Posterior Predictive Distribution - Approximation／Estimation by Sampling

how do we estimate the posterior predictive distribution 𝐏(𝑋ˆ|𝑋)?

steps:

1. sample from the posterior distribution 𝐏(𝜃|𝑋) to get a value 𝜃_𝑖
2. sample from the sampling distribution 𝐏(𝑋ˆ|𝜃_𝑖) to get a sample 𝑋ˆ_𝑖

repeat steps 1 and 2 many times

draw a histogram of the 𝑋ˆ_𝑖‘s, and that would give us an approximation of our posterior predictive distribution

there are 2 sources of uncertainty, 1 in each step:

• we have uncertainty over the parameter value (1^st step)
• we have uncertainty over the data generating process (2^nd step)

Video