Bayesian Computational Statistics

Pathway

Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.

1. Level 0

   26 pages
   • level 0Adaptive empirical-Bayes credible-set calibration and failure9 results

     This page fixes the mildly ill-posed Gaussian sequence model and its empirical-Bayes marginal-likelihood selector.

     3 definitions, 4 lemmas, 1 theorem, 1 counterexampleExamples & counterexamples →
   • level 0Banach Gaussian priors, RKHS support, and concentration18 results

     This page develops centered Borel Gaussian random elements in real separable Banach spaces.

     3 definitions, 5 lemmas, 10 theoremsExamples & counterexamples →
   • level 0Bayesian models, priors, and posterior laws4 results

     This page defines a proper Bayesian model as a prior probability paired with a parameter-measurable sampling kernel.

     2 definitions, 1 theorem, 1 corollaryExamples & counterexamples →
   • level 0Computational foundations: factorization, KL, and optimization8 results

     This page develops a supplied jointly measurable latent-variable density and its observed marginal, with pointwise conditional latent densities only on positive finite fibres.

     1 definition, 2 algorithms, 1 lemma, 4 theoremsExamples & counterexamples →
   • level 0Control variates, antithetics, and stratification9 results

     This page develops unbiased fixed-coefficient control-variate estimators with known control means, with a scalar optimum for positive-variance controls and a vector optimum…

     3 definitions, 1 lemma, 5 theoremsExamples & counterexamples →
   • level 0Cross-validation risk, model selection, and comparison uncertainty6 results

     For IID observations, a fixed weighted split schedule estimates the weighted risks at its training sizes; an independent randomized schedule averages those targets over its randomization law.

     2 definitions, 1 algorithm, 1 lemma, 2 theoremsExamples & counterexamples →
   • level 0Deterministic quasi-Monte Carlo and discrepancy10 results

     For a finite nonempty multiset of points in the half-open unit cube, star discrepancy measures the largest error on an anchored box.

     3 definitions, 1 algorithm, 4 lemmas, 2 theoremsExamples & counterexamples →
   • level 0Exact permutation tests and random-permutation Monte Carlo5 results

     This page treats permutation inference when a finite group acts measurably on the data space and every null law is invariant under that action.

     2 definitions, 1 algorithm, 2 theoremsExamples & counterexamples →
   • level 0General-state Markov kernels, invariance, and reversibility2 results

     For a probability measure π and a pointwise probability kernel P on any measurable state space, the joint flow defined in General-state reversibility by symmetric flow measure is a probability measure on the product sigma-algebra.

     1 definition, 1 theoremExamples & counterexamples →
   • level 0General-state MCMC ergodicity and mixing21 results

     This page separates invariance from convergence for a general-state Markov kernel.

     5 definitions, 13 lemmas, 3 theoremsExamples & counterexamples →
   • level 0Generalized Bayes loss updates and temperature5 results

     For a fixed observation, a measurable real loss, a probability prior and a positive temperature, the generalized Bayes update is the prior reweighted by e^-wℓ when its normalizer is finite and positive.

     1 definition, 1 lemma, 2 theorems, 1 remarkExamples & counterexamples →
   • level 0Gibbs sampling and scan schemes13 results

     An exact Gibbs block update retains the unselected coordinates and draws the selected block from a supplied proper full conditional kernel.

     3 definitions, 2 algorithms, 4 lemmas, 2 theorems, 2 corollariesExamples & counterexamples →
   • level 0Hellinger geometry and posterior consistency4 results

     This page defines root-L2 Hellinger distance for laws with supplied densities against a common sigma-finite measure and proves its invariance under a change of common dominating measure.

     1 definition, 2 theorems, 1 counterexampleExamples & counterexamples →
   • level 0IID bootstrap and jackknife for smooth statistics15 results

     Prerequisite context is provided by resampling-algorithms-and-sampling-units, the first- and second-order limits in delta-method-and-asymptotic-normality, confidence-set…

     1 definition, 7 algorithms, 4 lemmas, 3 theoremsExamples & counterexamples →
   • level 0IID Monte Carlo integration and error5 results

     For a real IID sequence of integrand outputs, IID Monte Carlo sample mean, variance, and standard error defines the sample mean, sample variance, and plug-in standard error.

