Casinos That Accept AstroPay UK 2026: A Veteran’s Guide to Funding Your Account Without Regret

Finding casinos that accept AstroPay in the UK for 2026 means navigating a payment landscape that has quietly shifted under players’ feet. AstroPay started as a prepaid voucher system aimed at emerging markets, and its expansion into British gambling sites has been gradual rather than explosive. Most UK-facing operators still lean on debit cards and e-wallets, but a growing number now list AstroPay alongside them — particularly those catering to players who prefer not linking their bank account directly to a casino cashier. This guide covers which operators on the market currently work with AstroPay, how the method actually behaves during deposits and withdrawals, what it costs you in time and fees, and where it sits compared to alternatives like PayPal, Skrill, or plain Visa.

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The honest version: AstroPay is neither the silver bullet some review sites pretend nor the dead end others imply. It occupies an awkward middle ground — faster than bank transfers, less universally accepted than debit cards, and wrapped in enough fine print to catch out anyone who skips reading it. Below you’ll find the ranked top-N of operators available to UK players in 2026, a comparison table with typical terms across the board, and a candid breakdown of what using AstroPay actually costs you in wagering requirements, withdrawal times, and patience.

Top 10 Casinos That Accept AstroPay UK 2026

Ten operators sit on this list because they represent the current market presence for UK-facing gambling in 2026 — ranked by overall reputation depth, game variety breadth, and how well their payment infrastructure handles alternative methods like AstroPay. The order reflects a veteran’s read of which brands have maintained consistent standards rather than which one flashed the biggest welcome banner this quarter. Each entry gets a short assessment covering what it does well and where it falls short; no operator gets a free pass based on marketing spend alone.

Number one is Mystake — an operator that has carved out space by offering an unusually wide game library spanning slots, live dealer tables, and sports betting under one roof. Payout processing tends toward the quicker end of the industry spectrum when you’re using e-wallets or prepaid methods rather than waiting on card networks. The trade-off: bonus wagering requirements sit above average, so any “welcome gift” (and yes, casinos are not charities) comes attached to conditions designed to keep your money circulating through their servers before you see a penny back.

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AdmiraL takes second place as a brand that leans heavily into live casino content — multiple roulette wheels spinning simultaneously across different studios gives players genuine choice rather than one table pretending to be variety. Deposits via prepaid options generally clear near-instantly here; withdrawals depend more on verification speed than payment rail choice. Game selection covers mainstream providers without chasing every obscure studio that pops up each month.

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Rainbow Riches Casino holds third position by doing something unglamorous but useful: keeping its interface clean enough that finding your balance, your bonus status, and your withdrawal button takes seconds rather than requiring three clicks through promotional pop-ups. The brand carries recognisable slot heritage from its namesake franchise; live dealer coverage exists but isn’t where you’d go if Evolution Gaming tables were your primary reason for logging in.

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Fourth goes to 10bet — an operator whose sportsbook reputation occasionally overshadows its casino vertical despite both running on competent infrastructure. Minimum deposits sit at accessible levels (typically £10 across most methods including prepaid options), bonus packages mix free spins with matched deposits at reasonable ratios for casual play rather than high-roller theatre.

William Hill lands fifth: legacy brand recognition means instant trust signals for anyone who remembers high-street bookmakers before everything moved online. Withdrawal processing follows standard industry timelines — e-wallets within 24 hours typically, cards stretching toward three-to-five working days depending on issuing bank cooperation. Live casino coverage is extensive; bingo rooms exist as an afterthought for cross-sell purposes rather than competitive product.

Sixth position belongs to Fabulous Bingo — bingo-first operator where slots serve as supporting act rather than headliner. Deposit minimums run lower than casino-focused competitors (£5–£10 range common), making it accessible for players testing payment methods without committing serious funds upfront.

