Mean-Tilted Relaxed Quantile Regression: Fixed-Content Interval Functionals and Generalized-Bayes Computation

Probability content does not by itself determine interval placement.

We study the interval functional induced by relaxed quantile regression (RQR), whose residual-product check loss estimates two unlabeled raw roots without preassigning endpoint quantiles.

Under explicit conditions, the unrestricted population minimizer is the unique contiguous content-c interval whose retained mean equals the population mean.

A fixed mean tilt preserves content while shifting that retained mean to mu + delta.

Interior admissible tilts index all finite-root interior content-c windows, with boundary members obtained as qualified one-sided limits.

Equal-tailed and shortest-contiguous intervals therefore have distribution-specific recovery tilts.

We construct loss-based generalized posteriors using a pseudo-asymmetric-Laplace normal-exponential augmentation.

A fixed-rate mean-tilted sampler covers static regression under proper Gaussian priors, with ordinary RQR obtained exactly at zero tilt.

The implemented ordinary branch also supports a conditional-Gaussian regularized-horseshoe adapter using the Nishimura-Suchard augmentation (RHS-NS), and a frozen deep echo-state-network feature matrix is a deterministic nonlinear-design specialization of the same static scan.

A dynamic linear extension replaces coefficient blocks by alternating root-specific forward-filtering backward-sampling steps: the stacked state prior is Gaussian, but the joint augmented observation kernel is quartic.

Current software and empirical evidence concern ordinary RQR; nonzero-tilt algorithms are derived but not yet implemented or validated.

All updates concern interval-root functionals under a loss and prior, not a response likelihood or posterior-predictive responses.