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Can't find it anywhere. I know Gamma is the conjugate prior for the exponential distribution (one parameter) but for the sum of exponential distributions (the hypoexponential distribution), I can't find it anywhere.

STAN says it looks like a smooth gamma but I would like to stay in closed form if possible.

Derek Fulton
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    How much does a closed form solution matter? In many Bayesian analyses the posterior is intractable, this is unfortunate but we learn to live with it. In my own experiences, I'd say most modern Bayesian analyses are intractable – jcken Jun 11 '20 at 07:13
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    Do you know for sure that it exists? Not every family of distributions admits a conjugate prior. – N. Virgo Jul 30 '20 at 02:14
  • Do not know if it exists. How would I go about investigating whether it exists? – Derek Fulton Jul 31 '20 at 09:57

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The Hypoexponential distribution is the distribution of sum of $k$ independent random variables $X_i$, each exponential with rate $\lambda_i$. The link above have expressions for the density function (an interesting one using the matrix exponential function which avoids the need for specialcasing of cases with some $\lambda_i=\lambda_j,\quad i\not= j$). This expressions make it clear that the hypoexponential family not is an exponential family.

There is a relationship between being an expoential family and having a conjugate prior, see Aside from the exponential family, where else can conjugate priors come from?. There are a few other examples, but they need the existence of a sufficient statistic of fixed dimension. The hypoexponential family does not have that, so the answer must be that a conjugate prior do not exist.

See also Exponential family and conjugate priors

kjetil b halvorsen
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