Bayesian Statistics

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7.3 Point estimates

If, for some reason, it is necessary to give a point estimate $\theta _0$ for the unknown parameter $\theta $, then we can obtain one in various different ways from the posterior $\Theta |_{\{X=x\}}$. Common choices are:

1. The mean $\theta _0=\E [\Theta |_{\{X=x\}}]$.
2. The median $\theta _0$ such that $\P [\Theta |_{\{X=x\}}\leq \theta _0]=\P [\Theta |_{\{X=x\}}\geq \theta _0]=\frac 12$.
3. The mode $\theta _0=\argmax _{\theta \in \Pi } L_{\Theta |_{\{X=x\}}}(\theta )$.

When doing so, we should be wary that $\theta _0$ contains much less information than the full posterior distribution $\Theta |_{\{X=x\}}$. If $\Theta $ is close to $\theta _0$ with high probability then we can hope it provides a reasonable approximation, but there is no guarantee in any case, and we should plot the distribution of $\Theta |_{\{X=x\}}$ (and, ideally, the corresponding sampling distributions) to assess this.

Note that some or all of these point estimates may fail to be well defined, dependent upon the distribution of $X$. For example the Cauchy distributions does not have a well-defined mean. Formulae for means and variances are listed on the reference sheets in Appendix A; modes can usually be obtained (for continuous distribution) via differentiation; medians are usually obtained either via symmetry or numerically.

Example 7.3.1 In Example 7.2.2 the mean, median and mode of $\Theta |_{\{X=x\}}\sim \Normal (0,1.5^2)$ are all equal to zero, due to the symmetry of the normal distribution. In Example7.2.3, in which $\Theta |_{\{X=x\}}\sim \Gam (3,4)$, they are

All of these are reasonable point estimates for $\Gam (0,1.5^2)$. The distribution of $\Gam (0,1.5^2)$ spreads out across a range of parameters, and we should make this clear in our analysis, perhaps by giving an HPD interval alongside the point estimate.

Lastly, consider if we had obtained a posterior distribution with the p.d.f. from Exercise 1.1:

There is no reasonable way to summarize this distribution with a point estimate. In a case like this we should decline to give a point estimate, even if we are asked for one.