Palgrave Handbook of Econometrics: Applied Econometrics

122 Metastatistics for the Non-Bayesian Regression Runner

0

50

100

150

200

0 .2 .4 .6 .8 1

x

Very informative prior

Less informative prior

0

5

10

15

20

0 .2 .4 .6 .8 1

x

Posterior using very informative prior

Posterior using less informative prior

Figure 3.2 Different prior and different posterior distributions

The mode of the posterior distribution occurs at:

α+h− 1

α+δ+N− 2

.

This can be fruitfully compared to the usual non-Bayesian maximum likelihood
(or method of moments) estimator, which is merely the sample mean:

h

N

.

The difference between the posterior mode and the usual non-Bayesian
estimator is that the former “adds”α−1 heads to the numerator and “adds”α+δ− 2
observations to the denominator.^34
To see what effect this has, the right-hand panel of Figure 3.3 shows the
resulting posterior distributions updated with 200 coin tosses, 100 of which are
heads.
For a slightly different type of comparison one can consider two situations:

Prior Data

Beta(99,99) 2 heads, 2 tails

Beta(2,2) 99 heads, 99 tails

In this case, although the experiments are very different, our conclusions are
exactly the same (see Figure 3.3).
The role of the prior distribution and the sufficiency of the posterior distribution
or likelihood are among the longest-standing debates in metastatistics. While a
complete review is impossible, some of the most frequently enumerated difficulties
are:

1. There is no way to verify whether the prior one has chosen adequately char-
   acterizes one’s beliefs. Also, there is no unique way to translate ignorance or