A Bayesian Cox analysis must place a prior on both regression coefficients and the baseline cumulative hazard. A prior on \(\beta\) combined only with partial likelihood is a generalized Bayesian procedure; a full posterior requires an event-time likelihood and a baseline-hazard prior.
Full semiparametric posterior
Definition. Let \[ \Pi=\Pi_\beta\otimes\Pi_{\Lambda} \] be a prior on \(\beta\in\mathbb R^p\) and nondecreasing cumulative hazards \(\Lambda\). Under independent censoring, the posterior is \[ \Pi(\mathrm{d}\beta,\mathrm{d}\Lambda\mid O^{(n)}) \propto \prod_{i=1}^n \left[ \{e^{\beta^\top Z_i}\mathrm{d}\Lambda(X_i)\}^{\Delta_i} e^{-e^{\beta^\top Z_i}\Lambda(X_i)} \right] \Pi_\beta(\mathrm{d}\beta)\Pi_\Lambda(\mathrm{d}\Lambda). \]
If censoring parameters are variation-independent and the prior factors, censoring-law terms cancel from this posterior. Dependent censoring requires a joint model or an identified weighted likelihood; it cannot be repaired by merely changing the prior on \(\Lambda\).
Piecewise exponential conjugacy
Example. Partition time into intervals \(I_k=(a_{k-1},a_k]\), let \(\lambda_0(t)=\alpha_k\) on \(I_k\), and assign independent priors \[ \alpha_k\sim\operatorname{Gamma}(a_k,b_k) \] in shape-rate form. Conditional on \(\beta\), define \[ D_k=\sum_i\int_{I_k}\mathrm{d}N_i(t),\qquad E_k(\beta)=\sum_i\int_{I_k} Y_i(t)e^{\beta^\top Z_i(t)}\,\mathrm{d}t. \] Then \[ \alpha_k\mid\beta,O^{(n)} \sim\operatorname{Gamma} \{a_k+D_k,\ b_k+E_k(\beta)\}. \]
Proof. The likelihood kernel for \(\alpha_k\) is \[ \alpha_k^{D_k}e^{-\alpha_kE_k(\beta)}. \] Multiplication by the gamma prior kernel \(\alpha_k^{a_k-1}e^{-b_k\alpha_k}\) gives the stated posterior parameters.
This representation supports Gibbs or Metropolis-within-Gibbs computation and makes prior regularization of sparse-event intervals visible. Letting the partition become too fine without scaling the gamma parameters can create a prior that changes with arbitrary discretization.
Process priors for cumulative hazards
Definition. A gamma-process prior has independent increments satisfying \[ \Lambda(B)\sim\operatorname{Gamma}\{\alpha(B),c\} \] for measurable time sets \(B\), where \(\alpha\) is a finite base measure on bounded intervals. More generally, a neutral-to-the-right prior makes multiplicative survival decrements over disjoint intervals independent.
Proposition. A gamma process is almost surely nondecreasing and purely atomic, while its expectation is \[ E\{\Lambda(B)\}=\frac{\alpha(B)}{c}. \] It nevertheless has full weak support around many continuous cumulative hazards when the base measure charges every interval.
Proof. Independent gamma increments are nonnegative, so cumulative sums are nondecreasing. The Lévy measure of the gamma process has infinitely many positive jumps, giving pure atomicity. The expectation follows from the gamma mean. Approximation of a continuous increasing function by step functions, together with positive gamma density for every finite collection of positive increments, gives positive mass to finite-dimensional neighborhoods; tightness extends this to weak neighborhoods.
Posterior consistency and contraction
Theorem (testing-and-prior-mass template). Let \(d\) be a metric on conditional event-time laws. Suppose for a sequence \(\varepsilon_n\downarrow0\):
- the prior assigns at least \(e^{-Cn\varepsilon_n^2}\) mass to a Kullback-Leibler neighborhood of \((\beta_0,\Lambda_0)\);
- outside a sieve of prior complement at most \(e^{-(C+4)n\varepsilon_n^2}\), the model has entropy \[ \log N(\varepsilon_n,\mathcal P_n,d) \le Cn\varepsilon_n^2; \]
- independent censoring retains positive observation probability on \([0,\tau]\).
Then for sufficiently large \(M\), \[ \Pi\{d(P_{\beta,\Lambda},P_{\beta_0,\Lambda_0}) >M\varepsilon_n\mid O^{(n)}\} \xrightarrow{P_0}0. \]
Proof outline. The entropy bound constructs exponentially powerful tests for annuli outside the target ball. The Kullback-Leibler prior mass lower-bounds the posterior denominator with high probability. The sieve-tail bound controls prior mass not covered by tests. Summing tested annuli yields a vanishing posterior numerator relative to the denominator. The argument is the censored-data version of the general theorem developed in testing, entropy, and posterior contraction.
