# Closed form posterior for a mixtures of two univariate Gaussians

Giving a univariate Gaussian mixture model $$\pi_1N(x|\mu_1,\sigma_1)+(1-\pi_1)N(x|\mu_2,\sigma_2),$$ are there any priors for $$\pi_1$$, $$\mu_1$$, $$\sigma_1$$, $$\mu_2$$, $$\sigma_2$$ which gives a closed form posterior?

• It depends what you call closed form. A conjugate prior gives a closed form posterior mixture but with $2^n$ terms (Dielbolt & Robert, 1994). – Xi'an Feb 12 at 16:51
• $2^n$ where $n$ is the number of observed independent random variables $x_1, \ldots, x_n$ as stated in jstor.org/stable/2345907, am I right? – Alessandro Jacopson Feb 12 at 18:02
• $n$ is the sample size. – Xi'an Feb 12 at 18:22