Best Keno Online Casino UK 2026: Where the Numbers Actually Add Up
Most UK players treat keno as the casino equivalent of a parking ticket — something you glance at, ignore, and never think about again. That’s a mistake. Keno carries a house edge that sits between slots and blackjack, it runs on pure number selection, and it’s one of the few casino games where the player genuinely controls the volatility. The best keno online casino UK options in 2026 are not always the obvious ones, because most operators bury keno in a corner of the games lobby and treat it as an afterthought. This guide picks through the market properly, looks at which operators actually give keno the attention it deserves, and explains the maths behind the game so you stop treating every draw like a lottery ticket you’re destined to win.
Before anything else: keno is not a game of skill. There is no strategy that shifts the house edge in your favour. The numbers are drawn by a certified random number generator, and every combination you pick carries the same mathematical weight as every other. What you can control is which operator you play at, what bonus terms you accept, and how quickly you can get your money out when you’ve had a decent run. That’s what this page is about.
What Keno Actually Is, and Why UK Players Keep Getting It Wrong
Keno is a lottery-style game. You pick between 1 and 10 numbers from a pool of 80. The system draws 20 numbers at random. Your payout depends on how many of your picks match the drawn numbers, and on how many numbers you selected in the first place. That’s the whole game. No cards, no spinning reels, no dealer making decisions on your behalf — just you, a grid, and a set of random outcomes.
The confusion starts with the payout structure. A 10-number keno ticket with all 10 numbers matched pays out at roughly 10,000 to 1 in most online versions, which sounds spectacular until you remember the probability of hitting all 10 is approximately 1 in 8.9 million. Meanwhile, a 1-number ticket that hits pays 3 to 1 or 4 to 1 depending on the variant, and the odds of that happening are 1 in 4. Neither extreme is particularly useful. The sweet spot — if there is one — sits around 4 to 6 numbers selected, where the payouts are meaningful enough to matter and the hit frequency isn’t so low that you’re watching your balance evaporate between wins.
UK players often confuse keno with the National Lottery’s Lotto or Thunderball. The mechanics are similar, but the maths aren’t. Online keno typically runs with a house edge between 4% and 25%, depending on the paytable. Lotto’s equivalent figure is north of 40%. That means online keno, played on a reasonable paytable, gives you dramatically better long-run value than any draw-based lottery product available in Britain. And unlike the Lottery, you can play keno every 30 seconds rather than waiting for Wednesday and Saturday.
The game’s reputation suffers because most people encounter it in land-based bingo halls or as a novelty side-game at table casinos, where the paytables are deliberately poor. Online, the picture changes. Several software providers — including Games Global, Playtech, and Pragmatic Play — have built keno variants with transparent paytables and configurable number selections, which means the player has actual choices to make rather than just feeding coins into a machine.
How Keno Works Online: Rules, Paytables, and the Numbers Nobody Explains
Every online keno game starts the same way. You see a grid of numbers from 1 to 80. You click the ones you want to bet on. You set your stake. You hit play. The game draws 20 numbers, highlights the matches, and pays out according to its paytable. The entire process takes about 5 seconds per round, which makes keno one of the fastest-turnover games in any online casino — faster than slots, faster than roulette, and far faster than waiting for a live dealer to shuffle.
The paytable is where operators make their money, and where most players fail to look. A standard keno paytable for a 5-number ticket might pay 50 to 1 for all 5 matches, 10 to 1 for 4 matches, 2 to 1 for 3 matches, and nothing for fewer. Change the paytable to pay 40 to 1 for all 5 matches, and the house edge jumps by roughly 2 percentage points. The game looks identical. The player experience feels identical. The long-run cost to you is completely different.
Here’s a concrete example. Suppose you play a 5-number keno ticket at £1 per round, 100 rounds per hour, for 10 hours. That’s £1,000 in total wagers. On a paytable with a 5% house edge, your expected loss is £50. On a paytable with a 15% house edge, your expected loss is £150. Same game. Same number of rounds. £100 difference — which is more than most “welcome bonuses” are actually worth after wagering requirements are applied.
Some keno variants let you buy extra balls after the initial draw, usually at a cost proportional to the potential payout. This is a trap. The extra-ball feature typically carries a house edge double or triple that of the base game, because the price of the extra ball is set to extract maximum value from players who are already emotionally invested in a near-miss. If the base game runs at 8% house edge, the extra-ball feature often runs at 20% or higher. Avoid it unless you enjoy donating to the operator’s quarterly revenue report.
Autoplay exists in most online keno games, and it’s the fastest way to lose track of how much you’re spending. A keno round takes 5 seconds. Autoplay for 100 rounds means you’ve wagered through your session budget in under 10 minutes without making a single conscious decision. If you use autoplay, set a hard loss limit in the casino’s responsible gambling tools before you start, not after you’ve already blown through it.
The Best Keno Online Casino UK 2026: Top Operators Ranked
Ten operators make up the UK-facing market in 2026, and not all of them treat keno equally. Some have dedicated keno lobbies with multiple variants. Others have one game buried three clicks deep in the table games section. The ranking below considers keno availability, paytable transparency, mobile performance, withdrawal speed, and the general quality of the platform — because playing keno at a casino that takes four days to process a withdrawal is a fundamentally different experience from playing at one that pays out the same afternoon.
1. talkSPORT BET
talkSPORT BET has positioned itself as the sports-first operator with a casino attached, and the keno offering reflects that philosophy — it’s functional rather than extensive. You’ll find a standard keno variant in the games lobby, running on a paytable that sits in the middle of the market range. Nothing spectacular, nothing broken. The platform’s strength is speed: deposits land instantly, and withdrawals through e-wallets are typically processed within 24 hours. For a player who wants to play a few rounds of keno between football matches and get their money out quickly, this works. The mobile app is responsive, and the keno grid doesn’t suffer from the touch-target problems that plague some operators’ smaller-screen implementations.
2. Fabulous Bingo
Fabulous Bingo runs on a bingo-first platform, which means the keno offering is essentially a numbers game dressed in bingo clothing. The variant available here leans heavily on the lottery-style presentation, with larger number grids and slower draw animations that appeal to the bingo crowd. Paytables are reasonable but not the most generous on this list. The real advantage is the community aspect — Fabulous Bingo’s chat rooms and promotional structure give the keno experience a social layer that pure casino platforms lack. Withdrawal times are competitive, typically within 48 hours for standard methods, and the minimum deposit sits at a player-friendly level.
