# Jacobian Adjustment : Mixture Poisson-Multivariate-Lognormal

Hello stan users :)

I’d like to fit a mixture of Poisson and Multivariate log-normal, such as

Y_{ij} \sim Poisson(\lambda_{ij})\\ \lambda_{ij} \sim MultiLogNormal(\mu_{ij}, \Sigma)

Because the multivariate log-normal is not implemented in stan, I thought about writing my model as

Y_{ij} \sim Poisson(\lambda_{ij})\\ log(\lambda_{ij}) \sim MultiNormal(\mu_{ij}, \Sigma) + \frac{\delta}{\delta \lambda_{ij}} log(\lambda_{ij})

Is this really similar to the former? I think that I might miss a subtility due to the multivariate nature of my model.

However, if I understand the manual correctly, jacobian adjustment would not be necessary if I write the model like this, because I sample the \lambda_{ij} parameter before transforming it.

Y_{ij} \sim Poisson(exp(\lambda_{ij}))\\ \lambda_{ij} \sim MultiLogNormal(\mu_{ij}, \Sigma)

What do you think would be the right way to implement the Poisson-multivariate log-normal model??

Thank you very much, and have a good day :)
Lucas

2 Likes

Call poisson_log_lpmf(y | log_lambda) for the likelihood, put your multivariate normal prior on log_lambda (quite possibly with the non-centering trick) and don’t worry about the Jacobian.

2 Likes

Thank you very much Ben!!