Gaussian optimum with Poisson error: large peak width results in difficulties in specifying the model

Modeling
1

Hi!

I am working toward a model of a Gaussian optimum of fitness that I can use to model how some phenotypic traits are selected and estimate their optimal value. The model features a Gaussian optimum for the expected fitness of the individuals:
\lambda(z) = W_{max} * \exp(-\frac{(z-\theta)^2}{2 \omega^2}),
which translates into realised fitness with some (possibly zero-inflated, but let’s keep it simple) Poisson error:
W(z) \sim \mathcal{P}(\lambda(z)).

A difficulty in this model is that selection is typically weak, and thus \omega is relatively large, compared to the variance of z (supposed mean-centered and scaled to a variance of 1 throughout) and I’m struggling defining an efficient model and a good prior, especially for the peak width (omega-related parameter). I’d appreciated any idea in how to best deal with this?

A final note before going to the model that I have : I do know the model as specified above can more efficiently translated into a constrained GLMM. However, because the parameters have a biological interpretation and should respond differently to different covariates or random effects, we would very like to keep the model explicit as we will build up more complexity into (making it less and less equivalent to a GLMM).

Here is the model that I have so far:

// Code for explicit model
functions { 
}  
data {
    int<lower=1> Nobs;          // Total number of observations
    int<lower=0> W[Nobs];       // Fitness variable
    vector[Nobs] z;             // Phenotype variable
} 
transformed data { 
} 
parameters { 
//     real<lower=0> tau;       // inverse-peak width (tau = 1 / (2*omega^2))
    real<lower=0> width;        // peak width
    real theta;                 // Values of the optimum parameter
    real log_Wmax;              // Values of the max fitness parameter
} 
transformed parameters { 
} 
model { 
    vector[Nobs] lambda;       // Values of the expected fitness
    vector[Nobs] eta;

    // Likelihood
    eta = log_Wmax - ((z - theta) .* (z - theta))./width;
    lambda = exp(eta);
    W ~ poisson(lambda);
    
    // Priors
    width ~ cauchy(0,5);
    theta ~ normal(0,10);
    log_Wmax  ~ normal(0,10);
}
generated quantities { 
    // Getting the quantities as they are defined in theoretical model
    real omega = sqrt(width/2);
    real Wmax = exp(log_Wmax);
}

Here are the points that I have difficulties with in the model:

  • Is there a more efficient way to specify the likelihood? I was going for a vectorised formulation as much as possible and thought writing the Gaussian optimum on the log scale then only taking the exponential of it would help computation efficiency, is this a good idea or not needed at all?
  • I’m hesitating between dividing by a “width” parameter or multiplying by its inverse (named “tau” and currently commented out). The issue is that width would be typically large (say between 10 and 100), but very similar to omega which we are interested with, while tau would be typically small but defining a prior that allows it to be 0 (e.g. like the Cauchy used here) might mean that the posterior mean of omega can be infinite/undefined.
  • Which leads us to my final worry: the parameter width (or equivalently tau) is quite sensitive to the prior distribution and most of them seem to create a relatively strong bias in the estimates. For example, using the Cauchy prior above, I obtain estimates around 5-6 on average for a true value around 4. I tried using a log-normal prior on either width or tau, but this makes the HMC very unhappy with a lot of divergent transitions. I would very much appreciate any pointer toward good practice here.

(EDIT: I forgot to say that the other priors -especially for theta- are relatively informative because the variable z is mean-centered and scaled to a variance of 1, so we know the ballpark of where theta should live and I thought having such a slightly informative prior would have the fit altogether)

If this helps, I’m attaching a fake dataset along with its true values.

#True values
omega       <- 4
Wmax        <- 6
theta       <- -0.5   

df <-
structure(list(Pheno = c(0.786157710243689, -0.816106924162414, 
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0.54810680837722, 0.0232626650502272, 1.06174593527589, 0.0907053226154761, 
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