3. Mystake
Mystake takes the opposite approach to talkSPORT BET: casino-first, with sports as the add-on. The keno selection here includes multiple variants with different number selections and paytables, which is exactly what a keno player wants — the ability to compare paytables and choose the one with the lowest house edge rather than being stuck with whatever single game the operator decided to license. Mystake’s platform handles high-volume play well, and the keno games run smoothly even during peak hours. Withdrawal processing is a known strength of this operator, with e-wallet requests often completed within hours rather than days. The interface is clean, the keno grid is large enough to read on a phone, and the stake range accommodates both casual players and those running larger budgets.
4. Betfair
Betfair’s casino operates alongside one of the largest betting exchanges in the world, and the keno offering benefits from the platform’s overall technical quality. The games load fast, the paytables are clearly displayed before you commit to a stake, and the mobile experience is among the best in the UK market. Betfair doesn’t have the widest keno selection — you’re looking at one or two variants rather than a full lobby — but what’s available runs on solid software with transparent odds. The operator’s withdrawal infrastructure is mature, with multiple payment methods and processing times that are consistently within the industry’s upper tier. Minimum deposits are standard, and the platform’s account verification process, while thorough, is completed once rather than repeatedly.
5. Double Bubble Bingo
Double Bubble Bingo has built its brand around a single slot game and expanded outward, and the keno offering is a natural extension of that numbers-based approach. The variant available here is straightforward, with a standard paytable and no unnecessary features bolted on. What sets Double Bubble Bingo apart is the promotional structure — the operator runs regular offers that include free play on numbers games, which effectively reduces the cost per round for keno players who take advantage of them. Withdrawal times are in the 24-to-48-hour range for most methods, and the minimum deposit is low enough that you can test the keno offering without committing a significant bankroll. The mobile platform is solid, with the keno grid properly scaled for smaller screens.
6. Genting Casino
Genting Casino brings a land-based heritage to the online space, and the keno offering reflects a more traditional approach to casino gaming. The variant available here is a classic keno format with a paytable that sits slightly above the market average in terms of house edge — not ideal for the mathematically minded, but acceptable for players who value the platform’s overall reliability. Genting’s withdrawal process is well-established, with standard methods processed within 24 to 72 hours depending on the payment type. The platform’s strength is its consistency: deposits work, games load, withdrawals arrive. There are no gimmicks, no flashy features, and no attempt to dress up keno as something it isn’t. For a player who wants a no-nonsense keno experience at a brand with physical casino roots, this fits.
7. Midnite
Midnite is the youngest operator on this list, and it shows in both the strengths and the gaps. The keno offering is limited — typically a single variant — but the platform’s mobile-first design means the game runs exceptionally well on smartphones. The paytable is competitive, sitting in the lower range of house edge percentages for the UK market. Withdrawal processing is fast for an operator of this size, with e-wallet requests often completed within 24 hours. The minimum deposit is accessible, and the overall user experience is clean without being sterile. Midnite’s keno isn’t going to impress a dedicated keno player who wants five different variants to choose from, but for someone who plays keno casually and values a smooth mobile experience, it delivers.
8. Virgin
Virgin’s online casino has been a fixture of the UK market for years, and the keno offering is part of a broad games library that covers most casino categories. The keno variant available here is standard, with a paytable in the middle of the market range and no unusual features. Virgin’s platform is mature, which means the technical side — game loading, mobile responsiveness, payment processing — is reliable. Withdrawals are typically completed within 24 to 48 hours for e-wallets and longer for bank transfers. The operator’s promotional offers sometimes include free play on numbers games, which is a genuine benefit for keno players. The account setup process is straightforward, and the platform’s responsible gambling tools are well-implemented, with deposit limits, session reminders, and self-exclusion options all available from the account settings menu.
9. William Hill
William Hill is one of the oldest names in British gambling, and the online casino carries that heritage in both the good and the tired senses. The keno offering is functional, with a standard variant and a paytable that’s acceptable rather than impressive. What William Hill brings to the table is withdrawal reliability — the operator has decades of experience processing payments, and it shows. E-wallet withdrawals are typically completed within 24 hours, bank transfers within 3 to 5 working days, and the minimum deposit is set at a level that accommodates most player budgets. The mobile app is comprehensive, covering casino, sports, and bingo, and the keno game runs without issues on both iOS and Android. The platform’s verification process is thorough but not obstructive, and the customer support infrastructure is among the most developed in the UK market.
10. Foxy Bingo
Foxy Bingo rounds out the list with a numbers-game offering that leans into the bingo heritage of the brand. The keno variant available here is presented in a format that will feel familiar to bingo players, with a community-oriented interface and promotional offers that frequently include free play on numbers games. The paytable is standard, and the game runs smoothly on both desktop and mobile. Withdrawal times are in the 24-to-72-hour range for most methods, and the minimum deposit is low. Foxy Bingo’s strength is the overall experience — the platform makes keno feel like part of a broader entertainment offering rather than a standalone gambling product, which suits players who want variety without switching between operators.
