Swamping the priors

We started with a uniform prior, but that might not be a good choice. I can believe that if a coin is lopsided, x might deviate substantially from 50%, but it seems unlikely that the Belgian Euro coin is so imbalanced that x is 10% or 90%.

It might be more reasonable to choose a prior that gives higher probability to values of x near 50% and lower probability to extreme values.

As an example, I constructed a triangular prior, shown in Figure 4-2. Here’s the code that constructs the prior:

def TrianglePrior():
    suite = Euro()
    for x in range(0, 51):
        suite.Set(x, x)
    for x in range(51, 101):
        suite.Set(x, 100-x) 
    suite.Normalize()
Uniform and triangular priors for the Euro problem.
Figure 4-2. Uniform and triangular priors for the Euro problem.

Figure 4-2 shows the result (and the uniform prior for comparison). Updating this prior with the same dataset yields the posterior distribution shown in Figure 4-3. Even with substantially different priors, the posterior distributions are very similar. The medians and the credible intervals are identical; the means differ by less than 0.5%.

Posterior distributions for the Euro problem.
Figure 4-3. Posterior distributions for the Euro problem.

This is an example of swamping the priors: with enough data, people who start with different priors will tend to converge on the same posterior.

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