Fixed-dimensional Bernstein-von Mises limit
Theorem (semiparametric BvM for \(\beta\), schematic). Suppose \(p\) is fixed, the true Cox model satisfies the regularity conditions of pages 6 and 10, the prior density for \(\beta\) is continuous and positive at \(\beta_0\), and the baseline prior contracts sufficiently fast while remaining stable under least-favorable local perturbations. Then \[ \sup_B\left| \Pi\left\{ \sqrt n(\beta-\widehat\beta)\in B \mid O^{(n)} \right\} -N_p(0,\widetilde I^{-1})(B) \right| \xrightarrow{P_0}0. \] Consequently, regular posterior credible ellipsoids for \(\beta\) have asymptotically correct frequentist coverage.
Proof outline. Expand the likelihood along the least-favorable direction that changes \(\beta\) while adjusting \(\Lambda\) to remove nuisance-score correlation: \[ \ell_n(\beta_0+h/\sqrt n,\Lambda_h) -\ell_n(\beta_0,\Lambda_0) =h^\top\widetilde\Delta_n -\frac12h^\top\widetilde I h+o_{P_0}(1). \] Posterior contraction localizes \((\beta,\Lambda)\) to a set where the expansion is uniform. Prior stability permits the nuisance change of variables without a first-order prior distortion. Integrating the Gaussian kernel and controlling tails gives total-variation convergence centered at the efficient estimator.
Why contraction alone is insufficient. A nuisance posterior can be consistent and still induce first-order bias in \(\beta\). BvM requires the nuisance contraction rate and likelihood remainder to satisfy a no-bias condition, often resembling \[ \sqrt n\,\varepsilon_n^2\to0, \] plus stability in a least-favorable direction. Credible-set validity is therefore stronger than posterior consistency.
Survival-function BvM
Proposition. If a joint BvM theorem holds for the efficient coordinates of \((\beta,\Lambda_0)\), then for fixed \(z\) and \(t\), \[ \sqrt n\{S(t\mid z)-\widehat S(t\mid z)\} \mid O^{(n)} \] converges to a centered normal law with variance \[ \operatorname{Var}_{P_0} \left[ -S_0(t\mid z)e^{\beta_0^\top z} \{\widetilde\phi_{\Lambda,t} +\Lambda_0(t)z^\top\widetilde\phi_\beta\} \right]. \]
Proof. The map \[ (\beta,\Lambda)\mapsto \exp\{-e^{\beta^\top z}\Lambda(t)\} \] is differentiable, with derivative represented by the influence expression in brackets. Apply the posterior delta method to the joint Gaussian limit.
Sparse high-dimensional Bayes
Definition. A spike-and-slab prior may be written \[ \pi(s)\,\frac{1}{\binom ps} \sum_{|S|=s} \left\{\prod_{j\in S}g(\beta_j)\right\} \left\{\prod_{j\notin S}\delta_0(\mathrm{d}\beta_j)\right\}, \] with a dimension prior satisfying approximately \(\pi(s)\lesssim e^{-c s\log p}\). Continuous shrinkage alternatives include global-local priors, but exact support statements then require thresholding or decision rules.
Theorem (sparse contraction template). If \(\beta_0\) has support size \(s_0\), the risk-set information satisfies a sparse compatibility condition, the prior gives sufficient mass near \(\beta_0\), and \[ s_0\log p=o(n), \] then a well-calibrated sparse prior can contract at \[ \|\beta-\beta_0\|_2 =O_{P_0}\left(\sqrt{\frac{s_0\log p}{n}}\right) \] up to model-specific constants, jointly with baseline-hazard contraction.
BvM boundary. Full \(p\)-dimensional Gaussian BvM is impossible when \(p\gg n\). One may seek a BvM theorem for a fixed low-dimensional functional \(a^\top\beta\), requiring posterior sparsity, an approximately sparse inverse-information direction, and a bias condition often stronger than contraction, such as \[ s_0\log p=o(\sqrt n). \] Model-selection priors can also create nonuniform behavior near zero coefficients. The inferential claim must specify the functional and sparsity regime.
Computation and posterior checking
Useful algorithms include:
- Gibbs updates for piecewise exponential baseline hazards;
- Hamiltonian Monte Carlo for smooth finite-dimensional hazard bases;
- reversible-jump or birth-death methods for random partitions and sparse models;
- Laplace and variational approximations for large \(p\), followed by calibration checks because covariance is often underestimated.
Posterior predictive checking should compare event counts over time, risk-set residuals, survival calibration, and patterns of censoring. A good fit to partial likelihood alone does not validate the baseline hazard or absolute-risk predictions. The broader posterior, contraction, sparse-prior, and variational theory is developed in Bayesian statistics.