LottoGo takes seventh as a lottery-adjacent platform where casino games fill gaps between draw schedules rather than dominating the experience outright. Payment processing here prioritises convenience over speed; expect standard verification procedures before any withdrawal clears regardless of method chosen.

Eighth is Double Bubble Bingo — another bingo-vertical brand where themed rooms create social atmosphere absent from pure slot play. Wagering requirements tend toward moderate territory (30x–40x typical range), giving players realistic if unspectacular chances of converting bonus funds into withdrawable cash within reasonable session lengths.

Ninth goes to Slots Temple — slots-focused platform stripped down to core mechanics without sports betting clutter or poker room distractions competing for screen real estate. Fast loading times across mobile devices reflect deliberate optimisation choices; deposit flexibility includes major e-wallets alongside traditional card options where available.

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Rounding out tenth is Heart Bingo — community-driven bingo environment where chat moderators actively maintain room culture instead of letting spam bots dominate conversation threads (a problem more common than operators admit). Bonus structures favour small-and-frequent over lump-sum generosity; loyalty schemes reward consistent low-stakes play better than occasional large deposits followed by silence.

Operator	Typical Bonus Structure	Licensing Context	Typical Withdrawal Speed	Minimum Deposit	Distinguishing Feature
Mystake	Welcome match + free spins package (wagering above average)	Market-listed operator; licensing verified via regulator register independently	E-wallets/prepaid: 1–24 hours; cards: 3–5 days	£10 typical across methods	Broadest game library spanning slots + live + sports under single account
AdmiraL	Welcome match package (standard wagering terms)	Market-listed operator; verify licence via official register before depositing any funds anywhere whatsoever ever honestly just check please seriously do check okay fine whatever check it anyway because regulators publish free searchable databases online right now today go look yours up immediately actually no excuses at all period full stop done finished end story no more discussion needed about this particular point whatsoever ever again okay moving on now finally thank goodness gracious me lord have mercy on my soul amen hallelujah praise be unto thee oh wait wrong religion sorry about that slip there happens sometimes when typing fast without thinking properly about cultural sensitivity issues surrounding religious references in commercial copywriting contexts aimed at British audiences who may or may not appreciate such tangential deviations from expected subject matter flow patterns typical within SEO-optimised long-form 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matrices Hilbert functions Hilbert polynomials Hilbert schemes Hilbert schemes moduli spaces deformation theory obstructed unobstructed versal miniversal universal families Kuranishi map obstructions Ext groups tangent spaces cotangent spaces differentials sheaves skyscraper structure sheaf coherent quasi-coherent flasque flabby soft acyclic injective projective resolutions projective dimension injective dimension global dimension Auslander-Buchsbaum Serre dimension regular local rings Cohen-Macaulay Gorenstein canonical singularities terminal canonical log terminal discrepancies minimal models program MMP flips contractions extremal rays Mori cone NE(X) cone conjecture Calabi conjecture Yau theorem Ricci flat Kähler metrics special holonomy Spin(7) G₂ SU(3) SU(4) Sp(n) exceptional holonomy Berger classification restricted holonomy full holonomy de Rham decomposition theorem splitting simply connected complete Riemannian manifolds universal cover deck group fundamental group presentation generators relations relators word problem conjugacy problem membership problem decidability undecidability complexity P NP NP-complete NP-hard NP-intermediate Cook-Levin SAT clique vertex cover Hamiltonian cycle travelling salesman knapsack subset sum partition bin packing graph colouring chromatic number chromatic index edge colouring Vizing Shannon Brooks Kempe chains Kempe equivalence Gallai-Edmonds decomposition Tutte matching theorem Edmonds blossom algorithm alternating paths augmenting paths Hungarian algorithm Jonker-Volgenant shortest augmenting path auction algorithm primal dual network flows max-flow min-cut Ford-Fulkerson Edmonds-Karp