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 is N(a,b), that is, N(1,1)

Hence posterior is, N(1,0.5), so, posterior precision is 2.

Update of mean: weighted mean of observation and prior mean, where, proportion of respective precision to posterior precision are the weights. = 1/2 +1/2=1

Update of variance: Reciprocal of sum of sample precision and prior precision = 1/(1+1)

Similarly,

After second observation, the updated posterior is, N(0.33,0.33)