# Existence of conditional conjugate prior for the dispersion parameter of a negative binomial model

Is there a conditional conjugate prior distribution for the dispersion parameter of the negative binomial distribution given the mean rate parameter, if I parameterize it in this manner $$f(y_i; \mu, \kappa) := \frac{ \Gamma(y_i+1/\kappa) }{ \Gamma(y_i+1) \Gamma(1/\kappa) } \frac{(\kappa \mu)^{y_i}}{ (1+\kappa \mu)^{y_i+1/\kappa}}$$ for i.i.d. observations $Y_i \sim \text{NegBin}(\mu, \kappa)$ - i.e. the distribution is paramterized in terms of a mean rate parameter $\mu>0$ and a dispersion parameter $\kappa>0$ (with the distribution going towards a Poisson as $\kappa \rightarrow 0$)? I am ideally looking for a continuous conditionally conjugate density - rather than, say, a solution with point masses on a number of values for $\kappa$.

I have a scenario, where such a prior (or similarly a conjugate for something like $\log \kappa$ or any other convenient transformation) would be really useful, because I can easily obtain independent samples for $\log \mu$ and a conditional conjugate prior (or a mixture of such priors to be more flexible in expressing priors) would then make it very easy to also sample the dispersion parameter (and thus the full posterior).

It does not look to me like the answer is as simple as it's a gamma distribution - or at least by inspection of the likelihood I am failing to see it, but perhaps I am missing something obvious. Most of the information I managed to find was on different parameterizations that inconveniently (at least to me) do a change of variables (if you start out with the above parameterization) and mix up the mean rate and dispersion parameter of the parameterization above into a new parameter.

• Are you looking for an informative prior (one in which you can easily encode prior information) or something more-or-less noninformative like a reference or Jeffreys prior? Oct 11 '16 at 18:17
• I was primarily looking for conditional conjugacy (I know there is no conjugate prior, if both the mean rate and the dispersion parameter are unknown). Actually, for my own application I was primarily thinking of relatively vague priors, but did alude to the possibility of more informative ones in one of the comments. The ability of any single conjugate prior to encode specific prior information (e.g. a single gamma prior for a Poisson rate can be somewhat restrictive) does not seem like much of an issue to me, because mixtures of such priors can approximate most distributions pretty well. Oct 11 '16 at 18:54

If you define a prior distribution on $\kappa$ conditional on $\mu$ that has a density of the form $$f(\kappa|\mu) \propto \prod_{i=1}^{n_0} \Gamma(y^0_i+1/\kappa) \Gamma(1/\kappa)^{-n_0} \dfrac{\kappa^{\sum_{i=1}^{n_0} y_i^0}}{(1+\kappa\mu)^{\sum_{i=1}^{n_0} y_i^0 +n_0/\kappa}}$$ it is parameterised by $n_0$ and $(y_1^0,\ldots,y^0_{n_0})$ and is formally conjugate since the posterior has the same shape. But I see little methodological appeal in defining such a family of priors.
• Thank you very much for the kind and helpful answer. I am just now trying to work our, what difference it makes when each observation has a separate time during which observations can be made (so that $Y_i|t_i \sim \text{NegBin}(\mu t_i, \kappa)$ - I suppose the simple answer is that $f(\kappa|\mu t_i)$ would be as above with $\mu$ replaced by $\mu t_i$ for each observation and anything more general would be hard to state without assuming something specific about the $t_i$. Oct 19 '16 at 11:34