Update BetaPrime tests

[cscherrer]

No description provided.

[cscherrer]

Current status:

test dist::betaprime::tests::cdf_values ... ok
test dist::betaprime::tests::ln_pp_vs_ln_m_single ... ok
test dist::betaprime::tests::mean_when_beta_leq_one ... ok
test dist::betaprime::tests::mode_when_alpha_geq_one ... ok
test dist::betaprime::tests::draw_should_return_positive_values ... ok
test dist::betaprime::tests::mode_when_alpha_lt_one ... ok
test dist::betaprime::tests::new ... ok
test dist::betaprime::tests::mean_when_beta_gt_one ... ok
test dist::betaprime::tests::rv_impl::should_impl_debug_clone_and_partialeq ... ok
test dist::betaprime::tests::test_ln_m_cache_consistency ... ok
test dist::betaprime::tests::test_posterior ... ok
test dist::betaprime::tests::test_posterior_parameter_updates ... ok
test dist::betaprime::tests::variance_when_beta_leq_two ... ok
test dist::betaprime::tests::rv_impl::should_impl_parameterized ... ok
test dist::betaprime::tests::variance_when_beta_gt_two ... ok
test dist::betaprime::tests::test_bayes_law ... ok
test dist::betaprime::tests::ln_m_vs_monte_carlo ... FAILED   <------------------bad
test dist::betaprime::tests::test_sbc ... ok
test dist::betaprime::tests::ln_m_single_datum_vs_monte_carlo ... ok
test dist::betaprime::tests::ln_pp_vs_monte_carlo ... ok
test dist::betaprime::tests::ln_pp_single_vs_monte_carlo ... ok

With 1K samples it fails with

-1.5404450409471488 !~ -1.299563872668477

If I increase it to 1M I get

-1.5404450409471488 !~ -1.2802343101033689

and for 100M

-1.5404450409471488 !~ -1.2785370199497486

With more samples it's getting worse, so something is very wrong. But the implementation is simple, and I don't see any errors with the math. Here's the derivation

and implementation

    fn ln_m_cache(&self) -> Self::MCache {
        self.ln_beta_ab()
    }

    fn ln_m_with_cache(
        &self,
        cache: &Self::MCache,
        data: &DataOrSuffStat<usize, StickBreakingDiscrete>,
    ) -> f64 {
        let post = self.posterior(data);
        post.ln_beta_ab() - cache
    }