1}, V_{2}, ..., V_{m} Functions: V_{i} = f((U_{1}, ..., U_{n}, V_{1}, ..., V_{j}, V_{j+1}, ..., V_{m}) - f is a logical function. Probabilities are only associated with exoginous variables. Exoginous variables - variables are beyond our control.

Example 1. U_{1} → X U_{2} → Y P(U_{1}) = 0.5. P(U_{2}) = 0.5.

\begin{matrix}U_{1} & U_{2} & X & Y \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{matrix}

U_{1} - Study picks patient for treatment. U_{2} - Propensity to die. X = 1, Y = 1 ⇒ U_{1} = 1, U_{2} = 1. P(Y = 0 | do(X=0)) = 0. In this case the analysis shows that the patient would have still died even if the treatment was not applied. Here’s another model. Example 2.

U_{1} → X → Y ← U_{2} The function that defines Y: Y = XU_{2} + (1 - X)(1 - U_{1}) U_{2} - a genetic factor that if present kills people take the treatment, if absent kills people who do not take the treatment.

When U_{1} = 1, U_{2} = 1 and X = 1, Y = 1. If we force X to be 0, then our function will return Y to be equal to 0, which means that the patient would have survived, had he not received the treatment.