Question

Let X1​,X2​,…,Xn​ be i.i.d. Gaussian random variables, each having an unknown mean θ and known variance σ₀²​. If θ is itself selected from a normal population having a known mean μ and a known variance σ², determine the maximum a posteriori estimate of θ

Answer 1

To find the maximum a posteriori estimate of θ, use Bayesian inference and calculate the posterior probability density function using Bayes' theorem.

Step-by-step explanation:

To find the maximum a posteriori estimate of θ in this scenario, we can use Bayesian inference. The maximum a posteriori estimate is the value of θ that maximizes the posterior probability density function (PDF) given the observed data. In this case, the observed data is the sample of i.i.d. Gaussian random variables X1, X2, ..., Xn.

The posterior PDF of θ can be calculated using Bayes' theorem:

P(θ|X1, X2, ..., Xn) = P(X1, X2, ..., Xn|θ) * P(θ) / P(X1, X2, ..., Xn)

Since the variance σ² of the normal population from which θ is selected is known, we can calculate the likelihood function P(X1, X2, ..., Xn|θ) as the product of the individual Gaussian PDFs for each X:

P(X1, X2, ..., Xn|θ) = Πi=1 to n (1/σ˄ * exp(-(Xi-θ)² / (2σ˄²)))

The prior probability density function P(θ) can be specified based on any other information or assumptions about the distribution of θ. Finally, the denominator P(X1, X2, ..., Xn) is a normalizing constant that ensures the PDF integrates to 1. The maximum a posteriori estimate of θ can be found by maximizing the posterior PDF with respect to θ.