6.2. When do priors matter? When don’t they matter?

Question

What happens when enough data is collected?

Answer

All posteriors, independent of prior, converge to a narrow PDF including \((p_h)_{\text{true}}\)

Follow-ups:

• Which prior(s) get to the correct conclusion fastest for \(p_h = 0.4, 0.9, 0.5\)? Can you explain your observations?

• Does it matter if you update after every toss or all at once?

• Why does the “anti-prior” work well even though its dominant assumptions (most likely \(p_h = 0\) or \(1\)) are proven wrong early on?

  Answer

  The “heavy tails” (which in general means the probability away from the peaks; in the middle for the “anti-prior”) mean it is like uniform (renormalized!) after the ends are eliminated. An important lesson for formulating priors: allow for deviations from your expectations.

Different priors eventually give the same posterior with enough data. This is called Bayesian convergence. How many tosses constitute ``eventually”? Clearly it depends on \(p_h\) and how close you want the posteriors to be. How about for \(p_h = 0.4\) or \(p_h = 0.9\)?

Answer

• \(p_h = 0.4\) \(\Longrightarrow\) \(\approx 200\) tosses will get you most of the way.

• \(p_h = 0.9\) \(\Longrightarrow\) much longer for the informative prior than the others.

Choosing priors: a good reference is the Stan page on Prior Choice Recommendations.