Subtle error in some capture-recapture likelihoods

[mbjoseph]

There are some subtle errors in the likelihoods of some capture-recapture models (Chapter 6 of the Bayesian Population Analysis translations). I noticed the errors as a side effect of digging into a separate issues on the Stan Forums: https://discourse.mc-stan.org/t/capture-recapture-model-with-partial-or-complete-pooling/20393/32

Here's an overview of the model that should help clarify what the problem is:

State model

These models are two-component finite mixture models, where the components correspond to a partly observed binary latent variable z[i] for individuals i=1, ..., N. The two components are:

• z[i] = 0 the individual is not in the population
• z[i] = 1 the individual is in the population

This latent quantity is modeled as a Bernoulli: z[i] ~ Bernoulli(omega).

Observation model

There are T sampling occasions, and on each sampling occasion an individual is either detected or not. The observation model is y[i, t] ~ Bernoulli(p * z[i]) for t=1, ..., T, or summing the number of detections to use a binomial instead, y[i] ~ Binomial(T, p * z[i]).

Stan implementation

Marginalizing over the discrete state z leads to the following integrated observation model: https://github.com/stan-dev/example-models/blob/master/BPA/Ch.06/M0.stan#L29-L38

Essentially, when an individual is observed, we know z[i] = 1, and otherwise we have to sum over z[i] = 0 and z[i] = 1 in the likelihood.

The issue

For some of these capture-recapture models, there's an error in the likelihood associated with the case when an individual is in the population but not detected (z[i] = 1, and max(y[i, 1:T]) = 0).

The observation model conditional on z[i]=1 should be binomial_lpmf(0 | T, p), or equivalently bernoulli_lpmf(y[i] | p), where y[i] is an integer array of 0's of length T. The following models have a subtle error in that they incorrectly use bernoulli_lpmf(0 | p) instead:

• Mb: capture effects on detection
• Mt: time-varying detection
• MtX: time-varying detection + covariate effects
• Mtbh: time-varying detection + individual heterogeneity + previous capture effects
• Mth: time-varying detection + individual heterogeneity

For an example of a model with the correct likelihood, see M0: https://github.com/stan-dev/example-models/blob/master/BPA/Ch.06/M0.stan#L37

[mbjoseph]

Oops - I meant to tag @ito4303, who put together the original translations. Does this sound right to you @ito4303?

[ito4303]

Thank you for your comment. I have understood the issue and appreciate your fix.

[bbbales2]

Closed by #205