Helper Function to Fit Beta Distribution with Variance Inflation for est_initial_R
Source:R/est_initial_R.R
fit_beta_with_variance_inflation_R.RdFits a Beta distribution to Monte Carlo R/N samples by the method of moments,
keeping the sample mean and multiplying the sample SD by
variance_inflation (so the variance scales by its square). Values in
(0, 1) tighten, values > 1 widen, and 0 or 1 leave the spread unchanged. The
concentration is floored at 2 (SD capped at \(\sqrt{m(1-m)/3}\)) so a very
large factor cannot produce an invalid Beta, and at \(1/m\) so shape1 stays
at least 1: with prop_R means of 1e-3 to 1e-2, a factor of 13-100 otherwise gave
shape1 of ~0.004-0.06, a prior whose median sat tens of decades below its
mean (e.g. ETH median 5.5e-87 against mean 1.8e-3). The mean is kept
regardless; the factor is effectively capped where it would break that floor.
Before v0.100.0 the half-widths of the sample 95% CI were scaled linearly,
the lower bound floored at 1e-10 and the result passed to
fit_beta_from_ci(). With the mode-exact logit-scale fitter that
unreachable floored bound dominated the fit (prior mean ~2.5x the sample
mean for typical prop_R inputs), so the SD is now scaled directly, which is
what the factor was documented to do.