Normalizing constant / marginal likelihood in TMB?

[Katie Paulson]

Hello,

Is it possible to use TMB to estimate the normalizing constant (marginal likelihood) for a Bayesian model?

To be clear, I have something like:

p(theta | data) = p(data | theta) p(theta) / p(data)

with a likelihood defined for p(data | theta), and latent model defined for theta. I've been using TMB to fit this model and approximate the posterior distribution p(theta | data).

Now, I would like to also get the normalizing constant p(data). With other tools I know how to extract this value (e.g. R-INLA via mlik in the model output) -- is this also possible in TMB?

Thank you!

Katie Paulson

[Paul vdb]

`aghq` is an R package that uses quadrature to approximate the posterior using a TMB object. It has some similarities to INLA for using Laplace for inner marginalization but then using adaptive gauss hermite quadrature for the outer marginalization to get p(data).
awstringer1/aghq: Adaptive Gauss Hermite Quadrature for Bayesian Inference

[2102.06801] Stochastic Convergence Rates and Applications of Adaptive Quadrature in Bayesian Inference

See the vignette for:  `normalized_posterior$lognormconst` from the built object to get the normalizing constant.... I think but double check.

[2101.04468] Implementing Approximate Bayesian Inference using Adaptive Quadrature: the aghq Package

Related package in NIMBLE: Using nimbleQuad for maximum likelihood estimation and deterministic posterior approximation