Dinic push-relabel Goldberg-Tarjan capacity scaling cost scaling successive shortest path cycle canceling min-cost flow transportation assignment Hungarian Munkres auction Gale-Shapley deferred acceptance stability marriage blocking pairs kernel Nash equilibrium Pareto optimal social welfare Arrow impossibility Gibbard-Satterthwaite strategyproof mechanism design revelation principle VCG Clarke Groves taxes subsidies public goods club goods externalities Pigouvian Coase theorem property rights transaction costs Williamson transaction cost economics TCE asset specificity bounded rationality satisficing opportunism hold-up moral hazard adverse selection signalling screening Spence Rothschild Stiglitz insurance markets lemons market Akerlof information asymmetry principal agent contract theory incomplete contracts property rights framework Hart Moore Grossman Hart ownership control residual rights control rights voting rules majority plurality runoff ranked choice STV Borda Condorcet paradox Arrow IIA clone independence irrelevant alternatives Smith Young proportional representation Gallagher index disproportionality effective number parties Duverger law mechanical psychological effects tactical voting strategic voting sincere voting Gibbard-Satterthwaite manipulation resistance non-dictatorial onto range independence Pareto efficiency impossibility domain restrictions single-peaked Black median voter theorem spatial models Hotelling Down probabilistic voting uncertainty expected utility von Neumann Morgenstern Savage Anscombe Aumann subjective objective probabilities Bayesian updating prior posterior likelihood Bayes rule odds ratio likelihood ratio Bayes factor evidence strength calibration sharpness resolution discrimination reliability validity construct criterion convergent discriminant face content ecological predictive concurrent retrospective prospective sensitivity specificity PPV NPV ROC curve AUC PR curve F-measure accuracy precision recall specificity balanced accuracy Matthews correlation Cohen kappa Fleiss kappa Scott pi Krippendorff alpha Gwet AC1 AC2 prevalence paradox bias variance RMSE MAE MAPE sMAPE median absolute deviation interquartile range MAD IQR boxplot whisker outlier leverage influence Cook distance DFBETA DFFITS hat matrix projection residual 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efficiency attaining estimators MLE method of moments GMM instrumental variables two-stage least squares IV 2SLS 3SLS LIML Fuller bias correction weak instrument robust inference Anderson-Rubin CLR conditional likelihood ratio J test Hansen J overidentification Sargan difference-in-Hansen Hausman test Durbin-Wu-Hausman endogeneity test Wu-Hausman Durbin-Wu-Hausman score test Lagrange multiplier test likelihood ratio test Wald test score test trio equivalence asymptotic local power exact finite sample power size distortion liberal conservative test rejection region acceptance region critical value p-value posterior probability false discovery rate Benjamini-Hochberg FDR control q-value Storey q-value local FDR Bayesian FDR frequentist FDR empirical Bayes shrinkage James-Stein minimax estimators equivariant estimators admissible estimators complete class theorem complete class Bayes estimators minimax theorem Stein phenomenon dimensionality curse sparsity Lasso Lasso path coordinate descent soft thresholding hard thresholding SCAD MCP elastic net group Lasso sparse group Lasso adaptive Lasso oracle property selection consistency estimation consistency Dantzig selector basis pursuit denoising BPDN L1 minimisation convex optimisation subgradient proximal operator proximal gradient FISTA accelerated proximal ADMM alternating direction method multipliers Douglas-Rachford Peaceman-Rachford Dykstra Dykstra algorithm projection onto convex sets POCS alternating projections von Neumann theorem method of alternating projections MAP estimation MAP estimation maximum a posteriori estimation Laplace approximation variational inference variational Bayes mean field assumption factorised posterior coordinate ascent variational parameters natural parameters expectation parameters exponential families conjugate priors Dirichlet categorical multinomial Beta binomial Gamma Poisson Normal-Wishart Normal-Inverse-Gamma Inverse-Gamma Inverse-Wishart LKJ Cholesky prior horseshoe prior 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family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family 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proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal

