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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 $\sigma_1^2 = 0$.

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  • $\begingroup$ 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. $\endgroup$ – Xi'an Aug 22 at 22:05

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