Calculate pointwise log-likelihood in multilevel models with measurement error and identifiability hacks

Modeling
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2

I think in this case you might initially think to to give the pointwise log likelihood as

normal_lpdf(y_true[i] | alpha0 + alpha[group[i]] + beta * x[i,], sigma) + 
   normal_lpdf(y_obs[i] | y_true[i], y_err);

But that won’t work well with loo because it depends on the observation-specific parameter y_true. Fortunately we can get rid of the y_true here with

normal_lpdf(y_true[i] | alpha0 + alpha[group[i]] + beta * x[i,], sqrt(sigma^2, y_err^2));

I’m pretty sure that’s the pointwise log-likelihood that you want, but maybe @martinmodrak might be kind enough to double-check with his perspective. I don’t think that the post-sweeping matters at all to the likelihood.

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