If my observed dataset has weights (for example tracking multiplicity) is it possible to provide this either to pystan or pymc3, similar to the function signature (http://mc-stan.org/rstanarm/reference/stan_glm.html) in the rstanarm package:
stan_glm(formula, family = gaussian(), data, weights, subset,
na.action = NULL, offset = NULL, model = TRUE, x = FALSE, y = TRUE,
contrasts = NULL, ..., prior = normal(), prior_intercept = normal(),
prior_aux = exponential(), prior_PD = FALSE, algorithm = c("sampling",
"optimizing", "meanfield", "fullrank"), adapt_delta = NULL, QR = FALSE,
sparse = FALSE)
With Stan (in any of its interfaces, including PyStan), you can introduce weights within the model. For example, in a linear regression, that'd be e.g., instead of y[i] ~ normal(mu[i], sigma) you use target += weight[i] * normal_lpdf(y[i] | mu[i], sigma).
This gives you a well specified density if the weights are positive. We tend to prefer generative approaches.