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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.

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    $\begingroup$ 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 $\endgroup$
    – jcken
    Commented Jun 11, 2020 at 7:13
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    $\begingroup$ Do you know for sure that it exists? Not every family of distributions admits a conjugate prior. $\endgroup$
    – N. Virgo
    Commented Jul 30, 2020 at 2:14
  • $\begingroup$ Do not know if it exists. How would I go about investigating whether it exists? $\endgroup$ Commented Jul 31, 2020 at 9:57

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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

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