Objective AFT families: parameter measures and executable certificates
=====================================================================

Scope and provenance
--------------------
The fixed-error families and gamma Fisher information follow the local
manuscript "Objective Bayesian Inference for Parametric Accelerated Failure
Time Models", Pedro L. Ramos, draft dated August 25, 2026 (project 095).
That draft proved gamma propriety only after compact restriction of shape.
The unrestricted gamma argument below completes that boundary analysis for
the sufficient failure-design conditions used by this implementation.
It is an implementation derivation, not a claim that the old draft already
contained the theorem or that a numerical check constitutes a proof.

The Student-t certificate below is deliberately more conservative than the
draft's almost-everywhere discussion. Full rank and absence of a fit of ALL
failures do not exclude singular configurations fitting many repeated
observations. No almost-sure statement is used as a pointwise software check.

All likelihoods use original-time densities for failures and survival
probabilities for right censoring. Censoring is independent/noninformative
conditional on covariates. Priors are constructed for the complete-data
experiment, not for every possible censoring mechanism. Every design contains
an intercept. Rank and noninterpolation use relative tolerance 1e-10.

1. Fixed-error location-scale models
-----------------------------------
log(T) = X beta + sigma Z, sigma>0; p is the number of regression coefficients.
Z has standard normal (lognormal), minimum extreme value (Weibull), logistic
(log-logistic), Laplace (log-Laplace), or Student-t_df (log-t) distribution.
Student degrees of freedom are fixed, explicitly supplied, not estimated.
The Weibull survival is exp[-(t/exp(eta))^(1/sigma)].

With respect to d beta d sigma, the joint Jeffreys prior is sigma^(-(p+1)).
The beta-interest, scale-nuisance ordered reference construction with product
compact sets gives 1/sigma. Write the chosen measure as sigma^(-a), with a=1
or a=p+1. Changing the dimension p changes joint Jeffreys, not reference.
There is no MDI route here. The prior called Simple belongs only to GG.

Let d denote failures and X_F,y_F their design and log times. For the four
exponentially bounded errors, sufficient conditions are full rank p,
noninterpolation y_F not in col(X_F), and d>p. Their density is bounded by
K exp(-c|z|). Thus the failure-only likelihood dominates the censored
likelihood and its integrated kernel is bounded by
  C sigma^(-d-a) (1+sigma^p) exp(-c0/sigma).
This is integrable at both ends. The same argument certifies
E(||beta||^r sigma^s)<infinity for r>=0 and d+a>p+r+s+1.

For a positive marginal moment of beta_j or sigma, the boundary is exact:
write beta=sigma b and integrate over a fixed small ball of b on which
|b_j|>epsilon (for beta_j). As sigma tends to infinity, all standardized
residuals remain bounded and all censored survival factors are bounded away
from zero. The lower bound is C sigma^(p+r-d-a). Hence absolute moments
at or above r=d+a-p-1 diverge even with these censored observations.
Moment flags FALSE refer to lack of absolute integrability, not a Monte Carlo
estimate. Standard deviations require a finite second moment.

Exponential and Rayleigh fix sigma at 1 and 1/2, respectively. Both priors
then reduce to a flat measure on beta. Full rank X_F alone is sufficient;
exact interpolation is harmless with scale fixed. All polynomial beta
moments are finite by an exponential-envelope/coercivity argument.

2. Student-t: a sufficient pointwise certificate
-----------------------------------------------
Choose p failures whose design rows form an invertible matrix. Reserve them.
Among the other failures, construct B disjoint blocks of p+1 observations
whose augmented matrices [X_block, y_block] have rank p+1.
For each block, the infimum over beta of its maximum absolute residual is
strictly positive. Therefore, for 0<sigma<=1, at least one error-density
factor in that block is bounded by C sigma^(df+1); the remaining factors
are bounded. Integration over beta using the p reserved Student densities
contributes exactly a constant times sigma^p. Consequently the small-scale
normalizing integral is bounded by
  C integral_0^1 sigma^(p-d-a+B*(df+1)) d sigma.
The implemented sufficient condition is
  B*(df+1) > d+a-p-1.
The large-scale bound is C sigma^(p-d-a); d>p and a>=1 suffice there.

Each block is found by deterministic, column-scaled QR with a documented
tolerance. A greedy block search need not find every certifiable dataset;
failure is reported as not certified, never as a proof of impropriety.
This criterion allows factors and repeated rows when enough noninterpolable
blocks remain, without enumerating all subsets or trusting generic position.

Integrating |beta_j|^r over the reserved p factors is finite when r<df and
is bounded by C(1+sigma^r). Thus a sufficient coefficient-moment condition
is r<min(df,d+a-p-1). The sigma condition is r<d+a-p-1 once the small-scale
certificate passes. The large-scale divergence bound in section 1 still
applies. Coefficient moments above df but below the large-scale bound remain
unknown, not falsely declared infinite. These conservative tests can reject
proper Student posteriors, particularly with many covariates or few failures.

Counterexample to the weaker old-draft check: intercept only, uncensored
log times (0,0,0,1), df=1, reference prior. In |beta|<c*sigma near zero,
the three tied likelihood factors contribute sigma^-3, the fourth contributes
sigma^1, the prior sigma^-1 and beta integration sigma. The remaining scale
integral is proportional to integral sigma^-2 d sigma and diverges.
The design is nevertheless full rank and not all four observations interpolate.
The new certificate refuses this sample.

