Checking Whether A Coin Is Fair - Posterior Probability Density Function

Posterior Probability Density Function

One method is to calculate the posterior probability density function of Bayesian probability theory.

A test is performed by tossing the coin N times and noting the observed numbers of heads, h, and tails, t. The symbols H and T represent more generalised variables expressing the numbers of heads and tails respectively that might have been observed in the experiment. Thus N = H+T = h+t.

Next, let r be the actual probability of obtaining heads in a single toss of the coin. This is the property of the coin which is being investigated. Using Bayes' theorem, the posterior probability density of r conditional on h and t is expressed as follows:

$f(r | H=h, T=t) = \frac {\Pr(H=h | r, N=h+t) \, g(r)} {\int_0^1 \Pr(H=h |r, N=h+t) \, g(r) \, dr}. \!$

where g(r) represents the prior probability density distribution of r, which lies in the range 0 to 1.

The prior probability density distribution summarizes what is known about the distribution of r in the absence of any observation. We will assume that the prior distribution of r is uniform over the interval . That is, g(r) = 1. (In practice, it would be more appropriate to assume a prior distribution which is much more heavily weighted in the region around 0.5, to reflect our experience with real coins.)

The probability of obtaining h heads in N tosses of a coin with a probability of heads equal to r is given by the binomial distribution:

Substituting this into the previous formula:

$f(r | H=h, T=t) = \frac{{N \choose h}\,r^h\,(1-r)^t} {\int_0^1 {N \choose h}\,r^h\,(1-r)^t\,dr} = \frac{r^h\,(1-r)^t}{\int_0^1 r^h\,(1-r)^t\,dr} .$

This is in fact a beta distribution (the conjugate prior for the binomial distribution), whose denominator can be expressed in terms of the beta function:

As a uniform prior distribution has been assumed, and because h and t are integers, this can also be written in terms of factorials:

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