Updating prior in MCMC with new estimates for parameters

Question

I'm new to doing Bayesian analysis and I wanted to learn by using baseball data. I took a group of players and found their hits and at bats for various years and I want to be able to get estimates for their true batting average (theta) every year up to the current year. I was able to run an MCMC model using rjags for the first year using a binomial likelihood and beta(1,1) prior for theta (batting average). I got values for alpha, beta, and thetas for each player. How do I go about incorporating that into next years analysis? When I plug in the estimates for alpha and beta as that I got from the first analysis as the new prior the trace plot becomes a flat line for alpha and beta (the thetas do change).

I know I could probably incorporate all the data into a single model file, but I want to do a separate file for each year of estimates.

This is my model

model{ 
for (i in 1:P){ 
#likelihood
hits[i] ~ dbin(theta[i], atbats[i]) 
#prior 
theta[i] ~ dbeta(alpha1, beta1)}
alpha1 ~ dbeta(1,1)
beta1 ~ dbeta(1,1)} 

The estimates after running are a = 0.7, b = 0.9

On the next years set of data, what do I do with the information obtained for these estimates? and why?

Answer 1

As pointed out by @Xi'an in his comment, you want to use the full posterior distribution as the prior for your next analysis.

Option 1 involves somehow finding a way of summarizing the posterior you got in the first analysis so that it can be used in the second analysis. Especially for 1 or 2 parameters, an approximation with a mixture distribution with a few mixture components works pretty well. See e.g. the vignettes for the RBesT R package and the referenced papers.

Option 2 involves refitting the model with the old data + the new data. In complicated cases, this may be easier to do in practice.

These two options are completely equivalent (but in your case with several parameters and hyper-parameters, it may be pretty difficult to find a way to implement option 1 "correctly".