     1 definition, 4 theoremsExamples & counterexamples →
   • level 0IID posterior contraction by entropy and prior mass6 results

     This page proves the IID posterior contraction theorem from a same-metric packing bound, an exponentially small prior tail outside a measurable parameter sieve, and local…

     1 definition, 4 lemmas, 1 theoremExamples & counterexamples →
   • level 0Laplace approximation and deterministic quadrature5 results

     For a compact measurable domain and a unique interior mode, the Laplace theorem gives the leading asymptotic equivalent under a positive amplitude, a positive- definite…

     1 definition, 2 lemmas, 2 theoremsExamples & counterexamples →
   • level 0Parametric Bernstein–von Mises limits and misspecification6 results

     This page proves a finite-dimensional total-variation Bernstein–von Mises theorem from compact-uniform LAN, a continuous positive prior density at an interior point, and posterior contraction at the LAN parameter rate.

     1 lemma, 3 theorems, 2 corollariesExamples & counterexamples →
   • level 0Posterior consistency at a KL projection9 results

     This page states almost-sure posterior concentration around the full set of KL-minimizing parameters in a dominated IID model.

     2 definitions, 5 lemmas, 1 proposition, 1 theoremExamples & counterexamples →
   • level 0Posterior Prediction, Credible Sets, and Bayes Actions12 results

     Fix a supplied measurable posterior probability kernel and a future probability kernel.

     4 definitions, 2 lemmas, 6 theoremsExamples & counterexamples →
   • level 0Rejection sampling4 results

     Rejection sampling draws proposals from a normalized proposal law and accepts each proposal according to a target-to-proposal density ratio bounded by a finite envelope.

     1 algorithm, 3 theoremsExamples & counterexamples →
   • level 0Resampling schemes and the sampling-unit contract8 results

     This page defines the empirical law, size-n bootstrap, delete-d replicates, subsampling without replacement, and the sampling-unit contract.

     2 definitions, 3 algorithms, 2 lemmas, 1 theoremExamples & counterexamples →
   • level 0State-space models and exact Bayesian filtering4 results

     For a finite-horizon state-space model on standard-Borel state and observation spaces, this page defines filtering, predictive, and full-path smoothing laws as conditional kernels.

     1 definition, 1 algorithm, 2 theoremsExamples & counterexamples →
   • level 0Statistical, resampling, simulation, and numerical error ledger5 results

     For a scalar analysis, name the exact target and every successive reference target before assigning error labels.

     1 definition, 3 theorems, 1 corollaryExamples & counterexamples →
   • level 0Synthetic likelihood and indirect inference6 results

     For a measurable summary vector with finite second moments, this page defines its exact binding mean and covariance and the ideal Gaussian synthetic-likelihood density when the covariance is positive definite.

     2 definitions, 1 algorithm, 1 lemma, 1 theorem, 1 remarkExamples & counterexamples →
   • level 0Weak nonparametric Bernstein–von Mises limits13 results

     This page proves the product-prior weak Bernstein–von Mises limit in the Gaussian white-noise model.

     1 definition, 8 lemmas, 1 theorem, 3 corollariesExamples & counterexamples →

2. Level 1

   13 pages
   • level 1Adaptive MCMC convergence limits6 results

     For a jointly measurable family of fixed Markov kernels with a common invariant target, diminishing adaptation controls the change between adjacent kernels in uniform total variation.

     2 definitions, 2 lemmas, 1 theorem, 1 corollaryExamples & counterexamples →
   • level 1Bootstrap failure, m-out-of-n methods, and subsampling8 results

     This page separates the resampling algorithms from their validity claims.

     1 definition, 3 lemmas, 4 theoremsExamples & counterexamples →
   • level 1Feynman–Kac targets and sequential particle filters8 results

     Define finite-horizon Feynman–Kac path measures on standard Borel state spaces, derive their proposal path Radon–Nikodym weights, and give the exact sequential importance resampling procedure.

     1 definition, 2 algorithms, 2 lemmas, 3 theoremsExamples & counterexamples →
   • level 1Gaussian-process posterior contraction10 results

     This page proves posterior contraction for three separate experiments using Gaussian prior concentration, local likelihood comparisons, exponential tests and a posterior denominator bound.

