Monte-Carlo Hellinger distance between a fit and its target

Estimates the squared Hellinger distance H^2(f, g) = 1 - integral sqrt(f(x) g(x)) dx as 1 - mean(sqrt(f(x) / g(x))) over n_mc fresh draws x from the fit g, for every regime. The draws are independent of any sample used to fit, so the estimate is unbiased, with standard error se. The target's log_density must be supplied and normalised; otherwise the Monte Carlo integral is biased by the missing \(\sqrt{Z(f)}\). When the target's normalised property is not TRUE, a warning is issued and the returned value is flagged.

Usage

hellinger_mc(fit, n_mc = 5000L, seed = NULL)

Arguments

fit

A gmm_fit whose target carries a log_density.

n_mc

Number of Monte Carlo draws from the fit.

seed

Optional integer seed for the draws.

Value

A list with components

• h2 - estimate of H^2(f, g),

• se - Monte Carlo standard error,

• n_mc - number of draws with a finite density ratio,

• trustworthy - TRUE when the target is declared normalised.

See also

Other diagnostics: bic_aic(), ess_summary(), ess_trace(), gmm_anneal_path(), gmm_conditional_entropy(), gmm_entropy(), gmm_evidence(), gmm_fit_ensemble(), gmm_fit_quality(), gmm_independence_graph(), gmm_mutual_information(), kld_trace(), proxy_functional_ci()

Examples

fit <- fit_proxymix(banana_target(), N = 3L, regime = "kld",
                    is_size = 2000L, max_iter = 25L, seed = 1L)
hellinger_mc(fit, n_mc = 1000L, seed = 1L)
#> $h2
#> [1] 0.005064436
#> 
#> $se
#> [1] 0.003431512
#> 
#> $n_mc
#> [1] 1000
#> 
#> $trustworthy
#> [1] TRUE
#>