C4: Credibility Theory

[New at pd]

Hi everyone,
When do we know that the posterior mean is equal to the Bayesian premium? Is it only the time when we have poisson/gamma, normal/normal, and binomial/beta? Any help is appreciated.

[Anonymous]

The posterior mean is the mean of a parameter having the posterior distribution. The Bayesian premium (when the loss function is mean square difference) is the mean of the predictive distribution. The two are equal whenever the mean of the predictive distribution is the parameter having the posterior distribution. This is true in each of three examples you mentioned, and in many other cases. For example, in the Poisson/gamma case, the mean of the predictive Poisson is "lambda", the parameter estimated by the posterior gamma.

[New at pd]

Thanks, here's another one:

Is it true that, whenever we have the case where the prior and posterior disturibution have the same form, then the Bayesian Estimate is equal to Buhlmann's Estimate?

As always, any help is greatly appreciated.

[Anonymous]

No, you also need that the prior is the conjugate prior of the model.

[roz]

That's the same thing.

You need the 'premium' from the posterior distribution (which is the mean of the predictive distribution) to be linear in the data. This is not necessarily the case even if the prior is conjugate to the model. Only works in a few cases (all in the linear exponential family).



<font size=-1>[ This Message was edited by: roz on 2001-09-18 20:52 ]</font>