Computes Infinitesimal Jackknife variance estimates for the fixed effects
(Giordano & Broderick, 2023; Ji, Lee & Rabe-Hesketh, 2024) from a single
MCMC run. The IJ influence of observation i is
I_i = n * cov_post(beta, loglik_i) — the posterior covariance between the
fixed-effect draws and the per-observation (conditional) log-likelihood draws
— and the variance estimate is
Details
V_IJ = (1 / (n (n - 1))) sum_i (I_i - Ibar)(I_i - Ibar)',
with a cluster-robust version that aggregates influences within cluster j
as I_j = (J / n) sum_{i in j} I_i and replaces n by the number of clusters
J.
Caveat for hierarchical models
The influences use the conditional per-observation log-likelihood (given the
random effects), whereas the IJ of Ji, Lee & Rabe-Hesketh (2024) is derived
for a marginal model. For coefficients identified by within-cluster
variation (e.g. slopes) the conditional IJ agrees well with the Yang-Wang-He
sandwich. For coefficients identified by between-cluster variation (the
intercept of a random-intercept model) it can under-estimate the variance
— up-weighting a cluster mostly shifts that cluster's random effect rather
than the fixed effect — and it is noisier; how much depends on the
random-effect-to-noise ratio. For valid fixed-effect inference in the
hierarchical model fitted by bqmm, prefer the default Yang-Wang-He sandwich
(ywh_adjust()); method = "ij" is provided for benchmarking.