*Ecological Archives* E087-159-A1

**William A. Link and Richard J. Barker. 2006. Model weights and the foundations of multimodel inference. ***Ecology* 87:2626–2635.

Appendix A. Inconsistency of posterior mean when prior depends on sample size.

As an illustration of the consequences of having priors depend on sample size, consider the case of a Binomial random variable *X* consisting of *N* independent Bernoulli trials with success parameter *p*. Suppose *p* has a beta prior distribution with parameters *a* and *b*, denoted . The posterior distribution of *p* is with mean

this is a weighted average of the prior mean and the maximum likelihood estimator The important feature is that as , the weight on the prior mean goes to zero, provided that *a* and *b* are fixed.

Now suppose that is fixed, but that (*a* + *b*) = *kN*, for a fixed value of *k*. Then the posterior mean becomes

The weight on the prior mean does not go to zero as . Indeed, the posterior mean converges to something different than the MLE, hence is not consistent.

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