# Incorporating population priors into MLE fits with few/limited samples

I am fitting Beta distributions to data resulting from each of many experiments using maximum likelihood. My goal is for each experiment, given iid data $y_{1:k}$, fit a Beta distribution, and then given an observation $y_{k+1}$, predict its probability given the fitted parameters: $p(y_{k+1}|\alpha,\beta)$.

For each new experiment, I would like to fit a Beta distribution to the resulting data. The problem is that at the beginning of each experiment, I have limited data, and I would like to incorporate prior information so as not to have initially spurious fits (I rerun fits as more data become available).

What would be the most sound approach to incorporate prior knowledge into my fits?

Possible approaches I've considered: (1) Apply a population prior based on fitting a Beta distribution across all experiments: in this scenario I essentially get a prior distribution $p(y|\alpha,\beta)$, and then wish to get a posterior using the data $y_{1:k}$ from that experiment -- it's not clear to me how to incorporate this given that $\alpha$ and $\beta$ are computed as point estimates: multiplying an MLE fit for the experiment data by the prior, or $p(y_{k+1}|y_{1:k},\theta) \propto l(\theta^*|y_{1:k})p(y|\alpha, \beta)$, doesn't seem right.

(2) Quantify distributions for $\alpha$ and $\beta$ across each experiment, use these as priors for $\alpha$ and $\beta$, and compute Bayesian predictions for $p(y_k|y_{1:k-1}):=\int_{\alpha,\beta} p(y_k|\alpha,\beta)p(\alpha,\beta|y_{1:k}) d \alpha d\beta$, or possibly compute a MAP estimate for $\alpha$ and $\beta$.

(3) Include some burn-in samples from the other experiments and (a) just use MLE or (b) use a weighted average incorporating proportion of samples from the burn-in population vs. experiments.

Is there a different, "optimal" approach?

• From a Bayesian point of view, the natural way to learn from other experiments is to build a hierarchical model where each experiment is associated with different parameters $(\alpha_i,\beta_i)$ but where these parameters are distributed with the same hyperprior. Through the hierarchy the past experiments feed information on the current parameter $(\alpha_t,\beta_t)$. – Xi'an Feb 8 '15 at 8:55