Question on Bayesian Learning and Probability Theory

I have some difficulties at solving a traditional problem where we have two hats, hat A and B, where there are black and white balls in each hats but the experimenter does not know the proportion of $\displaystyle $a$$ black balls in hat A and proportion $\displaystyle $b$$ for hat B. Let $\displaystyle $0 \le a, b \le 1$$ with $\displaystyle $a \ne b$$. The proportion of hat A is $\displaystyle $p_0 \in (0,1)$$. The experimenter draws randomly some balls (with replacement) to determine which of the hats he is drawing from. After each $\displaystyle $k$$ draws, the experimenter updates his beliefs using Bayes' rule. Denote $\displaystyle $p_k$$ where $\displaystyle $k=1,2,...,$$ the experimenter posterior probability after k balls have been drawn.

My questions are:

(1) why is $\displaystyle $p_k$$ a random variable that can take k+1 values? I don't see where the +1 comes from... that's probably because I am not sure how to define the probability space.

(2) is this a martingale due to the replacement of a ball back into the hat after each draws?

Re: Question on Bayesian Learning and Probability Theory

Hey chamar.

Hint: Think how many balls are in the hat and remember the possibility of a zero entry (i.e. no balls).

It should be a normal martingale if intuitively no past information gives us any better advantage of getting the new expectation value, and also that the expectation value doesn't change (as opposed to sub and super martingales).

For this process, if the balls are put back where they came from and the drawing process is strictly random, then it should be a martingale.

Re: Question on Bayesian Learning and Probability Theory

Thanks for the help chiro. yes that's what I was thinking... but can $\displaystyle $p_k$$ be stochastic?

Re: Question on Bayesian Learning and Probability Theory

Thanks for the help chiro. yes that's what I was thinking... but can $\displaystyle $p_k$$ be stochastic?

Re: Question on Bayesian Learning and Probability Theory

Absolutely: It can have a distribution and the point with this problem is that it really has to since you don't know the distribution (in terms of balls) and hence don't know the ratio.

This is a great example of using the Bayesian method for a problem that is well suited (in terms of what information you know and the models you can use).