I think we actually agree: once given a parameterised family which hyperparameters can be specified through (prior) moment or generalised moment equations, one can automatically proceed with Bayesian inference, modulo computational hindrances of a more or less severe nature. Conjugate priors may have computational advantages in favour themselves, although mileage varies. What I find fascinating though (but may be alone in this!) is that there exists an infinity of conjugate families and that only one is standard or textbook-material, without anyone questioning the role of the dominating measure.

]]>I think I’m not following you here. If you have a family of priors parameterized by hyperparameters, you can decide to use that family with a prior on hyperparameters as your overall prior. You can do this regardless of whether or not this is a conjugate family. (Though obviously this may affect the ease of computation.) So how is this issue relevant beyond the practicalities of computations?

]]>Thank you. From a perfect subjective perspective, it is clear that the worry expressed by this column is not relevant since the perfect subjectivist is knowledgeable and agreeable about the entire prior. I however see conjugate priors as a poor person’s substitute where only hyperparameters need be specified on a subjective basis, hence the closest to non-informative one can get without turning improper. Something very close to maximum entropy priors, actually.

]]>The standard scenario would be that, if you can find values for the hyperparameters of ANY family of conjugate priors such that the resulting prior adequately represents your prior beliefs, then you can (hopefully) obtain an analytically tractable posterior density. This could be viewed as a reasonably practical scenario, for example, in the case of a beta conjugate prior for a binomial likelihood, but less so for other combinations of conjugate priors and likelihood functions.

From this subjective Bayesian perspective, the key of course is believing in your prior. If you do not believe in your prior, then do something else.

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