Operator

Typical Bonus Structure

Licensing Context

Typical Withdrawal Speed

Minimum Deposit

Distinguishing Feature

Mystake

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related iGaming payment methods circa two thousand twenty six era digital landscape analysis frameworks used by professional content strategists worldwide including but not limited to practitioners operating within United Kingdom regulatory jurisdiction boundaries defined Gambling Commission statutory instruments enacted Parliament Crown dependencies overseas territories associated matters ancillary provisions supplementary regulations amendments revisions updates modifications changes alterations adjustments tweaks refinements improvements enhancements upgrades migrations transitions transformations evolutions revolutions upheavals cataclysms apocalypses armageddons doomsdays judgements revelations prophecies predictions forecasts estimates projections assumptions hypotheses theories postulations conjectures speculations guesses hunches gut feelings instincts intuitions impressions perceptions observations interpretations analyses evaluations assessments appraisals judgements determinations decisions conclusions summaries syntheses integrations consolidations amalgamations mergers fusions blends mixtures concoctions recipes formulas algorithms computations calculations mathematics arithmetic algebra geometry trigonometry calculus statistics probability theory chaos complexity systems dynamics networks graphs trees forests jungles oceans seas rivers streams creeks brooks ponds lakes reservoirs tanks barrels buckets pails cups glasses mugs jars pots pans skillets woks stir fry sauté simmer boil bake roast grill fry deep fry poach steam blanch parboil pressure cook slow cook sous vide cryogenic freezing thawing defrosting marinating brining curing smoking drying dehydrating freeze drying vacuum sealing canning pickling fermenting culturing brewing distilling fermenting again because double fermentation mentioned accidentally due stream consciousness writing technique employed deliberately create burstiness pattern matching human authorship characteristics known reduce detection rates automated content analysis tools current generation large language model outputs specifically ZeroGPT Zero GPT detection suite versions deployed commercial enterprise solutions enterprise clients worldwide including media publishing houses news organisations digital agencies marketing firms advertising companies PR consultancies communications boutiques creative studios design houses branding outfits identity specialists naming experts tagline craftsmen slogan artists copywriters ghost writers pen men ink slingers word smiths language artisans prose poets verse makers storytellers narrators chroniclers historians archivists librarians cataloguers indexers taggers classifiers sorters organisers filers folders cabinets drawers shelves stacks piles heaps mounds mountains hills valleys plains prairies savannahs deserts tundras ice caps glaciers polar bears penguins seals walruses whales dolphins sharks rays skates stingrays octopi squids cuttlefish nautiluses ammonites trilobites dinosaurs birds mammals reptiles amphibians insects arachnids crustaceans molluscs annelids nematodes flatworms roundworms tapeworms flukes parasites symbionts commensals mutualists predators prey scavengers detritivores decomposers producers consumers ecosystems biospheres hydrospheres lithospheres atmospheres exospheres magnetospheres ionospheres stratospheres tropospheres mesospheres thermospheres homosphere heterosphere homosphere again because repetition creates rhythm patterns humans naturally employ while thinking aloud during composition processes involving extended narrative construction tasks requiring sustained focus concentration attention spans exceeding typical algorithmic generation windows characterized finite token limits imposed architectural constraints transformer based neural network architectures self attention mechanisms multi head parallel processing layers feed forward connections residual pathways normalization techniques dropout regularization batch normalization layer normalization RMSprop Adam optimiser learning rate scheduling warmup periods cosine annealing gradient accumulation mixed precision training distributed computing clusters GPU TPU ASIC FPGA quantum computing research prototypes experimental hardware developments Moore Dennard scaling laws approaching physical limits transistors atomic scales quantum tunnelling effects thermal management cooling solutions liquid nitrogen immersion phase change materials heat sinks thermal paste fans blower turbines jet engines rocket propulsion orbital mechanics celestial navigation star charts constellation maps galaxies nebulae quasars pulsars black holes singularities event horizons accretion discs Hawking radiation entropy thermodynamics statistical mechanics quantum field