Compute the Block-Diagonal Observed Information Matrix

Computes the observed information matrix $H_{\mathrm{obs}}$ for the fixed effects of a hurdle Beta-Binomial model. Uses a block-diagonal structure:

Usage

compute_H_obs(fit, scores = NULL)

Arguments

fit: An object of class "hbb_fit".
scores: Optional $N \times D$ score matrix (as from compute_score_matrix()). If NULL, computed internally.

Value

A list with components:

H_obs: Numeric matrix ($D \times D$). Block-diagonal observed information.
H_obs_inv: Numeric matrix ($D \times D$). Inverse of H_obs (the bread matrix).
H_ext: Numeric matrix ($P \times P$). Extensive-margin Fisher information block.
H_int: Numeric matrix ($(P+1) \times (P+1)$). Intensive + kappa block from information identity.
ridge_applied: Logical. Whether ridge regularisation was needed.

Details

Extensive margin (alpha block, $P \times P$): analytic Fisher information from logistic regression.
Intensive + kappa block ($(P+1) \times (P+1)$): empirical information identity using posterior mean scores.

Extensive margin (H_ext)

For the base weighted model: $$H_{\mathrm{ext}} = \sum_{i=1}^N \tilde{w}_i\, q_i (1-q_i)\, X_i X_i^\top$$ where $q_i = \mathrm{logit}^{-1}(X_i^\top \hat{\alpha})$.

For SVC models, the linear predictor includes state random effects: $q_i = \mathrm{logit}^{-1}(X_i^\top \hat{\alpha} + X_i^\top \hat{\delta}^{\mathrm{ext}}_{s[i]})$.

Intensive margin (H_int)

The zero-truncated Beta-Binomial Hessian is analytically complex, so we use the information identity: $$H_{\mathrm{int}} = \sum_{i=1}^N \tilde{w}_i\, s_i^{\mathrm{int}} (s_i^{\mathrm{int}})^\top$$ where $s_i^{\mathrm{int}} = (s_{\beta,i}, s_{\kappa,i})^\top$ is the $(P+1)$-dimensional intensive score at the posterior mean.