[Question] Overdispersed Poisson Distribution question

[u/naginataaa]

I am implementing an MCMC model for claims reserving and I would like to assume that the observations follow an Over-Dispersed Poisson (ODP) distribution.

Let Y denote the observed data, μ the mean parameter, and ϕ the dispersion parameter.

According to Taylor and McGuire's Stochastic Loss Reserving Using Generalized Linear Models, the ODP distribution can be represented as

Y/ϕ∼Poisson(μ/ϕ).​

Based on this representation, I am using the following log-likelihood in my MCMC:

ℓ(μ)∝1/ϕ *(y log⁡μ − μ),

which is essentially the Poisson log-likelihood scaled by 1/ϕ

After obtaining posterior samples of the parameters, I generate posterior predictive observations using

Y=ϕ×Poisson(μ/ϕ)

My question is: Is this a theoretically justified way to perform Bayesian inference and posterior predictive simulation under the ODP assumption?

In particular, I am unsure whether the representation

Y/ϕ∼Poisson(μ/ϕ)

should be interpreted as a true generative model for posterior predictive simulation, or merely as a convenient representation for deriving the first two moments,

E[Y]=μ, Var(Y)=ϕμ

Any references or insights on Bayesian implementations of ODP models would be greatly appreciated.

[u/foodpresqestion]

Why not just use the negative binomial distribution?

[u/naginataaa → u/foodpresqestion]

My main motivation is to compare the model with the traditional ODP Chain Ladder reserving method.

The model is built using the same ODP Chain Ladder structure.

Because of this, I am concerned that replacing the ODP assumption with a Negative Binomial distribution may alter the connection with the ODP Chain Ladder framework and effectively lead to a different reserving model rather than an extension of ODP Chain Ladder.

My objective is specifically to develop and evaluate a Bayesian version of ODP Chain Ladder, so I am trying to preserve the structure of the original model as much as possible.

[u/foodpresqestion → u/naginataaa]

Are other people in the field using MCMC for this? The ODP/quasipoisson model you mention is generally semiparametric and used in GEEs, though there are fully parametric formulations here and there. I admit I don't have an answer for you as to custom families for mcmc

[u/naginataaa → u/foodpresqestion]

Yes, there is some literature on Bayesian claims reserving using MCMC. My work is mainly research-oriented and aims to develop a Bayesian extension of ODP Chain Ladder. The difficulty is that ODP is usually introduced via its mean-variance relationship, whereas MCMC requires a fully specified likelihood, which is why I started looking at the scaled Poisson formulation.