Fit analytical distribution to posterior samples for generating more waves

This function allows "updating" of the prior with a posterior distribution from the same family as the prior with updated parameters.

Usage

posterior_fit_analytical(posteriors_df, priors_list)

Arguments

posteriors_df: a dataframe of posteriors that have been selected by ABC this may include columns for scores, weight and/or simulation outputs ( abc_component_score, abc_summary_distance, abc_weight, abc_simulation ) as well as columns matching the priors input specification.
priors_list: a named list of priors specified as a abc_prior S3 object (see priors()), this can include derived values as unnamed 2-sided formulae, where the LHS of the formula will be assigned to the value of the RHS, plus optionally a set of constraints as one sided formulae where the RHS of the formulae will resolve to a boolean value.

Value

an abc_prior S3 object approximating the distribution of the posterior samples from the same family as the prior.

Details

This takes weighted posterior samples $\{(\theta^{(i)}, w^{(i)})\}$ for each parameter $\theta_j$ and fits an analytical distribution function $Q_j(\theta_j)$ that approximates the posterior marginal for that parameter.

The resulting empirical distribution $Q_j$ is a dist_fns object that includes the support constraints from the original prior. The set of all fitted marginal distributions $\{Q_j\}$ forms the new proposal list for the next wave.

Additionally, the weighted covariance matrix $R$ of the samples in the MVN space (defined by the prior copula) is calculated: $$ R = \text{Cov}_{w}(\Phi^{-1}(P_1(\theta_1)), \dots, \Phi^{-1}(P_d(\theta_d))) $$ where $\Phi^{-1}$ is the quantile function of the standard normal. This covariance matrix is stored as an attribute ("cor") of the returned proposal list and is used to induce correlation structure when sampling new proposals from the empirical distributions in subsequent waves.

Examples


fit = example_smc_fit()
proposals = posterior_fit_analytical(fit$posteriors, fit$priors)

proposals
#> Parameters: 
#> * mean: unif(min = 4.9, max = 5.05)
#> * sd1: unif(min = 1.81, max = 2.22)
#> * sd2: unif(min = 0.898, max = 1.09)
#> Constraints:
#> * mean > sd2