I am interested in understanding the differences between Bayesian and Frequentist estimation in the context of hierarchical models.

Consider $n$ subjects, where for subject $i$ there are $k_i$ successes out of $m_i$ attempts, adequately modeled with a binomial distribution with parameter $\theta_i$. In a hierarchical model one could consider (if adequate for the problem at hand) $\theta_i$ coming from a Beta distribution with parameters $a$ and $b$, or a re-parametrization of them, such as mean $\mu$ and standard deviation $\sigma$. To estimate the group mean $\mu$ (random effects) in a Bayesian setting, one would consider a reasonable prior for $\mu$ and $\sigma$ and then estimate the posterior $p(\mu|k_1,\ldots,k_n,m_1,\ldots,m_n)$ by sampling or using a 2D grid for the joint posterior of $\mu$ and $\sigma$ (making use of products of beta-binomials to express the likelihood term) and then marginalizing with respect to $\sigma$.

Would it be correct to state that the frequentist estimation of $\mu$ and $\sigma$ in this hierarchical model is the couple of parameters ($\mu_{ML}$,$\sigma_{ML}$) that maximizes the likelihood term (which I could obtain again on a 2D grid) or should I apply some corrections to regularize if $n$ is small? Is there an R package that I could use to get the maximum likelihood estimate for the beta-binomial model, or is it common to first apply a logit transformation and then use normal models for the population mean?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.