I still don’t understand the setup. If experience is data, then so is q_weighted, how can it have a posterior distribution?
Are you after estimating the expected q_weighted in some more complicated model? Eg assume that people die with some probability q_i, which depends on some covariates,
\text{logit}(q_i) = X_i \beta + \varepsilon_i
where X_i is known, and \beta and \sigma = \text{std}(\varepsilon) are parameters. These you can estimate from the data using a multilevel model and then obtain a posterior for q_weighted.
But if you are new to Bayesian modeling, I would really recommend working through parts of a textbook first otherwise you will run into conceptual and practical problems all the time. Gelman and Hill is a great intro.