1. Structure of generic bayesian inference procedure
Design a proper prior p(θ): prior design is important for scientific applications, and various types of prior is used in different cases
Non-informative prior, Spike-and-slab prior, ...
Likelihood term L(X∣θ), i.e. data generating mechanism takes account into contribution of dataset(observations)
Posterior π(θ∣X)∝L(X∣θ)p(θ) can be obtained in a closed form for some prior designs(conjugate priors), while most of the case we only can calculate 'kernel' of the distribution since π(θ∣X)=∫L(X∣θ)p(θ)dθL(X∣θ)p(θ) , and integral of the nominator is usually hard to calculate.
Perform some statistical inferences on posterior distribution, for example calculate the posterior mean or credible interval.
2. Bayesian inference on posterior distribution
Posterior mean: Eπ[θ]=∫π(θ∣X)θdθ≈∑θ=1NN1π(θ∣X)θ (monte carlo estimate)