5
$\begingroup$

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?

$\endgroup$

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.