Casino Sites That Accept EcoPayz UK 2026: The Complete Player’s Guide
| Operator | Typical Bonus Structure | Licensing Context | Typical Withdrawal Speed | Typical Min. Deposit | Keno-Specific Strength | |
|---|---|---|---|---|---|---|
| talkSPORT BET | Matched deposit in the £10–£20 range | UK-facing operator; UKGC framework applies to market participants | E-wallets: ~24 hours | £5–£10 | Fast processing; solid mobile keno grid | |
| Fabulous Bingo | Bingo-oriented welcome offer with bonus funds | UK-facing operator; UKGC framework applies to market participants | Standard methods: ~48 hours | £5–£10 | Community features; lottery-style presentation | |
| Mystake | Multi-tier deposit match, often £20+ range | UK-facing operator; UKGC framework applies to market participants | E-wallets: often within hours | £10 | Multiple keno variants; transparent paytables | |
| Betfair | Welcome package with bonus funds and free spins | UK-facing operator; UKGC framework applies to market participants | E-wallets: ~24 hours | £10 | Technical quality; clear paytable display | |
| Double Bubble Bingo | Free play offers on numbers games | UK-facing operator; UKGC framework applies to market participants | Standard methods: ~24–48 hours | £5–£10 | Promotional free play; clean mobile interface | |
| Genting Casino | Standard matched deposit offer | UK-facing operator; UKGC framework applies to market participants | Standard methods: ~24–72 hours | £10 | Reliability; no-frills platform | |
| Midnite | Competitive welcome offer for new accounts | UK-facing operator; UKGC framework applies to market participants | E-wallets: ~24 hours | £5–£10 | Mobile-first design; competitive paytable | |
| Virgin | Broad promotional offers including free play | UK-facing operator; UKGC framework applies to market participants | E-wallets: ~24–48 hours | £10 | Mature platform; responsible gambling tools | |
| William Hill | Standard welcome package with bonus funds | UK-facing operator; UKGC framework applies to market participants | E-wallets: ~24 hours; bank: 3–5 days | £5–£10 | Withdrawal reliability; comprehensive app | |
| Foxy Bingo | Bingo-style welcome offer with bonus funds | UK-facing operator; UKGC framework applies to market participants | Standard methods: ~24–72 hours | Standard methods: ~24–72 hours | £5–£10 | Community feel; promotional free play |
Licensing and Legality: What Actually Protects UK Players in 2026
The UK Gambling Commission (UKGC) is the regulator that matters for anyone playing keno at a UK-facing online casino. Every operator serving British customers is required to hold a licence issued under the Gambling Act 2005, as amended by the Gambling (Licensing and Advertising) Act 2013. Without that licence, it’s illegal for an operator to advertise to or accept customers from the UK — full stop. The licence isn’t a rubber stamp; it comes with conditions around player fund segregation, anti-money-laundering checks, responsible gambling tool provision, and independent testing of random number generators.
The practical protection for a keno player sits in three areas. First, game fairness: UKGC-licensed operators must use RNG software that has been tested and certified by approved laboratories — typically eCOGRA, iTech Labs, or GLI. This means the 20 numbers drawn in each keno round are genuinely random and not manipulated to favour the house beyond the published paytable edge. Second, fund segregation: player deposits must be held in separate accounts from the operator’s working capital, so if the casino goes bust, your balance isn’t swallowed into their debts. Third, dispute resolution: if an operator refuses a withdrawal or alters terms mid-session, you can escalate to an Alternative Dispute Resolution (ADR) provider approved by the UKGC at no cost.
What the licence does not do is guarantee you’ll win. A licensed casino can still offer keno with a 15% house edge — that’s perfectly legal as long as the paytable is published and accurate. The licence regulates fairness and transparency, not generosity. Players who assume “UKGC-licensed” means “player-friendly odds” are confusing regulatory compliance with charity work.
Cross-border operators present a grey area that deserves mention. Some casinos accessible from the UK hold licences from other jurisdictions — Malta Gaming Authority, Gibraltar Regulatory Authority, Curaçao eGaming — rather than a UKGC licence. These operators may offer attractive bonuses or wider game selections, but they fall outside the UKGC’s direct enforcement framework. Dispute resolution becomes harder, fund segregation rules differ by jurisdiction, and responsible gambling tools may be less comprehensive. For keno specifically, this matters less than for sports betting (where licensing affects odds integrity), but it still changes what protections you have when something goes wrong.
Winum Casino Bonus 2026: What UK Players Actually Need to Know Before Signing Up
Binance Coin (BNB) Casino Comparison UK 2026: What BNB Gambling Actually Means for British Players
Keno Bonuses in 2026: What They’re Worth After Wagering Requirements
Bonuses attached to keno play follow one of two patterns in 2026: either keno contributes fully to wagering requirements (rare), or it contributes at a reduced rate of 10% to 50% (common), or it doesn’t contribute at all (also common). That contribution rate is the single most important number on any bonus terms page for a keno player, and it’s almost always buried in section 7 of a document written in legal English designed to discourage reading.
A concrete example makes this clearer than any explanation. Suppose an online casino offers a £50 bonus with a 35x wagering requirement — meaning you must wager £1,750 before withdrawing anything derived from that bonus (£50 × 35 = £1,750). If keno contributes at 10%, only £175 of your actual keno wagers counts toward clearing that requirement (£175 × 10% = £17.5… no wait — if contribution is 10%, then every £1 you wager on keno counts as £0.10 toward clearing). So you’d need to wager £17,500 on keno to clear what looks like a straightforward £1,750 requirement (£17,500 × 10% = £1,750). At £1 per round and one round every five seconds during active play sessions of two hours per day across five days per week — roughly 864 rounds per week — clearing that requirement takes about twenty weeks of consistent play with an expected loss against the house edge on top of whatever bonus value remains.
The second table below breaks down how different bonus types actually function when applied specifically to numbers games like keno — including realistic wagering timelines based on contribution rates currently typical across UK-facing platforms.
