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In Polya's urn, we have $b$ black balls and $w$ white balls at time $t$. At time $t+1$, we have $b+1$ black balls with probability $\frac{b}{w+b}$ and $w+1$ white balls with probability $\frac{w}{w+b}$.

Is there a way to generalize the probability function? Can you construct a generalized Polya's urn such that the probability of $b+1$ black and $w+1$ white balls at $t+1$ is $f(b, w)$, where $f: \{\mathbb{N}^+\}^2 \rightarrow \{0, 1\}^2$?

ux74bn1
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1 Answers1

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It is possible to use any function $f:\mathbb N^+ \times \mathbb N^+ \to [0,1]$ to represent

  • $f(b,w)$ as the probability of moving from $b$ black balls and $w$ white balls to $b+1$ black balls and $w$ white balls
  • $1-f(b,w)$ as the probability of moving from $b$ black balls and $w$ white balls to $b$ black balls and $w+1$ white balls

and given the starting position, then to find the distribution of the number or proportion of black balls or of white balls a given number of steps later. With the original Polya urn you have $f(b,w)=\frac{b}{b+w}$ and $1-f(b,w)=\frac{w}{b+w}$, but any other function can be used instead

A further generalisation could be to allow transitions to other states, not just adding a single ball with each step, or to have more than two colours, though these might require a more complicated function or a switch to a Markov-chain type of analysis

Henry
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  • What if you also generalize the updating of the number of balls at each round? Rather than increment by 1, you use a function g() which takes in the number of balls of each color at $t$, and outputs the number at $t+1$. – ux74bn1 Dec 06 '18 at 16:10
  • @ux74bn1 - Yes, though you might then find a Markov chain a more useful approach – Henry Dec 06 '18 at 19:34