How to do Bayesian updating on biased information?

Question

You have a coin that you can flip, but you can't see. It's a weighted $3$-sided coin taken (uniformly) randomly from some small known collection of $100$ weighted coins. However, we don't know how each coin in the collection is weighted. The sides of the coin are green, red, and black. You throw the coin $5$ times. If at any point the black side comes up, you stop this experiment, and you don't get to see what the coin is. If all $5$ throws are non-black, then you get to see the coin. For each new experiment, you pick a random coin from your collection (with replacement). So, $\mbox{P(we are playing with coin №}\mbox{1)} =\frac{1}{100}$. When we flip it and see "color", then $\mbox{P(coin №}\mbox{1 | color)} = \mbox{P(color | coin №}\mbox{1)} \frac{\mbox{P(coin №}\mbox{1)}}{\mbox{P(color)}}$ by Bayes Theorem. We obviously know $\mbox{P(coin №}\mbox{1)}$, and we know $\mbox{P(color}\mbox{1)}$ from running these experiments for a long time. But how do we estimate $\mbox{P(color | coin №}\mbox{1)}$? How do we account for the fact that we are not going to see coins that are weighted heavily towards the black side very often? And when we do see a coin, it's only because of some luck? What does updating your hypotheses look like in this case?

Answer 1

The key insight is that you do not have 100 separate models for the coins 1 to 100, but one model which contains the probabilities for all coins. So if we write as $g_i$ the probability that if coin $i$ is tossed, it gives green, and with $r_i$ the probability that if coin $i$ is tossed, it gives red (the probability to get black when coin $i$ is tossed is then just $1-g_i-r_i$, so we don't need a separate variable for that), then the probability function for the model is given by the joint probability function $$P(g_1,r_1,g_2,r_2,\ldots,g_{100},r_{100})$$ While for the prior you'll probably choose $$P_0(g_1,r_1,g_2,r_2,\ldots,g_{100},r_{100}) = P(r_1,g_1)P(r_2,g_2)\cdots P(r_{100},g_{100})$$ this product structure will definitely not remain as soon as a toss results in black.

Your events are the "non-black events" like "coin 1 resulted in two reds and three greens" or "coin 2 resulted in one red and four greens", and the special event "a black result was tossed" which doesn't include any other details of the coin. Now with the complete model containing the probabilities for all the coins, it is of course no problem to calculate the probability for each event, including the special event "black was tossed", and therefore the normal Bayesian update rule can be used without problems on the complete model probability function.