---
title: "Valid inference under the asymmetric Laplace likelihood"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Valid inference under the asymmetric Laplace likelihood}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", eval = FALSE)
```

## The problem

The asymmetric Laplace distribution (ALD) is a convenient *working* likelihood
for quantile regression: its mode-as-quantile and the check-loss connection make
Bayesian computation straightforward (Yu and Moyeed, 2001). But it is
misspecified for almost any real data-generating process, and a misspecified
likelihood produces a posterior whose spread is the *wrong* asymptotic variance
for the quantile-regression estimator. Naive credible intervals from such a
posterior do not have correct frequentist coverage.

## The correction

Yang, Wang and He (2016) restore validity with a multiplicative sandwich that
re-uses the posterior covariance as the "bread":

$$
V_\text{adj} = \Sigma_\text{post}\, G\, \Sigma_\text{post},
$$

where \(\Sigma_\text{post}\) is the posterior covariance of the fixed effects and
\(G\) is the meat — the variance of the asymmetric-Laplace working-likelihood
score. With score \(s_i = \sigma^{-1} x_i\,(\tau - \mathbf{1}\{r_i<0\})\) on the
conditional residuals \(r_i\), the meat is
\(G = \sigma^{-2}\sum_g\big(\sum_{i\in g} x_i\psi_i\big)\big(\cdot\big)'\)
(cluster-robust on the grouping factor; the default), or its independence
analogue.

Using \(\Sigma_\text{post}\) as the bread is what makes this correct for a
**mixed** model: the posterior covariance already encodes the multilevel
pooling, so the adjustment keeps the random-effect contribution to fixed-effect
uncertainty while fixing the misspecified ALD scale. Under correct
specification \(G \approx \Sigma_\text{post}^{-1}\) and the correction reduces to
\(\approx \Sigma_\text{post}\).

```{r}
vcov(fit, adjusted = TRUE)    # corrected (multiplicative, cluster meat)
vcov(fit, adjusted = FALSE)   # naive posterior covariance
confint(fit, adjusted = TRUE)
```

## Why not the plain Koenker sandwich?

The textbook fixed-effects sandwich
\(\tau(1-\tau)D_1^{-1}D_0D_1^{-1}/n\) (available internally as
`compute_ywh_sandwich()` and validated against `quantreg`) is computed on
residuals with the random effects removed, so it drops the between-cluster
variance and **under-covers** the mixed-model fixed effects. A simulation
bake-off (`tools/bakeoff.R`) confirmed this: across homoscedastic and
heteroscedastic two-level designs at several quantiles, the Koenker form covered
the fixed intercept at only 0.72–0.92, while the multiplicative form above
covered at 0.95–1.00 — at or just above nominal everywhere.

## Scope and caveats

* Validity is claimed for the **fixed-effect block**. Variance components retain
  their model-based posterior summaries.
* The correction is mildly **conservative** (slightly over-nominal) under weak
  misspecification — the price of guaranteed validity.
* It is a large-sample / many-clusters argument; with very few clusters the
  cluster-robust meat is noisy.

## References

Yang, Y., Wang, H. J. and He, X. (2016). Posterior inference in Bayesian
quantile regression with asymmetric Laplace likelihood. *International
Statistical Review*, 84(3), 327-344.