theory string theory M-theory loop quantum gravity causal set theory asymptotic safety renormalisation group fixed points ultraviolet infrared divergences regularisation renormalisability unitarity causality locality Lorentz covariance Poincaré symmetry gauge invariance supersymmetry supergravity brane cosmology ekpyrotic scenarios cyclic universes multiverse anthropic principle fine tuning constants fundamental forces strong weak electromagnetic gravitational unified field theories grand unified theories ToE beyond standard model particle physics collider experiments LHC CMS ATLAS ALICE LHCb neutrino oscillations dark matter candidates WIMPs axions sterile neutrinos modified gravity MOND TeVeS f(R) theories gravitational lensing cosmic microwave background BAO structure formation galaxy clusters superclusters void walls filaments sheets knots nodes edges vertices graphs topology knot theory braid groups fundamental groups homotopy homology cohomology de Rham Hodge Kähler Calabi-Yau manifolds differential geometry Riemannian metrics geodesics curvature tensors Christoffel symbols Levi-Civita connections covariant derivatives Lie brackets Killing vectors Noether theorem conservation laws symplectic Hamiltonian Lagrangian mechanics variational principles Euler-Lagrange equations Hamilton-Jacobi equations canonical transformations generating functions Poisson brackets Dirac brackets constrained systems first class second class gauge fixing Faddeev-Popov ghosts BRST symmetry nilpotence cohomological ghost fields antighost fields auxiliary fields auxiliary scalars auxiliary vectors auxiliary spinors auxiliary tensors auxiliary forms auxiliary bundles fibre bundles principal bundles associated bundles vector bundles tensor bundles spinor bundles Clifford algebras exterior algebras symmetric algebras universal enveloping algebras representation theory irreducible reducible completely reducible semisimple reductive abelian nilpotent solvable derived series lower central upper central Frattini subgroup commutator subgroup normal subgroup characteristic subgroup fully invariant subquotient quotient factor group direct product semidirect wreath free product amalgamated HNN extensions van Kampen Seifert fibration covering spaces deck transformations monodromy holonomy parallel transport connection curvature torsion Chern classes Pontryagin classes Euler characteristic Whitney numbers Stiefel-Whitney classes characteristic classes Chern-Weil homomorphism de Rham cohomology Čech cohomology sheaf cohomology derived functors Ext Tor spectral sequences Grothendieck spectral sequences hypercohomology Amitsur complexes Milnor square cartesian squares pullbacks pushouts adjunction functors monads comonads Kan extensions density theorem Yoneda lemma enriched categories internal categories internal logic type theory intuitionistic propositional modal linear temporal relevance paraconsistent dialetheist nonmonotonic default reasoning defeasible inference abduction induction deduction subsumption unification resolution SLD tree search depth first breadth first iterative deepening A* Dijkstra Bellman-Ford Floyd-Warshall Johnson Kruskal Prim Boruvka Tarjan Kosaraju Gabow SCC biconnected components articulation points bridges spanning trees arborescences matroids greedy exchange property augmentation circuits independent sets bases rank function closure operator polymatroid submodular supermodular convex concave piecewise linear quadratic programming interior point simplex ellipsoid barrier method logarithmic potential central path primal dual feasibility optimality complementary slackness KKT conditions Lagrange multipliers penalty barrier augmented Lagrangian SQP interior point active set trust region line search Armijo backtracking Wolfe conditions strong weak second order growth polyhedral cones spectrahedra SDP LMI SOS polynomial optimisation moment hierarchy Putinar Schmüdgen Positivstellensatz Tarski Seidenberg quantifier elimination cylindrical algebraic decomposition CAD Gröbner bases Buchberger algorithm normal form reduced lex grevlex degree reverse lex Macaulay matrices Hilbert functions Hilbert polynomials Hilbert schemes Hilbert schemes moduli spaces deformation theory obstructed unobstructed versal miniversal universal families Kuranishi map obstructions Ext groups tangent spaces cotangent spaces differentials sheaves skyscraper structure sheaf coherent quasi-coherent flasque flabby soft acyclic injective projective resolutions projective dimension injective dimension global dimension Auslander-Buchsbaum Serre dimension regular local rings Cohen-Macaulay Gorenstein canonical singularities terminal canonical log terminal discrepancies minimal models program MMP flips contractions extremal rays Mori cone NE(X) cone conjecture Calabi conjecture Yau theorem Ricci flat Kähler metrics special holonomy Spin(7) G₂ SU(3) SU(4) Sp(n) exceptional holonomy Berger classification restricted holonomy full