What is a good analogy to illustrate the strengths of Hierarchical Bayesian Models? I'm relatively new to bayesian statistics and have been using JAGS recently to build hierarchical bayesian models on different datasets. While I'm very satisfied of the results (compared to standard glm models), I need to explain to non-statisticians what the difference with standard statistical models is. Especially, I would like to illustrate why and when HBMs perform better than simpler models. 
An analogy would be useful, especially one that illustrates some key elements:


*

*the multiple levels of heterogeneity

*the need for more computations to fit the model

*the ability to extract more "signal" from the same data


Note that the answer should really be an analogy enlightening to non-stats people, not an easy and nice-to-follow example.
 A: I would like to illustrate an example as to modelling relating to cancer rate(As in Johnson and Albert 1999). It will touch first and third element of your interest.  
So the problem is predicting cancer rates in various cities. Say we have data of number of people in various cities $N_i$ and number of people who died with cancer $x_i$. Say we want to estimate cancer rates $\theta_i$. There are various ways to model them and as we see problems with each of them. We will see how heirachical bayes modelling can overcome some problem. 
1. One way is to do estimation seperately but we will suffer from sparse data problem and would be an underestimate of the rates as for low $N_i$. 
2. One more approach to manage the problem of sparse data would be to use same $\theta_i$ for all cities and tie the parameters but this is also a very strong assumption. 
3. So what could be done is all $\theta_i$'s are similar in some way but also with city specific variations. So one could model in such a way that all $\theta_i$'s are drawn from a common distribution. Say $x_i \sim  Bin(N_i,\theta_i)$ and $\theta_i \sim Beta(a,b) $ 
A full joint distribution would be then $p(D,\theta,\eta|N)= p(\eta)\prod_{i=1}^N Bin(x_i|N_i,\theta_i)Beta(\theta_i|\eta)$ where $\eta = (a,b)$. 
We need to infer $\eta$ from data. If it is clamped to a constant then information will not flow between $\theta_i$'s and they will be conditionally independent. But by treating $\eta$ as unknowns we allow cities with less data borrow statistical strength from cities with more data. 
The main idea is to more bayesian and setting priors on priors as to model uncertainty in hyperparameters. This allows flow of influence between $\theta_i$'s in this example.
A: When you are ill, you observe symptoms but what you want is a diagnosis. If you are not a physician I guess that you can simply find the diagnosis that best matches your symptoms. But what Ph HBM would do is to look at your symptoms, their relative meaningfulness, how they fit/relate your different previous health problems, the one of your family, the current common diseases and environmental conditions, your weakness, your strength... and then he will combine of these stuffs using its knowledge to update what he guess of your health conditions and will give you the more likely diagnosis. 
I am sure that this analogy achieves its limit pretty soon but I think that it can give a good intuition of what one would expect from a HBM, do you ? (and I did not find a better one)