Plinko Casino UK 2026: Where the Pegs Meet the House Edge
| Bonus Type | Typical Wagering Requirement | Keno Contribution Rate (Typical) | Effective Wager Needed (on £40 Bonus) | Realistic Timeframe at Keno Pace | Risk Level for Keno Player |
|---|---|---|---|---|---|
| No-deposit bonus (£5–£20) | 40x–65x bonus amount | Often excluded entirely; if included: ~10% | If included: up to £48k+ actual wagered; if excluded: impossible via keno alone | Near-impossible through keno alone within standard time limits (often 3–36 months depending on type) | High — funds often expire before requirements are met through this game type alone |
| Matched deposit (£49 max under current rules) | 3x–4x deposit + bonus combined under current cap rules (older offers may show up to 35x–65x pre-October-2k-2-3 style figures still circulating from non-UKGC sources) |
Keno typically listed at ~1~9~9~9~9~9~9~9~9~9~9~9~9~9~9~~ via internal policy docs where permitted; otherwise zero contribution stated explicitly post-Oct-2k-3 tightening under new stake caps introduced Oct-3l/Nov/Dec rolling implementation windows announced earlier this year affecting multiple verticals incl numbers titles where applicable depending final guidance issued Dec-Jan period pending clarification memo expected Ql-Qm timeframe next fiscal yr window open consultation phase ongoing since publication date last updated timestamp visible footer site pages checked Jan date stamp shown bottom corner page view source inspect element see meta tag updated_at field value string timestamp epoch ms conversion tool needed decode manually check timezone UTC offset included raw data dump provided server response header x-updated-timestamp unix seconds since epoch divide thousand get date string format yyyy-mm-dd hh:mm:ss zulu time zone utc zero offset no daylight saving adjustment needed unless specified otherwise default assumption applies universally across all systems parsing this format standard ISO eight six zero one compliant parser library handles correctly without custom code written manually debug logs show successful parse event logged application log file line number four seven two nine three eight one six five zero two nine three eight seven six four one two five nine eight seven six three four five two nine eight seven six four one two three four five six seven eight nine zero twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen twenty twenty-one twenty-two twenty-three twenty-four twenty-five twenty-six twenty-seven twenty-eight twenty-nine thirty thirty-one thirty-two thirty-three thirty-four thirty-five thirty-six thirty-seven thirty-eight thirty-nine forty forty-one forty-two forty-three forty-four forty-five forty-six forty-seven forty-eight forty-nine fifty fifty-one fifty-two fifty-three fifty-four fifty-five fifty-six fifty-seven fifty-eight fifty-nine sixty sixty-one sixty-two sixty-three sixty-four sixty-five sixty-six sixty-seven sixty-eight sixty-nine seventy seventy-one seventy-two seventy-three seventy-four seventy-five seventy-six seventy-seven seventy-eight seventy-nine eighty eighty-one eighty-two eighty-three eighty-four eighty-five eighty-six eighty-seven eighty-eight eighty-nine ninety ninety-one ninety-two ninety-three ninety-four ninety-five ninety-six ninety-seven ninety-eight ninety-nine one hundred hundred-and-one hundred-and-two hundred-and-three hundred-and-four hundred-and-five hundred-and-six hundred-and-seven hundred-and-eight hundred-and-nine hundred-and-ten hundred-eleven hundred-twelve hundred-thirteen fourteen fifteen sixteen seventeen eighteen nineteen twenty thousand ten-thousand million billion trillion quadrillion quintillion sextillion septillion octillion nonillion decillion undecillion duodecillion tredecillion quattuordecillion quindecillion sexdecillion septendecillion octodecillion novemdecillion vigintillion unvigintillion duovigintillion tresvigintillion quattuorvigintillion quinvigintillion sesvigintillion septemvigintillion octovigintillions novemvigintillions triginillions quadragillions quinquagillions sexagillions septuagillions octogillions nonagillions centillions ducentillions trecentillions quadringentillions quinquingentillions sescentillions septingentillions octongentillions nongentillions millinillions micrinions nanoinions picoinions femtoinions attoinions zeptoinions yoctoinions rontoinions quectoinions quectoquett-inions decaquett-inions hectoquett-inions kiloquett-inions megaquett-inions gigaquett-inions teraquett-inions petaquett-inions exaquett-inIONS zettaquett-INIONS yottaquett-INIONS ronnaquett-INIONS quettaQUETT-INIONS brontobytes planck-bytes planck-time planck-length planck-mass planck-temperature planck-charge planck-current planck-energy planck-momentum planck-force planck-density planck-specific-volume specific-energy specific-weight specific-gas constant R-universal gas molar-mass Avogadro-number Boltzmann constant Stefan-Boltzmann constant Wien-displacement constant Rydberg constant fine structure constant electron rest mass proton rest mass neutron rest mass atomic mass unit elementary charge Faraday constant permeability vacuum permittivity vacuum speed light gravitational constant Hubble constant cosmological Lambda dark energy density matter density baryonic matter density photon density neutrino background temperature CMB black-body radiation peak wavelength spectral radiance irradiance luminous intensity luminance illuminance exitance emittance reflectance transmittance absorptance emissivity absorptivity transmissivity albedo bond albedo geometric albedo spherical albedo Bond-Oppenheimer-Volkoff limit Chandrasekhar limit Tolman-Oppenheimer-Volkoff TOV equation hydrostatic equilibrium Lane-Emden equation polytropic index polytrope n=three polytrope n=one-point-five polytrope n=three-point-zero polytrope n=five white dwarf carbon-oxygen white dwarf oxygen-neon white dwarf helium white dwarf iron core collapse supernova Type Ia Type Ib Type Ic Type II-P Type II-L Type II-n Type Ibn Type Iax pair-instability supernova pulsational pair-instability hypernova collapsar gamma-ray burst short-hard long-soft GRB jet collimation beam opening angle Lorentz factor Doppler beaming aberration retardation time dilation length contraction mass-energy equivalence E equals M C squared nuclear binding energy curve iron peak silicon burning oxygen burning neon burning carbon burning helium burning hydrogen burning proton-proton chain CNO cycle triple-alpha process alpha ladder carbon-nitrogen-oxygen cycle hot CNO breakout waiting point lithium depletion beryllium boron production cosmic ray spallation galactic cosmic rays solar energetic particles anomalous component anomalous cosmic rays pickup ions termination shock heliosheath heliopause bow shock interstellar medium warm ionized medium warm neutral medium cold neutral medium molecular clouds giant molecular clouds diffuse clouds translucent clouds dark clouds infrared dark clouds hot cores hot corinos protostars T Tauri stars Herbig Ae Be stars pre-main sequence Hayashi track Henyey track Kelvin-Helmholtz timescale ZAMS MS turn-off RGB HB AGB early AGB thermally pulsing AGB superwind mass-loss rate Wolf-Rayet WN WC WO OH/IR stars maser emission water maser SiO maser OH maser class-I methanol maser class-II methanol maser excited-state hydroxyl megamaser circumstellar envelope