I have been studying a Bayesian hierarchical model. In that model all I am dealing is with the estimation of parameters. In Bayesian analysis, loosely speaking, we update our prior knowledge (in light of new evidences/data) to posterior knowledge. But in hierarchical model I don't see any prior knowledge or any prior distribution.

My question is what is the relation between Bayesian analysis and Bayesian hierarchical analysis?

I suppose the latter is a subset of former but I am still confused how are these two related? Is it enough for any statistical model which uses Bayes theorem to be categorized under Bayesian analysis/statistics?


1 Answer 1


You probably need to dig a bit more on how inference using bayesian theory.

Very loosely speaking, whenever you're estimating model parameters you need two ingredients: the likelihood & the prior distribution (wether it is informative or not). As for a vector of parameters $\theta \epsilon \Theta \subseteq R^K$ inference is from $\theta$ 's posterior distribution which is porportional to $f(\theta | X_{(n)}) \propto L(X_{(n)} | \theta) f(\theta)$.

Like so, in any given models ypu must assign a prior to every given parameter you're making inference on.

Finally, in the particular case of hierarchical models you're basically dealing with a problem of the form $y|\theta_1$,...,$\theta_n$. Then you must set a collection of priors on each $\theta_i$ and depending of the levels of your model you operate to get to your posterior.

A very simple example can be found at chapter 2 of Gamerman, Stochastic Simulation for bayesian Analysis dealing specifically with this model.


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