Posterior distribution of logistic regression coefficient


I have a binary logistic regression with the following properties

Consider the logistic regressionfor binarydataYi ∈{0,1} and the covariate vector xi =(xi,1,xi,2,…,xi,p) . Under the logistic regression assumption, the sampling distribution of Yi is (1). We assume a normal prior for βj for j = 1,…,n as in (2), and βj’s are independent of each other. (μj,σj2) are prior parameters and need to be specified.

Prob(Yi = 1|β) = exp(x⊤i β)/ 1+exp(x⊤i β) βj ∼N(μj,σj2) (2)

I have to find the posterior distribution for β, i.e., p(β|y, μ1, . . . , μp, σ12, . . . , σp2),.