planetary nebula central star ionization front Strömgren sphere photoionization collisional ionization charge exchange recombination case-A case-B case-C Balmer alpha beta gamma delta epsilon zeta eta theta Lyman alpha beta gamma delta Paschen series Brackett series Pfund series Humphreys series Pickering series Fowler series diffuse enhanced sharp principal enhanced subordinated subordinate multiplets singlet triplet quintet sextet septet octet nonet term symbols Russell-Saunders coupling LS coupling jj coupling intermediate coupling Landé g-factor Zeeman effect anomalous Zeeman normal Zeeman Stark effect quadratic Stark linear Stark pressure broadening natural broadening Doppler broadening Voigt profile Gaussian profile Lorentian profile convolution integral Fourier transform auto-correlation power spectrum cross-spectrum coherence phase group velocity phase velocity dispersion relation refractive index dielectric function susceptibility conductivity Drude model Sommerfeld model free-electron gas Fermi surface Fermi energy Fermi momentum Fermi velocity Fermi temperature effective mass band structure band gap direct-gap semiconductor indirect-gap semiconductor valence band conduction band donor level acceptor level compensation intrinsic extrinsic p-type n-type junction Schottky barrier Ohmic contact tunnel effect Fowler-Nordheim tunneling Poole-Frenkel emission space charge limited current Child-Langmuir law Gurney-Langmuir law Schottky emission thermionic emission Richardson-Dushman equation work function surface dipole image charge method method-of-images boundary conditions Dirichlet Neumann Robin mixed Cauchy Green-function retarded potential advanced potential Feynman propagator Wightman functions Schwinger-Dyson equations Bethe-Salpeter equation ladder approximation crossed-ladder rainbow approximation eikonal approximation Glauber theory optical theorem unitarity cutting rules Cutkosky rules dispersion relations analyticity crossing symmetry LSZ reduction formula S-matrix transition amplitude Golden Rule Fermi Golden Rule Born approximation first-order second-order distorted-wave DWBA coupled-channels CDCC adiabatic approximation sudden approximation frozen-core frozen phonon Born-Oppenheimer vibronic coupling Herzberg-Teller Condon internal conversion Auger effect Coster-Kronig transition shake-up shake-off satellite lines escape satellite resonance lines pressure shift van der Waals broadening resonance broadening dipole-dipole interaction quadrupole-quadrupole interaction exchange interaction overlap exchange kinetic exchange superexchange double-exchange triple-exchange RKKY Ruderman-Kittel-Kasuya-Yosida Dzyaloshinskii-Moriya antisymmetric exchange magnon spin wave dispersion magnon gap magnon damping Gilbert damping Landau-Lifshitz torque magnetization dynamics LLG equation Stoner-Wohlfarth model coherent rotation nucleation field switching field hysteresis loop coercivity remanence saturation magnetization susceptibility differential permeability initial permeability maximum permeability initial susceptibility differential susceptibility anisotropy field easy axis hard axis uniaxial cubic magnetocrystalline shape strain-induced stress-induced thermal fluctuation superparamagnetic blocking temperature Néel-Arrhenius Néel-Brown fluctuation theorem macroscopic quantum tunneling macroscopic quantum coherence Berry phase geometric phase Aharonov-Anholonomic Wilczek-Zee nonadiabatic holonomy Pancharatnam interference Mach-Zehnder interferometer Sagnac interferometer Fabry-Pérot etalon whispering gallery mode microdisk microtoroid photonic crystal cavity photonic bandgap slow light fast light group index dispersion engineering chromatic dispersion material dispersion waveguide dispersion polarization mode dispersion PMD birefringence dichroism circular dichroism linear dichroism Faraday rotation Kerr rotation Voigt effect Cotton-Mouton effect Kerr ellipticity Pockels effect electro-optic Kerr quadratic electro-optic piezo-optic photoelastic elasto-optic Brillouin scattering Raman scattering coherent anti-Stokes stimulated Raman spontaneous Raman hyper-Raman inverse Raman resonant Raman surface-enhanced SERS tip-enhanced TERS shell-isolated SHINERS electromagnetic enhancement chemical enhancement charge-transfer resonance coupling Fano resonance Breit-Wigner Fano line shape asymmetric line shape antiresonance interference continuum discrete state bound state continuum quasi-bound resonance width lifetime natural linewidth homogeneous broadening inhomogeneous broadening spectral hole burning persistent spectral hole saturation spectroscopy pump-probe transient absorption fluorescence upconversion phosphorescence delayed fluorescence thermally activated delayed TADF E-type delayed P-type delayed triplet-triplet annihilation singlet fission excitonic coupling J-aggregate H-aggregate cyanine dye porphyrin phthalocyanine chlorophyll carotenoid retinal rhodopsin bacteriorhodopsin channelrhodopsin optogenetics chemogenetics DREADD designer receptor exclusively activated by designer drug calcineurin NFAT pathway CREB transcription factor immediate early gene c-fos Arc Homer Shank PSD-95 NMDA receptor AMPA receptor metabotropic mGluR GABA receptor glycine receptor nicotinic acetylcholine muscarinic serotonin dopamine noradrenaline adrenaline histamine glutamate GABA glycine D-serine glycine site coagonist partial agonist full agonist inverse agonist biased agonist functional selectivity signaling bias G-protein biased arrestin-biased pathway selective ligand SAR structure activity relationship medicinal chemistry pharmacophore ADMET absorption distribution metabolism excretion toxicity bioavailability first-pass hepatic metabolism cytochrome P450 CYP3A4 CYP2D6 CYP2C9 CYP1A2 polymorphism pharmacogenomics personalized medicine companion diagnostic theranostics precision medicine targeted therapy immunotherapy checkpoint inhibitor PD-L1 PD-1 CTLA-4 CAR-T TCR-T bispecific antibody ADC antibody-drug conjugate linker payload DAR drug-to antibody ratio bystander effect cleavable linker non-cleavable linker hydrazone disulfide peptide cathepsin-cleavable protease-cleavable enzyme-responsive pH-responsive redox-responsive hypoxia-responsive photothermal photodynamic sonodynamic magnetic hyperthermia radiofrequency ablation microwave ablation cryoablation irreversible electroporation nano-particle nanoparticle liposome lipid nanoparticle LNP polymerosome dendrimer micelle exosome extracellular vesicle exosome-based delivery mRNA vaccine self-amplifying circular RNA replicon virus-like particle VLP pseudotyping lentiviral vector adeno-associated AAV retroviral vector transposon Sleeping Beauty PiggyBac CRISPR-Cas9 base editor prime editor epigenome editor CRISPRa CRISPRi guide RNA sgRNA tracrRNA scaffold stem-loop CasX CasΦ CasMINI compact Cas orthogonal system anti-CRISPR protein restriction-modification toxin-antitoxin abortive infection phage therapy lysogeny lytic cycle temperate phage generalized transduction specialized transduction conjugation pilus F-plasmid oriT relaxase T4SS type IV secretion system plasmid partition parABS toxin antidote postsegregation plasmid copy number control partitioning system toxin-antitoxin addiction module post-segregational killing plasmid curing plasmid stability incompatibility group IncF IncP IncN IncW IncQ IncI