Modelling single value based on two likelihoods

[pcstiles]

Hello all,

I am currently working on a spatio-temporal project where the final outcome is the product of two modelled quantities that are irregularly sampled in space and time, but have the same spatial and temporal scales. My initial thought was to produce two separate models where the fitted values are multiplied together to create the final outcome, but there are obviously some issues with how to aggregate uncertainty in that case. I know INLA can handle multiple likelihoods in one model, but I was wondering if it was possible to use INLA to build a model of a single quantity (i.e., single intercept) that is based on two likelihoods? If not, how would you suggest handling the uncertainty of the two models when multiplying their fitted values? The two likelihoods in my case are pooled binomial and zero-inflated Poisson.

Thank you in advance for your insight!

Pascale

[Helpdesk]

you're likely looking at a joint model where some components are shared
/are the same, between the models, and you have one set of observations
for each.

this is 'straight forward' in theory, but you need to know what you're
doing. the spde-book have examples of this kind, please check that one
online. please build your joint model in small steps to keep control.

here is the example you asked for, a single intercept observed as 1 in a
binomial and 2 in a poisson (of'course, the example does not make any
sense, but its a proof of concept)

> y=matrix(NA,2,2)
> diag(y) = c(1,2)
> y
[,1] [,2]
[1,] 1 NA
[2,] NA 2
> r=inla(y ~ 1, data = list(y=y), family=c("binomial", "poisson"))
> r$summary.fixed
mean sd 0.025quant 0.5quant
0.975quant
(Intercept) 0.8331420422 0.6297788024 -0.5375458096 0.8833363934
1.931141256
mode kld
(Intercept) 0.988442421 3.596643302e-06

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Håvard Rue
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[Pascale Stiles]

Thank you for pointing me in the right direction! I will read through that section of the book and come back to this space if I have more questions.

Best,

Pascale