# Bayesian analysis of data

I have a big dataset in the form: $X_1, X_2, X_3, X_4, Y$. All the $X_i, i \in {1,...,4}$ come from different unknown distributions and $Y$ follows a bernoulli distribution, so it can take only values in $\{0, 1\}$. The problem is how I can predict the value of $Y$ given the observations for $X_i$. From what I know, this can be solved using binary logistic regression. Also, we can set a threshold of 0.95, lets say, for the probability of Y=1 in order to estimate the value of Y (i.e. if P(Y=1)>=0.95, then we estimate that Y is 1) with some given confidence.
The question that arises here is whether we can use (hierarchical?) Bayesian models to approach the same problem of estimating the value of Y. Any suggestions? Also, is there any paper, book chapter, or tutorial that addresses the same problem? From what I have seen, the statistics textbooks cover the trivial case of well-known distributions where we can calculate everything analytically. But what about my case where I have multidimensional experimental data that can follow an arbitrary distribution? Thanks.

If you just want to predict $Y$ then an explicit joint model for $X$ and $Y$ when $X$ is 4 dimensional is probably overkill. This is especially true if you truly don't have any information about the distribution of the $X_i$'s or the relationship between $Y$ and $X$. An exception would be if there is significant missingness in $X$, or perhaps measurement error in $X$. Bayesian approaches to binary regression will appear in any decent textbook on applied Bayesian methods; my usual recommendation is Bayesian Data Analysis by Gelman, Carlin, Stern & Rubin.