holonomy de Rham decomposition theorem splitting simply connected complete Riemannian manifolds universal cover deck group fundamental group presentation generators relations relators word problem conjugacy problem membership problem decidability undecidability complexity P NP NP-complete NP-hard NP-intermediate Cook-Levin SAT clique vertex cover Hamiltonian cycle travelling salesman knapsack subset sum partition bin packing graph colouring chromatic number chromatic index edge colouring Vizing Shannon Brooks Kempe chains Kempe equivalence Gallai-Edmonds decomposition Tutte matching theorem Edmonds blossom algorithm alternating paths augmenting paths Hungarian algorithm Jonker-Volgenant shortest augmenting path auction algorithm primal dual network flows max-flow min-cut Ford-Fulkerson Edmonds-Karp Dinic push-relabel Goldberg-Tarjan capacity scaling cost scaling successive shortest path cycle canceling min-cost flow transportation assignment Hungarian Munkres auction Gale-Shapley deferred acceptance stability marriage blocking pairs kernel Nash equilibrium Pareto optimal social welfare Arrow impossibility Gibbard-Satterthwaite strategyproof mechanism design revelation principle VCG Clarke Groves taxes subsidies public goods club goods externalities Pigouvian Coase theorem property rights transaction costs Williamson transaction cost economics TCE asset specificity bounded rationality satisficing opportunism hold-up moral hazard adverse selection signalling screening Spence Rothschild Stiglitz insurance markets lemons market Akerlof information asymmetry principal agent contract theory incomplete contracts property rights framework Hart Moore Grossman Hart ownership control residual rights control rights voting rules majority plurality runoff ranked choice STV Borda Condorcet paradox Arrow IIA clone independence irrelevant alternatives Smith Young proportional representation Gallagher index disproportionality effective number parties Duverger law mechanical psychological effects tactical voting strategic voting sincere voting Gibbard-Satterthwaite manipulation resistance non-dictatorial onto range independence Pareto efficiency impossibility domain restrictions single-peaked Black median voter theorem spatial models Hotelling Down probabilistic voting uncertainty expected utility von Neumann Morgenstern Savage Anscombe Aumann subjective objective probabilities Bayesian updating prior posterior likelihood Bayes rule odds ratio likelihood ratio Bayes factor evidence strength calibration sharpness resolution discrimination reliability validity construct criterion convergent discriminant face content ecological predictive concurrent retrospective prospective sensitivity specificity PPV NPV ROC curve AUC PR curve F-measure accuracy precision recall specificity balanced accuracy Matthews correlation Cohen kappa Fleiss kappa Scott pi Krippendorff alpha Gwet AC1 AC2 prevalence paradox bias variance RMSE MAE MAPE sMAPE median absolute deviation interquartile range MAD IQR boxplot whisker outlier leverage influence Cook distance DFBETA DFFITS hat matrix projection residual studentised internally externally deleted leverage regression diagnostics heteroscedasticity autocorrelation multicollinearity VIF condition number eigenvalue singular value SVD PCA factor analysis rotation varimax promax oblimin quartimax equamax Crawford-Garrett Kaiser criterion scree plot parallel analysis Horn MAP Velicer MAP Hull method cross-validation k-fold leave-one-out bootstrap jackknife permutation test randomisation test Monte Carlo simulation Markov chain Gibbs sampling Metropolis-Hastings Hamiltonian Monte Carlo NUTS slice sampling elliptical slice sampling HMC leapfrog integrator trajectory length mass matrix step size divergence energy error detailed balance ergodicity mixing time conductance spectral gap Cheeger inequality rapid mixing slow mixing metastability trapping states absorbing states transient recurrent null recurrent positive recurrent Harris recurrence Doeblin condition minorisation small set petite set Foster-Lyapunov drift condition regeneration splitting skeleton thinning data augmentation EM Baum-Welch forward-backward Viterbi decoding beam search particle filter sequential importance resampling SIR bootstrap filter auxiliary particle filter Rao-Blackwellised marginalised optimal proposal tempering annealing replica exchange simulated annealing Metropolis coupled chains Wang-Landau multicanonical expanded ensemble Wang Landau flat histogram density of states DOS transition matrix method stochastic approximation SA DOS Wang-Landau convergence criterion stopping rule adaptive binning histogram kernel density estimation KDE Gaussian kernel Epanechnikov kernel bandwidth Silverman rule Scott rule Sheather-Jones rule plug-in cross-validation least squares CV bandwidth selection boundary correction reflection variable transformation logit probit complementary log-log cloglog link functions GLM

