Answer Happy

Problem 3 Suppose we have two sensors with known (and different) variances v₁ and v2, but unknown (and the same) mean μ. Suppose we observe n₁ observations y(¹) ~ N(µ, v₁) from the first sensor and no observations y²) ~ N(u, v₂) from the second sensor. (For example, suppose (2) is the true temperature outside, and sensor 1 is a precise (low variance) digital thermosensing device, and sensor 2 is an imprecise (high variance) mercury thermometer.) Yi µ Let D represent all the data from both sensors. What is the posterior p(µ|D), assuming a non-informative prior for µ (which we can simulate using a Gaussian with a variance of ∞)? Give an explicit expression for the posterior mean and variance.