3.14 Coin guess using Bayes’ rule

We can verify that posterior odds of 2:1 in favour of coin 1 being heads are correct by using Bayes’ rule. Event \(A\) from Bayes’ rule in this example is coin 1 being a head and event \(B\) is coin 1 being a tail, since these are the options that player 2 must decide between. Event E is the knowledge that the result was not a double tails, since this is the information that player 1 reveals. This results in the following application of Bayes’ rule: \[\text{posterior odds of coin 1 head to coin 1 tail}=\text{LR} \times \text{prior odds of coin 1 head to coin 1 tail},\] where the LR equals the following ratio of conditional probabilities \[\text{LR}=\frac{\text{probability of no double tails conditioned on coin 1 head}}{\text{probability of no double tails conditioned on coin 1 tail}}.\] To apply Bayes’ rule, we need to assign the probabilities underlying the LR and the prior odds of coin 1 head to coin 1 tail.

The LR in this example considers the following question: how much more (or less) probable is it to toss something other than double tails if coin 1 is a head compared to if coin 1 is a tail? We can inspect the numerator and denominator of the LR separately.

LR - numerator: This is the probability of no double tails conditioned on coin 1 being a head. If coin 1 is a head, then it is certain that a double tails will not be tossed. That means that this probability is 1.

LR - denominator: This is the probability of no double tails conditioned on coin 1 being a tail. Out of every 5000 tosses in which coin 1 is tails, we expect 2500 to lead to coin 2 being heads and 2500 to lead to coin 2 being tails. The outcomes are evenly split because the probability is 0.5.

LR: These probabilities result in an LR of \(\frac{1}{0.5}=2\). In other words, an outcome other than double tails is twice as likely when coin 1 is a head compared to when coin 1 is a tail.

Prior odds: The second component of Bayes’ rule that we needed was the prior odds of coin 1 being a head. These odds are evens as they were given by the background information for the tosses of the coins, i.e. the coins are tossed so as to guarantee even odds of heads and tails. The prior odds of coin 1 showing a head are 1:1.

Bayes’ rule states that the posterior odds must be equal to the prior odds multiplied by the LR. With an LR of 2 and prior odds of 1:1, we obtain posterior odds of 2:1 in favour of a head; coin 1 is twice as likely to be a head when we know the outcome is not double tails. This result is the same as when we reasoned without using Bayes’ rule, and so we have verified that it satisfies Bayes’ rule.

In this example it was possible to calculate the posterior odds without using Bayes’ rule. This meant that the odds could be verified. In many real situations, the posterior odds are hard to quantify without using Bayes’ rule. Bayes’ rule offers an powerful and elegant solution to this problem since the prior odds and LR are often easier to quantify.