Closed form for Finite Gaussian Mixture Model when weights are known and prior variance can be 0

Suppose I have a normal likelihood $$x|\theta \sim N(\theta, \sigma^2_{known})$$ where the variance is known and a mixture prior $$\theta \sim p * N(\mu_1, \sigma^2_1) + (1-p) * N(\mu_2, \sigma^2_2)$$, where $$p$$ is known.

My prior is conjugate, so I known $$\theta | x$$ is also a Normal mixture with a closed form solution. However, my algebra skills are failing me and I keep getting nonsense answers. Does anyone known the closed form solution for the posterior?

There is one additional complication. I am writing R code to calculate this, and I want users to be able to make the first component in the mixture a point mass. The easiest way for me to do this is consider the first component of the mixture as a Normal random variable with $$\sigma_1^2 = 0$$. Are there closed form solutions for this case as well? I suspect the closed form solutions will break down when $$\sigma_1^2 = 0$$.

• Could you please expand on where you get stuck in deriving the posterior? It is a mixture indeed with posterior weights proportional to the prior weights times the marginal likelihoods. As for the limiting case of a point mass, there is no difficulty as the posterior is a mixture of a point mass and of a Normal distribution as well. – Xi'an Aug 22 at 22:05