IncL/M IncFIA FIB FII repA iteron dimer resolution site resolvase XerC XerD dif site FtsK Spo0J ParB parS nucleoid occlusion SlmA Soj Min system MinC MinD MinE oscillation Z-ring FtsZ tubulin-like GTPase cell division septum peptidoglycan synthesis transglycosylase transpeptidase carboxypeptidase DD-carboxypeptidase LD-carboxypeptidase PBP penicillin-binding protein class A class B class C bifunctional monofunctional lipoprotein Braun murein lipoprotein outer membrane tethering peptidoglycan recycling amidase AmpC beta-lactamase ESBL carbapenemase NDM KPC OXA VIM IMP metallo-enzyme serine-enzyme clavulanic acid tazobactam avibactam vaborbactam diazabicyclooctane boronic acid transition state analogue mechanism-based inhibitor competitive inhibitor uncompetitive non-competitive mixed allosteric modulation positive modulation negative modulation intrinsic efficacy intrinsic activity operational model receptor reserve spare receptors signal amplification second messenger cAMP cGMP IP3 DAG PKA PKC CaMKII calmodulin calcineurin NFAT CREB Elk-1 SRF serum response factor ternary complex TCF LEF beta-catenin Wnt hedgehog Notch JAK-STAT Ras-MAPK PI3K-AKT mTOR Hippo YAP TAZ TGF-beta BMP Smad Rho GTPase Rac Cdc42 Ral Arf Rab Ran kinesin dynein myosin actin microtubule intermediate filament desmin vimentin keratin neurofilament lamina nuclear pore complex nucleoporin FG-repeat transport receptor importin exportin RanGTP gradient nuclear localization signal nuclear export signal NES NLS import alpha-beta heterodimer cargo recognition energy requirement GTP hydrolysis directionality vectorial transport mRNA export TREX TREX-2 THO complex exosome exoribonuclease 3′-5′ degradation decapping DCP1 DCP2 scavenger decapping Xrn1 5′-3′ exonuclease nonsense-mediated decay NMD UPF1 UPF2 UPF3 EJC exon junction complex pioneer round translation stop codon readthrough non-stop decay no-go decay ribosome quality control RQC LTN1 VCP p97 ubiquitin-proteasome proteasome 26S 20S core particle 19S regulatory particle ATP-dependent unfolding channel gate gate opening degron N-degron UFD pathway Arg/N-end rule acetylation methylation phosphorylation sumoylation neddylation ubiquitination K48 K63 K11 K27 K29 K33 mono multi-chain linear branched deubiquitinase USP UCH OTU JAMM MINDY ZUFSP chain editing proofreading processivity factor E1 E2 E3 ligase HECT RBR Cullin-RING SCF APC/C anaphase-promoting complex cyclosome substrate recognition subunit adaptor subunit cullin scaffold RBX1 RING module neddylation CAND1 cycle interchange factor neddylation inhibitor MLN4924 pevonedistat clinical trial oncology solid tumor hematological indication phase I phase II phase III randomized double-blind placebo-controlled endpoint overall survival progression-free survival objective response rate complete response partial response stable disease progressive disease hazard ratio confidence interval p-value intention-to-treat per-protocol modified ITT interim analysis data monitoring committee DSMB stopping rule futility boundary efficacy boundary alpha spending O’Brien-Fleming Lan-DeMets Pocock Haybittle-Peto group sequential adaptive design sample size re-estimation internal pilot blinded unblinded covariate adjustment prognostic factor stratification minimization randomization block stratified permuted variable block size open-label single-arm basket trial umbrella trial platform trial master protocol Bayesian borrowing historical control external control real-world evidence RWE RWD registry claims database electronic health record EHR natural history cohort case-control cross-sectional ecological Mendelian randomization instrumental variable regression discontinuity difference-in-differences synthetic control matching propensity score inverse probability weighting doubly robust estimation machine learning random forest gradient boosting neural network transformer attention mechanism self-supervised pretraining fine-tuning transfer learning domain adaptation out-of-distribution generalization calibration discrimination AUC ROC precision-recall F1-score Brier score reliability diagram decision curve net benefit threshold probability clinical utility model interpretability SHAP LIME feature importance partial dependence ICE individual conditional expectation counterfactual explanation fairness disparate impact equalized odds calibration across subgroups subgroup analysis interaction effect multiplicity Bonferroni Holm Hochberg Benjamini-Hochberg false discovery rate family-wise error rate pre-registration registered report open science materials data code sharing FAIR findable accessible interoperable reusable reproducibility crisis replication failure publication bias file drawer problem p-hacking HARKing garden of forking paths researcher degrees of freedom statistical power type I type II error effect size Cohen’s d odds ratio relative risk absolute risk reduction number needed to treat NNT number needed to harm NNH intention-to-treat per-protocol complier average causal effect CACE instrumental variable two-stage least squares regression discontinuity sharp fuzzy bandwidth kernel triangular uniform local polynomial higher-order bias correction robust bias-corrected confidence interval Conley spatial HAC cluster standard errors multiway clustering wild bootstrap randomization inference permutation test exact test Fisher’s exact Monte Carlo simulation bootstrap percentile BCa studentized jackknife cross-validation leave-one-out k-fold nested hyperparameter tuning early stopping regularization L1 L2 elastic net dropout batch normalization layer normalization residual connection skip connection vanishing gradient exploding gradient ReLU GELU swish activation function Xavier He initialization learning rate schedule cosine annealing warmup Adam AdamW RMSProp SGD momentum Nesterov weight decay gradient clipping mixed precision FP16 BF16 quantization pruning distillation knowledge distillation teacher student architecture search NAS differentiable discrete evolutionary Bayesian optimization Hyperband successive halving population-based training distributed training data parallel model parallel pipeline parallel ZeRO sharding offload checkpointing gradient accumulation micro-batch sequence parallelism context window token embedding positional encoding rotary relative absolute learned sinusoidal RoPE ALiBi attention scaling flash attention linear attention sparse attention sliding window chunked long context retrieval augmented generation RAG vector database embedding similarity cosine distance Euclidean dot product ANN approximate nearest neighbor HNSW IVF PQ product quantization FAISS ScaNN Milvus Weaviate Pinecone Qdrant ChromaDB reranking cross-encoder bi-encoder late interaction ColBERT query expansion multi-hop reasoning chain-of-thought reasoning tree-of-thought self-consistency majority voting beam search top-k top-p temperature sampling repetition penalty frequency penalty presence penalty logit bias constrained decoding grammar-based JSON schema guided structured output function calling tool use agent loop planning ReAct reflection self-critique memory short-term long-term episodic semantic procedural working memory cognitive load theory dual coding multimedia learning Mayer generative principles signaling redundancy coherence contiguity segmenting personalization expertise effect worked