Package 'fibr'

Description

For each group-level (random-effect) coordinate $\alpha_j$ of a fitted hierarchical model, the prior fraction

$\pi_j = \frac{\text{prior precision}}{\text{prior precision} + \text{likelihood information}} = \frac{1/\sigma^2}{G_{FF,j}}$

is the share of that coordinate's posterior precision contributed by the prior rather than by its own data. It is the pooling factor of Gelman and Pardoe (2006); its complement $1-\pi_j$ is the shrinkage factor. The prior/likelihood balance it captures is the one Betancourt and Girolami (2015) tied to the optimal centred/non-centred parameterisation.

Interpretation. $\pi_j \approx 1$ means the coordinate is prior-dominated: its posterior is essentially the prior pushed through shrinkage, so the estimate is mostly regularisation toward the population and should not be over-interpreted unless the prior is one you would defend. $\pi_j \approx 0$ means the data speak. This is a prior-influence report, not a convergence diagnostic, and it is read-only: nothing is reparameterised or refit.

Scope and limits. The estimate is exact for the common GLM families (gaussian, bernoulli, binomial, poisson, negbinomial) with the standard (... | g) random-effect structure. For correlated random effects it reports the per-marginal fraction (using each coefficient's own sd); the full story there is the eigenvalues of a matrix pooling factor, and a message is emitted. Coordinates with no data (n_obs == 0) are flagged with $\pi = 1$. Smooths and GP terms have correlated coordinates and should be read with that caveat. The diagnostic says nothing about multimodality, aliasing, or likelihood mis-specification.

Usage

prior_fraction(x, ...)

## Default S3 method:
prior_fraction(x, lik_information, labels = NULL, ...)

## S3 method for class 'brmsfit'
prior_fraction(x, ndraws = 200L, ...)

Arguments

Value

A data frame of class fibr_prior_fraction with one row per coordinate and columns group, coef, level, n_obs, prior_sd, lik_info, pi. Has print and plot methods.

Methods (by class)

• prior_fraction(default): Manual path for any model. Supply the per-coordinate prior precision x ($1/\sigma^2$) and the per-coordinate likelihood information lik_information ($G_{FF,j} - 1/\sigma^2$); optionally a labels data frame to carry through. Use this to validate against the closed-form GLMM or to handle Stan fits this package does not parse.

• prior_fraction(brmsfit): Adapter for brms fits. Extracts the random-effect structure, per-coordinate prior SDs, and the family information at the posterior mean, and returns the per-coordinate prior fraction. Requires brms.

References

Gelman and Pardoe (2006), Technometrics 48(2):241–251. Betancourt and Girolami (2015), in Current Trends in Bayesian Methodology with Applications.

Examples

## Manual path (no model fit needed): supply the per-coordinate prior
## precision (1/sigma^2) and likelihood information (sum of per-observation
## Fisher information). This is the closed-form GLMM prior fraction.
sigma <- 1.5
lik   <- c(0.2, 1.0, 5.0)          # e.g. sum p(1-p) for three groups
prior_fraction(1 / sigma^2, lik_information = lik)


## brms path: which group-level estimates are prior-dominated?
if (requireNamespace("brms", quietly = TRUE)) {
  set.seed(42)
  dat <- data.frame(y = rpois(30, 5), site = rep(letters[1:10], 3))
  fit <- brms::brm(y ~ 1 + (1 | site), data = dat,
                   family = stats::poisson(), iter = 500, chains = 1,
                   refresh = 0)
  pf <- prior_fraction(fit)
  pf                                # summary: how many coordinates have pi > 0.8
  plot(pf)                          # pi vs. number of observations
}

Description

For each random effect on a changepoint location (omega_k_g = per-group deviation from the population changepoint at breakpoint k), computes the Fisher information decomposition at a subsample of posterior draws.

The key quantity is prior_frac:

$\text{prior\_frac}_{k,g} = \frac{G_\text{prior}}{G_\text{prior} + G_\text{lik}}$

where

$G_\text{prior} = \sigma_{\text{re},k}^{-2}, \quad G_\text{lik} = \sigma^{-2} \sum_{i:\,\text{group}=g} \!\!\!\left(\frac{\partial\mu_i}{\partial\omega_{k,i}}\right)^{\!2}$

• prior_frac $\to 1$: prior dominates – group changepoints are poorly identified from data relative to the shrinkage prior. The sampler is in the funnel regime and non-centred reparameterisation would help.

• prior_frac $\to 0$: likelihood dominates – centred parameterisation is efficient and mixing should be adequate.

• Mixed: flag individual groups for attention.

What to do with the results:

When prior_frac is high, re-fit with reparameterise = "omega":

fit_nc <- smoothbp(..., reparameterise = "omega")
fit_nc_ss <- smoothbp_ss(..., reparameterise = "omega")

This activates the non-centred HMC parameterisation in the Rust sampler: z[j] = beta_om[j] / sigma_re_om[k] is sampled with an N(0,1) prior, and beta_om[j] = z[j] * sigma_re_om[k] is reconstructed automatically. The sigma_re_om Gibbs step and the stored draws are unchanged – output is in the original (centred) parameterisation for easy interpretation.

Additional options if reparameterise = "omega" is insufficient:

1. Increase warmup and iter.

2. Check fit$n_divergent – many divergences confirm a remaining funnel.

3. Fix the changepoint for poorly-identified groups using omega = list(fixed(value)).

The prior_frac values quantify the severity: values above 0.8 indicate a serious funnel; 0.6-0.8 suggests moderate difficulty worth addressing.

The gradient is computed analytically from the sigmoid smooth-transition likelihood:

$\frac{\partial\mu_i}{\partial\omega_{k,i}} = -\Bigl[\delta_{k,i}\,\sigma_{ki}\bigl(1 + d_{ki}\,\rho_{ki}(1-\sigma_{ki})\bigr) + b_{1,i}\,\mathbf{1}_{k=1}\Bigr]$

where $d_{ki} = \tau_i - \omega_{k,i}$ and $\sigma_{ki} = \text{logistic}(d_{ki}\,\rho_{ki})$.

Usage

smoothbp_advisor(fit, n_draws = 200L, threshold_nc = 0.6, threshold_c = 0.4)

Arguments

Value

An S3 object of class fibr_smoothbp_advice. Contains one list element per breakpoint that has omega random effects, each with prior_frac_mean, prior_frac_q05, prior_frac_q95, recommendation, and delta (one entry per group). delta[j] is the recommended per-group NC mixing fraction $\delta_j = \sqrt{\text{prior\_frac}_j}$. Print and plot methods included.

References

Gelman and Pardoe (2006), Technometrics 48(2):241–251 (the pooling factor). Papaspiliopoulos, Roberts and Skold (2003), Bayesian Statistics 7; Tan and Nott (2013), Statistical Science 28(2):168–188 (the closed-form partial non-centring weight $\delta_j$).