# Is there any reference about the almost sure convergence of a posterior distribution to the posterior with non-informative prior?

I'm trying to show

$$\pi(\theta_n|X) \overset{a.s.}{\to} \pi(\theta|X)$$

where $$\theta_n \sim N(0, n)$$, and $$\pi(\theta) = 1$$(improper), and $$X$$ is normal.

Is there any reference, or hint to approach this problem?

• I think you are describing the bayesian central limit theorem which takes your idea a step further and says the posterior distribution will converge approximately to a normal distribution. Here's the first reference I could find with Google: www2.myoops.org/twocw/mit/NR/rdonlyres/Mathematics/… – Great38 Oct 5 '18 at 1:53