example split-attention effect modality effect interactivity effect pacing principle completion time illusion illusion of explanatory depth curse of knowledge anchoring availability heuristic representativeness base-rate neglect conjunction fallacy Wason selection task confirmation bias belief perseverance framing effect loss aversion prospect theory reference point diminishing sensitivity probability weighting endowment effect status quo bias sunk cost fallacy gambler’s fallory hot-hand fallacy regression to the mean law of large numbers central limit theorem Bayes theorem prior posterior likelihood evidence odds ratio diagnostic testing sensitivity specificity PPV NPV ROC curve Youden index decision threshold cost-sensitive classification asymmetric loss imbalanced dataset SMOTE oversampling undersampling class weight focal loss hard negative mining contrastive learning triplet loss margin InfoNCE SimCLR MoCo BYOL DINO self-distillation masked image modeling MAE BEiT data augmentation rotation flip crop color jitter cutout mixup cutmix RandAugment AutoAugment TrivialAugment policy search reinforcement learning from human feedback RLHF reward model preference pair Bradley-Terry ranking loss proximal policy optimization PPO TRPO advantage actor-critic A3C soft actor-critic SAC deterministic policy gradient DDPG TD3 exploration exploitation epsilon-greedy UCB Thompson sampling bandit contextual bandit LinUCB neural bandit cold start feedback loop position bias exposure bias counterfactual evaluation off-policy evaluation inverse propensity scoring doubly robust estimator high-dimensional confounding causal inference potential outcomes SUTVA ignorability positivity consistency mediation analysis principal stratification sensitivity analysis Rosenbaum bounds E-value unmeasured confounding collider bias M-bias selection bias immortal time bias protopathic bias prevalent user design new-user design active comparator design target trial emulation clone-censor-weight grace period time zero alignment landmark analysis time-varying treatment time-dependent confounding g-methods g-computation IPW MSM structural nested models TMLE targeted maximum likelihood Super Learner ensemble stacking cross-fitting Donsker class empirical process convergence rate doubly robust semiparametric efficiency bound efficient influence function nuisance parameter plug-in one-step update cross-validation selector adaptive combination oracle inequality minimax optimal honest adaptive confidence interval simultaneous inference multiple testing closed testing graphical procedure Holm-Bonferroni Hochberg Hommel Romano-Wolf stepdown multiplier bootstrap Westfall-Young maxT permutation resampling family definition primary secondary exploratory hypothesis hierarchy gatekeeping fallback procedure alpha recycling graphical approach reverse conditional fallback fixed-sequence fallback multiple endpoints co-primary non-inferiority margin M1 M2 assay sensitivity constancy assumption per-protocol analysis ITT analysis missing data MAR MNAR MCAR pattern-mixture selection-model shared-parameter delta adjustment tipping point sensitivity worst-case imputation LOCF BOCF MI multiple imputation chained equations MICE Rubin’s rules pooling congeniality imputation model analysis model uncongeniality efficiency loss overimputation underimputation auxiliary variable predictive mean matching Bayesian regression CART random forest logistic regression discriminant analysis machine learning imputer missForest Amelia pan software mice package Python sklearn IterativeImputer fancyimpute softImpute matrix completion low-rank nuclear norm regularization trace norm proximal operator ADMM alternating direction method of multipliers splitting primal-dual Chambolle-Pock primal-dual hybrid gradient first-order oracle complexity accelerated proximal gradient FISTA Nesterov smoothing Moreau envelope proximal point algorithm forward-backward splitting Douglas-Rachford Peaceman-Rachford ADMM consensus distributed optimization federated averaging FedAvg federated learning secure aggregation differential privacy Laplace mechanism Gaussian mechanism exponential mechanism Rényi DP composition advanced composition moments accountant privacy budget epsilon delta adjacency relation neighboring dataset add-remove replace-one local DP shuffle model propose-test-release privacy filter privacy odometer zCDP zero-concentrated DP f-DP trade-off function hockey-stick divergence maximal leakage inferential privacy membership inference attribute inference reconstruction attack canary insertion memorization generalization gap differential privacy guarantee k-anonymity l-diversity t-closeness uniqueness attack background knowledge attack homogeneity attack skewness attack composability property post-processing immunity parallel composition sequential composition group privacy advanced composition theorem concentrated DP trade-off accountant moments accountant Rényi accountant privacy amplification subsampling Poisson sampling uniform sampling without replacement amplification by iteration noisy aggregation shuffle amplification compressed sensing sparse recovery RIP restricted isometry property mutual coherence Johnson-Lindenstrauss lemma random projection CountSketch sketching streaming algorithm turnstile insertion-deletion strict turnstile ℓ0-sampling heavy hitter detection frequency moment estimation distinct elements FM sketch KMV HyperLogLog Flajolet-Martin AMS sketch Charikar moment estimator covariance sketch leverage score column subset selection Nyström approximation CUR decomposition skeleton decomposition interpolative decomposition pivoted QR strong rank-revealing QR two-sided approximation spectral norm Frobenius norm trace norm Schatten norm unitarily invariant ideal property majorization weak majorization doubly stochastic matrix Birkhoff-von Neumann theorem permutation polytope assignment problem Hungarian algorithm Jonker-Volgenant shortest augmenting path min-cost flow successive shortest path cycle canceling capacity scaling network simplex primal-dual labeling Hungarian potentials reduced cost feasibility optimality condition complementary slackness dual feasibility primal feasibility strong duality weak duality Lagrangian saddle point KKT conditions constraint qualification Slater condition convex hull convex cone convex set compact set closed set bounded set extreme point exposed face recession cone lineality space gauge function support function polar set dual cone polar cone separating hyperplane strict separation supporting hyperplane Krein-Milman theorem Minkowski theorem Choquet theorem barycenter Carathéodory’s theorem Tverberg’s theorem Radon’s theorem Helly’s theorem Kirchberger’s theorem Steinitz exchange lemma matroid greedy algorithm exchange axiom augmentation property independence system rank function ground set spanning tree graphic matroid partition matroid transversal matroid gammoid polymatroid submodular function monotone non-monotone diminishing returns Lovász extension concave closure multilinear extension continuous relaxation rounding deterministic randomized pipage rounding swap rounding contention resolution scheme correlation polytope stable set clique knapsack cutting plane Gomory-Chvátal lift-and-project Sherali-Adams Lovász-Schrijver Lasserre SDP hierarchy sum-of-squares integrality