3. Gamma AFT with unrestricted, estimated shape
------------------------------------------------
T_i ~ Gamma(k, rate=k*exp(-eta_i)), eta_i=x_i' beta, k>0.
The conditional mean is exp(eta_i). There is no extra free sigma.
Complete-data Fisher information is block diagonal:
  I_beta,beta = k X'X,
  I_k,k = n (trigamma(k)-1/k).
With beta as the interest block, product compact sets give
  pi_R(beta,k) proportional to sqrt(trigamma(k)-1/k).
Joint Jeffreys is
  pi_J(beta,k) proportional to k^(p/2)*sqrt(trigamma(k)-1/k).
The scale-dependent nuisance reference normalizer is constant in beta, so
the ordered reference construction does not supply a k^(p/2) factor.
These priors are rederived in this submodel; they are not the GG prior
evaluated on a lower-dimensional constraint.

THEOREM (sufficient unrestricted gamma propriety).
If X_F has full rank p, y_F is not in its column space, and d>p, both
posteriors above are proper on beta in R^p and k in (0,infinity).
There are no prior hyperparameters or artificial shape limits.

PROOF.
Write z_i=y_i-x_i'beta and phi(z)=exp(z)-1-z. The log-time density is
  c(k) exp[-k phi(z)], c(k)=k^k exp(-k)/Gamma(k).
Original-time Jacobians are positive data constants. Censoring factors are
at most one. By full rank, H(beta)=sum_F phi(y_i-x_i'beta) is coercive,
since phi(z)>=c|z|-C for fixed positive c,C. Noninterpolation gives
h0=inf_beta H(beta)>0.

For 0<k<=1, c(k)<=Ck. Also H(beta)>=c1||beta||-C1, whence
  integral ||beta||^r exp[-k H(beta)] d beta <= C_r k^(-p-r)
for every r>=0. The reference shape factor is asymptotic to 1/k;
the joint Jeffreys factor is asymptotic to k^(p/2-1). Therefore the
normalizing integrals at zero are bounded by powers
  reference: k^(d-p-1),  joint Jeffreys: k^(d-p/2-1),
both integrable under d>p.

For k>=1, c(k)<=C sqrt(k). Split k H(beta) into k H(beta)/2 twice.
One part is at least k h0/2; the other is at least H(beta)/2. Thus
  integral ||beta||^r exp[-k H(beta)] d beta <= C_r exp(-k h0/2).
The reference and joint shape factors grow at most polynomially
(respectively k^-1 and k^(p/2-1)). Every positive shape moment is finite.
The bound also permits polynomial coefficient moments subject to the
small-k threshold. This proves propriety and the stated finite moments.

For completeness the coefficient-moment threshold is exact, including
right-censored observations. Set beta=b/k, and choose a compact full-
dimensional ball with all x_i'b>epsilon>0 and |b_j|>epsilon for the desired
coefficient. Such a ball exists because the design contains an intercept.
Failure factors are bounded below by C k. For a finite censored time t_i,
the gamma CDF at k*t_i*exp(-x_i'b/k) tends to exp(-x_i'b)<1, uniformly
on this ball. Survival factors are therefore bounded below by positive
constants. Zero censoring contributes one. The beta Jacobian is k^-p
and |beta_j|^r contributes k^-r. The lower powers are the same as above.
Hence absolute coefficient moments exist exactly for
  reference: r<d-p; joint Jeffreys: r<d-p/2.
All positive shape moments exist. A finite first absolute moment does not
imply that exponentiated coefficients or lifetime means have finite means.
The implementation reports time-ratio medians/quantiles, not their means.

If shape is explicitly fixed, its prior factors are constants and both
routes reduce to flat beta. Full rank of the failure design then suffices,
with all polynomial beta moments finite; the scale is not additionally fitted.

4. Sampling, diagnostics and comparison
---------------------------------------
The existing GG model retains its unbounded Reference/Simple priors, its
separate propriety statement and unavailable Q mean/DIC. See gg-aft-theory.txt.
GG is not used to emulate the new priors or fixed-shape submodels.

Other models use fixed-basis stepping-out/shrinkage slice sweeps after a
unit-Jacobian quantile-intercept translation. An initial lognormal/least-
squares approximation and numerical optimization determine fixed geometry,
not the posterior. For nonsmooth Laplace targets, a positive design-based
fallback is permitted when a numerical Hessian is unsuitable. It changes
only the proposal basis. Covariate-column scaling is a constant-Jacobian
change and is reversed in all saved coefficients and predictions.

Exceptions with exact simulation: complete lognormal regression uses
sigma^2 ~ inverse-Gamma((n+a-p-1)/2,RSS/2), followed by its conditional
normal beta. Intercept-only exponential/Rayleigh with right censoring and
complete intercept-only known-shape gamma use transformed gamma draws.

WAIC/LOO use the observed original-time likelihood. Each LOO deletion must
pass the model-specific certificate under the SAME prior (including p and
fixed df/shape). Failure does not trigger a silent prior change or refit.
DIC is provided only when all required parameter means are certified and
the numerical deviance calculations succeed. Its usual additional finite-
deviance assumption and parameterization dependence remain documented.

Rank-normalized R-hat/ESS concern Monte Carlo mixing, not posterior propriety.
Quantile MCSE concerns precision, not existence of a parameter mean. An
apparently stable finite simulation cannot resolve an improper posterior.

References
----------
Fernandez and Steel (1999), Reference priors for the general location-scale
model, Statistics & Probability Letters 43:377-384,
doi:10.1016/S0167-7152(98)00276-4.
Fernandez and Steel (2000), Bayesian regression analysis with scale mixtures
of normals, Econometric Theory 16:80-101, doi:10.1017/S0266466600161043.
Neal (2003), Slice sampling, Annals of Statistics 31:705-767,
doi:10.1214/aos/1056562461.
Prentice (1974), A log gamma model and its maximum likelihood estimation,
Biometrika 61:539-544, doi:10.1093/biomet/61.3.539.
