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I am trying to analytically compute the posterior distribution for a simple dataset. I have a multi-variate Gaussian likelihood and a discrete uniform prior. I am multiplying my prior with my likelihood to get a Gaussian mixture with n terms. I am not sure how to then analytically compute the pdf from this, I don't have any experience with mixture models. Any help would be appreciated!

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    $\begingroup$ Please explain what you mean by "analytically compute." The pdf is a linear combination of Gaussians, requiring a sum (over the support of the prior) for its full, correct expression. Unless your case is special, no simplification is possible. $\endgroup$
    – whuber ♦
    Commented Dec 30, 2021 at 20:35
  • $\begingroup$ I suppose I mean to formally write it out, so that I may draw samples from it. From what you have written, though, if I have a discrete uniform prior between 0 and n, and a multi-variate gaussian likelihood, the posterior would be the sum of n multi-variate Gaussian distributions, correct? Is this a mixture model then? $\endgroup$
    – Max
    Commented Dec 30, 2021 at 22:02
  • $\begingroup$ It is hard to see how a formula would improve your ability to sample. Generally, one samples from a mixture by generating a multinomial value for the number of observations to obtain in each mixture component, and then sampling separately from each of those components (and, if necessary, randomly permuting the bunch). See stats.stackexchange.com/a/64058/919 for an explanation with a two-component mixture. With more work you could also invert the mixture's quantile function, as described at stats.stackexchange.com/a/411671/919. None of these need analytical formulas. $\endgroup$
    – whuber ♦
    Commented Dec 30, 2021 at 22:05
  • $\begingroup$ So to check my understanding, I have a two-component mixture: discrete uniform with parameters a and b, and multi-variate Gaussian with parameters mean and covariance matrix. I am not sure, then, what the proportion of each component is in my case. The provided example makes sense, but I am not sure how to extrapolate to my problem. I am sorry if these questions are simple, but I do not have a robust statistics background and this is my first exposure to mixture models. $\endgroup$
    – Max
    Commented Dec 30, 2021 at 22:20

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