     9 lemmas, 1 theoremExamples & counterexamples →
   • level 1Identifiability and approximation bias in computational inference4 results

     This page separates the full-data inferential target from changes introduced by algorithms, summaries, asymptotic approximations, and numerical computation.

     1 definition, 1 lemma, 2 theoremsExamples & counterexamples →
   • level 1Latent-variable EM and likelihood ascent4 results

     For a specified jointly measurable complete-data density, EM alternates a conditional latent-law calculation with maximization of the expected complete-data log density.

     1 algorithm, 2 theorems, 1 remarkExamples & counterexamples →
   • level 1Metropolis–Hastings kernels5 results

     This page constructs the Metropolis–Hastings transition kernel from a target probability and a general proposal kernel.

     1 algorithm, 3 lemmas, 1 theoremExamples & counterexamples →
   • level 1Moving and tapered block bootstrap for dependent data17 results

     This page defines overlapping moving blocks and tapered blocks for Hilbert-valued time series, proves their finite-sample conditional covariance identities, and verifies the taper normalization.

     2 definitions, 4 algorithms, 7 lemmas, 4 theoremsExamples & counterexamples →
   • level 1Multilevel Monte Carlo and cost theorem5 results

     This page defines a geometric hierarchy of approximations, coupled adjacent-level corrections, independent level sample blocks, and the finite MLMC estimator.

     1 definition, 1 lemma, 1 theorem, 2 corollariesExamples & counterexamples →
   • level 1Ordinary and self-normalized importance sampling7 results

     Ordinary importance sampling changes measure with a known target normalizer.

     1 definition, 2 algorithms, 4 theoremsExamples & counterexamples →
   • level 1Posterior propriety and dominated Bayes formula2 results

     For a fixed observation, the evidence is the integral of the nonnegative likelihood section against the prior measure.

     1 definition, 1 propositionExamples & counterexamples →
   • level 1Randomized quasi-Monte Carlo and replicate error20 results

     An RQMC rule makes each point marginally uniform and separately declares its low-discrepancy property; points within a rule need not be independent.

     5 definitions, 10 lemmas, 3 theorems, 2 corollariesExamples & counterexamples →
   • level 1Studentized permutation tests under weak parameter nulls21 results

     This page develops two-sample order-(r,r) U-statistics, their first projections, and second-moment control of the higher-order remainder.

     3 definitions, 1 algorithm, 10 lemmas, 6 theorems, 1 corollaryExamples & counterexamples →

3. Level 2

   4 pages
   • level 2Approximate Bayesian computation and summary-based inference5 results

     Approximate Bayesian computation starts with a proper prior, standard-Borel parameter and data spaces, a simulator probability kernel, and a measurable Euclidean summary.

     1 definition, 1 algorithm, 3 theoremsExamples & counterexamples →
   • level 2Conjugate families and Gaussian conditioning21 results

     This page develops conjugate updates only on fibers with finite positive normalizers.

     7 definitions, 6 lemmas, 8 theoremsExamples & counterexamples →
   • level 2Finite-horizon SMC error, CLTs, and degeneracy13 results

     Give the finite-horizon empirical-law and particle-count CLTs for sequential importance resampling, with multinomial selection and the residual-selection branch where finite-sample floor counts have a stable limit.

     2 definitions, 8 lemmas, 3 theoremsExamples & counterexamples →
   • level 2Particle MCMC: PMMH and particle Gibbs10 results

     Particle marginal Metropolis–Hastings (PMMH) is an exact pseudo-marginal Metropolis update when the likelihood estimator is nonnegative, measurable, and unbiased for a finite-horizon state-space likelihood.

     2 definitions, 4 algorithms, 4 theoremsExamples & counterexamples →

4. Level 3

   8 pages
   • level 3ABC-MCMC and ABC-sequential Monte Carlo6 results

     This page makes the simulator-based approximate target explicit on the augmented parameter-data space whenever the tolerance-event probability is positive.

     1 definition, 2 algorithms, 1 lemma, 2 theoremsExamples & counterexamples →
   • level 3Dirichlet-process priors and finite-partition updates8 results

     This page defines the Dirichlet process as a probability law on evaluation-measurable probability measures and proves a countably additive normalized-gamma construction from a stripwise Poisson random measure.

