# Relation between Bayesian analysis and Bayesian hierarchical analysis?

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?

http://www.stat.missouri.edu/~wikle/WikleBerlinerCressie1998.pdf - I have been studying a hierarchical model from this paper.

• A reference to the hierarchical model you have been studying would make the question easier to answer. Mar 7 '16 at 10:38
• stat.missouri.edu/~wikle/WikleBerlinerCressie1998.pdf - This is the paper which I have been following. Mar 7 '16 at 10:43

In my view, hierarchical modeling in a Bayesian setting mainly refers to the building of a complex prior structure. Consider a parameter of interest $\theta_{0}$ and your observation $(x_i)$.
Now, consider for example that you are adding a supplemental layer to your model $p(\theta_0|\theta_1)$ through hyperprior $p(\theta_1)$ on $\theta_1$, then $p(\theta_0)$ writes: $$p(\theta_0)=\int_R p(\theta_0|\theta_1)p(\theta_1)d\theta_1,$$ and so on for $\theta_2, \ldots$.
The same for the observation model : consider that your parameter of interest $\theta_0$ is not directly related to the observation but to an other parameter $\theta_1$ that is itself related to the observations: $$p((x_i)|\theta_0)=\int_R p((x_i)|\theta_1)p(\theta_1|\theta_0) d\theta_1.$$
To sum up, in principle, you can always (to the best of my knowledge) marginalize the hierarchical structures to get something as $p(\theta_0|x) \propto p(x|\theta_0) \cdot p(\theta_0)$ so to the simplest Bayes formulation. However, most of the time, integrals are intractable and we need to work with all the latent variables of the prior structure. So, IMHO, a hierarchical Bayesian model is only a decomposed Bayesian model (assuming that we call a Bayesian model something of the simplest form $p(\theta_0|x) \propto p(x|\theta_0) \cdot p(\theta_0)$).
Finally to answer your last question: "Is it enough for any statistical model which uses Bayes theorem to be categorized under Bayesian analysis/statistics" I would say no. A model can be qualified as a Bayesian model if it relies on Bayesian interpreation of probability https://en.wikipedia.org/wiki/Bayesian_probability and in particular if the posterior $p(\theta|x_i)$ makes any sense. Bayes theorem can be used in other contexts. See the related answer to the question Can frequentists use Bayes theorem?.