Bayesian Inference on Nongenerative Model

Modeling
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1

See below a Stan program, given the data I can run Bayesian inference to estimate the parameters, but I cannot sample new data using estimated parameters – i.e., the model in the below Stan program is NOT generative.

I am wondering since the model is not generative, can we trust the inference on parameters?

Any help or comments arre appreciated?

vishy

data {
  int<lower=1> T; // number of periods
  int<lower=1> N; // number of items
  matrix[T, N] X; // cash flow data
  vector[T] f; // quarterly returns
}

parameters{
  real<lower=-1, upper=1> alpha;
  real<lower=0, upper=3> beta;
  real<lower=0, upper=1> sigma;
  real<lower=0> tau;
  vector<lower=-1, upper=10>[T] delta; // discount rates
}

transformed parameters {
  vector<lower=-1, upper=10>[N] y; // discount rates
  vector<lower=-1, upper=10>[T] r; // discount rates
    
    r[1] = 1 / (1 + delta[1]);
    for (t in 2:T) {
        r[t] = 1 - delta[t - 1] / delta[t];
    }
   y = dot_product(X', delta)
}
model {
// inference
   r ~ normal(alpha + beta*f, sigma)
   y ~ normal(0, tau); 
   
// non informative priors
   alpha ~ cauchy(0, 1)
   beta ~ cauchy(0, 1)
   sigma ~ cauchy(0, 1)
   tau ~ cauchy(0, 1)
   alpha ~ cauchy(0, 1)
   delta ~ cauchy(0, 1)
}
2

You can absolutely draw new samples from the estimated posterior distribution of parameters; that’s essentially what the generated quantities block is for, you can use distribution functions with _rng to generated (pseudo-) random samples.

I think there’s a lot of confusion on the “generative” terminology, since it is often a convoluted way of making a statement about joint versus conditional probabilities, or making informal distinctions about models. It’s not clear to me in this instance what is the importance of classifying your model as generative or not.

Again, the distinction of generative vs not-generative is probably not a useful one here, it has no bearing on whether we can trust the inferred parameters or not – also, “trust” is also a subjective term: do you want to define a measure of confidence or precision in the estimate, or some sort of out-of-sample prediction?

So there are a few loose ends there that may be worth clarifying before making these kinds of statements.