     2 definitions, 2 lemmas, 3 theorems, 1 remarkExamples & counterexamples →
   • level 3Exact linear-Gaussian Kalman filtering4 results

     This page proves the finite-horizon Kalman recursion gives an exact filtering-kernel version for a linear-Gaussian state-space model.

     1 algorithm, 2 lemmas, 1 theoremExamples & counterexamples →
   • level 3Gaussian-process priors and finite posterior laws3 results

     For an arbitrary index set, a real mean function and symmetric positive-semidefinite covariance kernel define consistent finite Gaussian laws.

     1 lemma, 2 theoremsExamples & counterexamples →
   • level 3Hamiltonian Monte Carlo and leapfrog correction7 results

     This page defines the Gaussian-momentum Hamiltonian target and the finite leapfrog Metropolis update.

     2 definitions, 1 algorithm, 1 lemma, 3 theoremsExamples & counterexamples →
   • level 3Hierarchical Bayes and partial pooling7 results

     This page develops a finite-group hierarchical Bayesian model from its probability kernels.

     2 definitions, 1 lemma, 4 theoremsExamples & counterexamples →
   • level 3MCMC diagnostics and finite-run limitations10 results

     This page defines finite-array procedures for split R-hat, Blom rank normalization, folding, bulk and tail ESS, and indicator-based quantile MCSE.

     4 definitions, 6 algorithmsExamples & counterexamples →
   • level 3Variational inference and the evidence lower bound7 results

     Fixing a measurable nonnegative target density γ with finite positive normalizer Z gives the normalized target π=γ/Z and the ELBO for admissible densities q.

     1 definition, 1 algorithm, 1 lemma, 2 theorems, 1 counterexample, 1 remarkExamples & counterexamples →

5. Level 4

   7 pages
   • level 4Dirichlet normal-mixture posterior contraction13 results

     This page builds the density, finite-mixture approximation, scale-band entropy, Dirichlet simplex mass and local KL-mass estimates needed for the compact-truth Dirichlet normal-mixture contraction theorem.

     11 lemmas, 1 theorem, 1 corollaryExamples & counterexamples →
   • level 4Fixed-interval Gaussian smoothing and RTS recursion4 results

     This page defines the marginal law of a state given the full finite observation interval, proves the backward Gaussian conditional kernel, and states the Rauch–Tung–Striebel (RTS) algorithm and its exactness theorem.

     1 definition, 1 algorithm, 1 lemma, 1 theoremExamples & counterexamples →
   • level 4Functional BvM limits and credible-set calibration22 results

     This page develops scalar functional BvM limits and credible-interval calibration in Gaussian white noise, including the nonlinear delta-method condition.

     3 definitions, 11 lemmas, 7 theorems, 1 corollaryExamples & counterexamples →
   • level 4Langevin algorithms: ULA bias and MALA exactness17 results

     ULA applies a noisy gradient step without correction, while MALA uses the same Gaussian proposal with a Hastings acceptance rule and rejected self-transitions.

     1 definition, 2 algorithms, 10 lemmas, 4 theoremsExamples & counterexamples →
   • level 4Marginal likelihoods, Bayes factors, and empirical Bayes4 results

     For a finite set of dominated models with proper parameter priors and specified jointly measurable likelihood densities against a common sigma-finite data measure, each marginal likelihood is the prior-predictive density.

     2 definitions, 2 theoremsExamples & counterexamples →
   • level 4MCMC central limit theorems, MCSE, and long-run variance43 results

     This page defines covariance long-run variance, three stationary mixing coefficients, geometric and polynomial total-variation rates, one-step minorization, and the overlapping-batch-means estimator.

     6 definitions, 1 algorithm, 27 lemmas, 9 theoremsExamples & counterexamples →
   • level 4Particle smoothing, backward simulation, and Rao–Blackwellized filters11 results

     Backward simulation samples paths from filtering particles by reweighting each past particle with its transition density.

     1 definition, 3 algorithms, 4 lemmas, 3 theoremsExamples & counterexamples →

6. Level 5

   1 page
   • level 5Data assimilation: 3DVAR, 4DVAR, and ensemble filters13 results

     This page sets up finite-dimensional data assimilation and distinguishes forecast from analysis, online filtering from fixed-interval smoothing, and point estimates from posterior laws.

     1 definition, 3 algorithms, 6 lemmas, 3 theoremsExamples & counterexamples →