I need help with the following question:
Consider m observations $(y_1; n_1); ... ; (y_m; n_m)$, where $y_i \sim Bin(n_i; θ_i)$ are binomial variables.
Assume that $θ_i \sim w_1Beta(α_1; β_1) + w_2Beta(α_2; β_2)$ are mixture from two Beta distributions
$(w_1 + w_2 = 1)$
- Derive a Laplace approximation of the likelihood and each mixture component of the prior.
- Derive the empirical Bayes likelihood of the data by integrating out $\theta_i$ using the Laplace approximation, and leave the hyper-parameters $(w_j ; α_j ; β_j)$ (j = 1; 2)
- Derive the EM algorithm to estimate the hyperparameters (you may use also mixture of Gaussian prior to approximate mixture of Beta prior)