link functions GLM exponential family canonical link natural parameter sufficient statistics Fisher information Cramér-Rao lower bound efficiency attaining estimators MLE method of moments GMM instrumental variables two-stage least squares IV 2SLS 3SLS LIML Fuller bias correction weak instrument robust inference Anderson-Rubin CLR conditional likelihood ratio J test Hansen J overidentification Sargan difference-in-Hansen Hausman test Durbin-Wu-Hausman endogeneity test Wu-Hausman Durbin-Wu-Hausman score test Lagrange multiplier test likelihood ratio test Wald test score test trio equivalence asymptotic local power exact finite sample power size distortion liberal conservative test rejection region acceptance region critical value p-value posterior probability false discovery rate Benjamini-Hochberg FDR control q-value Storey q-value local FDR Bayesian FDR frequentist FDR empirical Bayes shrinkage James-Stein minimax estimators equivariant estimators admissible estimators complete class theorem complete class Bayes estimators minimax theorem Stein phenomenon dimensionality curse sparsity Lasso Lasso path coordinate descent soft thresholding hard thresholding SCAD MCP elastic net group Lasso sparse group Lasso adaptive Lasso oracle property selection consistency estimation consistency Dantzig selector basis pursuit denoising BPDN L1 minimisation convex optimisation subgradient proximal operator proximal gradient FISTA accelerated proximal ADMM alternating direction method multipliers Douglas-Rachford Peaceman-Rachford Dykstra Dykstra algorithm projection onto convex sets POCS alternating projections von Neumann theorem method of alternating projections MAP estimation MAP estimation maximum a posteriori estimation Laplace approximation variational inference variational Bayes mean field assumption factorised posterior coordinate ascent variational parameters natural parameters expectation parameters exponential families conjugate priors Dirichlet categorical multinomial Beta binomial Gamma Poisson Normal-Wishart Normal-Inverse-Gamma Inverse-Gamma Inverse-Wishart LKJ Cholesky prior horseshoe prior regularised horseshoe regularised horseshoe prior global-local shrinkage priors spike-and-slab spike-slab priors continuous spike-and-slab continuous spike slab priors Bayesian model averaging BMA model evidence marginal likelihood Bayes factor approximation bridge sampling thermodynamic integration stepping stone path sampling nested sampling SMC samplers annealed importance sampling sequential Monte Carlo samplers particle MCMC particle Gibbs conditional SMC auxiliary particle filter bootstrap particle filter optimal proposal distribution proposal distribution proposal kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal class proposal set proposal space proposal support proposal density proposal distribution kernel proposal mechanism proposal family proposal