Question

Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (vi) Suppose that ( of y with a -ab1. Suppose that you observe a realization Compute the posterior distribution value of 1. π(μ|1) and explain how it relates to π(μ). vii) Suppose now that you observe a second realization of y with a value of -1. Update the posterior π(p11) to incorporate this new information.

Answer 1

Let, prior of

: N(a, b); i.e., N(1, 1)

Posterior is: N(1,0.5), as n=1,y=1.

mean update= (1+1)/(1+1)= weighted mean of observation and prior mean, weights being proportion of their respective precision.

variance update= reciprocal of (prior precision+sample precision)=1/(1+1)

Similarly,

With a second observation, posterior becomes, N(0.33,0.33)