Sequential estimation of joint posterior


The GP model I have is bit unsolvable. Therefore I though about estimating only subset P1 of parameters first and then using resulting posterior as prior estimate remaining parameters P2. The model M1 to estimate P1 is:

data {
  int<lower=1> N0;
  int<lower=0,upper=1> y0[N0];
parameters {
  real<lower=0,upper=1> alpha;
  real<lower=0,upper=1> beta;
  vector<lower=0,upper=1>[N0] y0_raw;
model {
  y0_raw ~ beta(alpha, beta);
  y0 ~ bernoulli(y0_raw);

Some patients initially are sick(y0=0), some are healthy(y0=1). I am trying to represent the initial patient population just using parameters alpha and beta. My plan is to use the initial health status of patient population with alpha and beta. Right now they are difficult to interpret. I wonder if I could use some distribution instead of beta so the parameters are more interpretable and directly linked to the proportion of sick patients.

I appreciate any advice.

There a number of things you could try, as far as I see. You could assume a normal distribution on y0_raw but how well that would work depends on the variation in the data. If variation at this latent level is not very large, you might get a good estimate of the parameters.
That said, the Beta is a natural distribution to place over a random variable in (0, 1), so the other thing you can do is just look at the transformed parameters \mu = \alpha /( \alpha + \beta) and \sigma = \sqrt{\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha + \beta + 1)}} to get the mean and standard deviation of the latent variable.

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