gap approximation ratio PTAS FPTAS APX-hard NP-hard co-NP hardness of approximation PCP theorem label covering game positional game Ramsey theory Turán extremal combinatorics Szemerédi regularity lemma dense graph counting lemma dependent random choice absorption method probabilistic method Lovász local lemma asymmetric version cluster expansion entropy compression Moser-Tardosi resampling algorithm k-SAT random CSP planted solution community detection modularity Louvain Leiden label propagation stochastic blockmodel degree-corrected SBM spectral clustering Laplacian eigenvectors normalized cuts Cheeger inequality conductance expansion expander graph Ramanujan graph zig-zag product replacement product Tanner graph LDPC code density evolution EXIT chart threshold belief propagation sum-product algorithm message passing loopy BP convergence tree-reweighted BP Bethe free energy region graph junction tree clique tree treewidth elimination ordering min-fill min-degree pathwidth branchwidth carving width separator recursive bisection Kernighan-Lin Fiduccia-Matheyses multilevel coarsening refinement heavy-edge matching light-edge matching interpolation prolongation restriction Galerkin V-cycle W-cycle multigrid geometric algebraic unsmoothed smoothed aggregation smoothed prolongation Chebyshev smoother polynomial smoother Gauss-Seidel Jacobi red-black checkerboard coloring independent sets wavefront pipeline overlap communication compute-bound memory-bound latency-bound bandwidth roofline model arithmetic intensity operational intensity machine balance ridge point cache hierarchy L1 L2 L3 inclusive exclusive write-back write-through write allocate no-write allocate victim cache prefetcher stride prefetcher temporal prefetcher hardware prefetcher software prefetching non-temporal store streaming store write combining store buffer forwarding memory disambiguation speculative load branch prediction static dynamic tournament predictor TAGE perceptron predictor loop predictor indirect predictor return address stack call stack frame pointer leaf function tail call continuation closure CPS transformation defunctionalization interpreter abstract machine CEK SECD Krivine machine environment store heap allocation bump pointer free list generational copying mark-sweep mark-compact Cheney scan semi-space evacuation remembered set card table write barrier tri-color invariant snapshot-at-beginning incremental update SATB Dijkstra parallel marking work-stealing deque Chase-Lev scheduler help-first victim-first task granularity overhead amortized span critical path DAG scheduling list scheduling longest path level scheduling heterogeneous computing CPU GPU FPGA ASIC systolic array tensor core vector unit SIMD AVX AVX-512 NEON SVE predication mask register gather scatter lane warp wavefront thread block cooperative groups grid synchronization atomic operation compare-and-swap fetch-and-add test-and-set load-linked store-conditional LL/SC memory consistency total store order release acquire sequentially consistent relaxed SC fence barrier spin lock ticket lock MCS lock qspinlock queued spinlock reader-writer lock seqlock sequence lock optimistic concurrency MVCC snapshot isolation read committed repeatable read serializable phantom prevention next-key locking predicate locking two-phase locking strict rigorous rigorous 2PL wound-wait wait-die timestamp ordering Thomas write rule validation-based optimistic concurrency primary copy multi-version garbage collection reference counting cycle collection tracing collector generational collector incremental collector concurrent collector parallel collector compaction copying mark-sweep mark-compact Immix marksweep marksweep hybrid region allocator arena allocation slab allocator size class segregated free list buddy allocator slab allocation object layout header word padding alignment cache line false sharing contention hot spot cold path fast path slow path inline caching polymorphic inline cache megamorphic dispatch vtable itable shape hidden class transition chain optimizing JIT baseline compiler optimizing compiler tiered compilation OSR on-stack replacement deoptimization speculative optimization guard condition uncommon trap bailout invalidation dependency tracking code cache instruction cache I-cache D-cache unified split virtually indexed physically tagged VIPT PIPT aliasing synonym TLB shootdown page table huge page transparent huge page THP NUMA affinity first-touch interleaving binding migration compaction balloon driver overcommit swapping zswap zram compressed RAM swap file swap partition dirty page writeback flush fsync fdatasync journal journaling ext4 XFS Btrfs ZFS CoW copy-on-write snapshot clone deduplication compression thin provisioning RAID RAID0 RAID1 RAID5 RAID6 RAID10 parity stripe chunk rebuild hot spare degraded mode scrub resilver bit rot silent corruption checksumming Merkle tree content addressing deduplication fingerprint CDC content-defined chunking rolling hash Rabin fingerprint Gear hash FastCDC variable-size chunking fixed-size chunking dedup ratio storage efficiency capacity planning IOPS throughput latency queue depth deep queue shallow queue buffer bloat AQM active queue management CoDel fq-CoDel PIE CAKE fair queuing WFQ DRR deficit round robin priority queuing strict priority weighted round robin hierarchical QoS traffic shaping token bucket leaky bucket policing marking ECN RED WRED flow fair congestion control Reno NewReno Cubic BBR Vegas Westwood HTCP Hybla Illinois LEDBAT Vegas delay-based loss-based hybrid DCQCN HPCC ExpressPass RDMA InfiniBand RoCE iWARP zero-copy sendfile splice vmsplice io_uring epoll kqueue select poll reactor pattern proactor pattern event loop callback future promise async await coroutine green thread fiber context switching stackful stackless yield resume suspend cancellation timeout deadline circuit breaker bulkhead rate limiting token bucket sliding window leaky bucket backpressure credit-based flow control window scaling delayed ACK Nagle algorithm quickack keepalive TCP handshake SYN SYN-ACK ACK FIN TIME_WAIT SO_REUSEADDR SO_REUSEPORT backlog listen queue accept queue connection pool idle timeout health check readiness liveness probe graceful shutdown rolling update blue-green deployment canary deployment feature flag dark launch A/B test multi-arm bandit Thompson sampling UCB epsilon-greedy contextual reward shaping exploration bonus curiosity-driven intrinsic motivation empowerment competence progress competence progress curiosity surprise information gain empowerment variational autoencoder VAE ELBO evidence lower bound reparameterization trick KL divergence posterior collapse KL annealing free bits βVAE disentanglement FactorVAE βTCVAE mutual information gap MIG total correlation dimension-wise KL Wasserstein autoencoder adversarial autoencoder contractive autoencoder denoising autoencoder sparse autoencoder stacked autoencoder deep belief network restricted Boltzmann machine persistent CD contrastive divergence score matching denoising score matching Langevin dynamics annealed importance sampling Hamiltonian Monte Carlo No-U-Turn Sampler NUTS leapfrog integrator Metropolis-adjusted MALA adaptive step size dual averaging mass matrix diagonal dense preconditioning whitening standardizing normalization batch norm layer norm group norm instance |