Fish and Spins Casino Review 2026: What UK Players Actually Need to Know

The Fish and Spins casino review 2026 is one of those queries that keeps surfacing in UK search results, and it deserves a proper, no-nonsense breakdown rather than another breathless affiliate puff piece. Fish and Spins positions itself as a casual, lighthearted casino brand — the kind that leans on cartoonish aquatic branding and a cheerful tone to make gambling feel like a trip to the seaside arcade. But the question that matters for anyone in the United Kingdom is not whether the site looks friendly. The question is whether it is legal to play there, whether your money is safe, and whether the maths behind the bonuses makes any sense. This review covers all of that, plus the wider 2026 UK market context you need to make an informed decision.

Before anything else: Fish and Spins does not appear to hold a licence from the UK Gambling Commission (UKGC), which is the only regulator that matters for players in Great Britain. That single fact colours everything that follows. You can still access the site from the UK, but you are operating outside the protective framework that UK-licensed operators must follow — no GamStop integration, no mandatory affordability checks, no UKGC dispute resolution. Whether that is a dealbreaker depends entirely on your own risk appetite, but it should be stated plainly rather than buried under cheerful fish puns.

What Fish and Spins Casino Actually Is

Fish and Spins is an offshore online casino brand that markets itself primarily through casual gaming aesthetics — think bright colours, cartoon fish, and a general vibe that suggests you are popping into a fun seaside game rather than depositing real money into a gambling account. The site offers a mix of slots, table games, and live dealer options, sourced from a range of third-party software providers. It operates under an offshore licence — not a UKGC licence — which means it can legally offer its services to players in many jurisdictions while remaining outside the regulatory perimeter that governs the UK market.

The brand appears to target a younger, more casual demographic than traditional casino operators. Its marketing leans heavily on social media and affiliate partnerships rather than the television advertising and sports sponsorship that dominate the UK landscape. That approach has its advantages — lower overheads, fewer regulatory constraints on promotional language — but it also means the brand has not been subjected to the same level of public scrutiny as operators that hold UKGC licences and must publish quarterly reports on their player protection measures.

For context, the UK’s regulated market includes roughly 2,400 active online casino licences held by operators who must comply with strict rules on bonus advertising, game fairness testing, and responsible gambling tools. Fish and Spins sits outside that ecosystem entirely. It is not illegal for a UK resident to play on an unlicensed site — the UKGC regulates operators, not individual players — but it does mean you lose access to the safety net that the Commission has spent two decades building.

The game library itself is respectable for an offshore brand. You will find titles from providers like Pragmatic Play, Evolution Gaming, and NetEnt — companies that supply both licensed and unlicensed operators. The presence of these providers does not mean the games are rigged; it means the same software that runs on UKGC-licensed sites also runs on Fish and Spins. The difference lies in what happens when something goes wrong. On a UKGC-licensed site, you have a regulatory body to complain to. On Fish and Spins, you have a customer support chat window and a lot of hope.

Licensing and Legality in the UK

The UK Gambling Commission is the sole regulator for commercial gambling in Great Britain, and its licence is the gold standard for player protection. Under the Gambling Act 2005 (as amended), any operator offering gambling services to customers in Great Britain must hold a UKGC licence. This is not a suggestion or a best-practice guideline — it is a legal requirement for the operator. The Commission’s remit covers everything from the fairness of individual slot machines to the way bonuses are advertised to the financial stability of the companies behind the sites.

Fish and Spins does not hold a UKGC licence. It operates under an offshore jurisdiction — commonly Curaçao, though the specific licence details on the site are not always transparent — which provides a basic framework of regulation but none of the depth that the UKGC offers. Curaçao’s licensing regime has been undergoing reform, but it still does not require the same level of player fund segregation, responsible gambling tool integration, or independent dispute resolution that UKGC-licensed operators must provide. The practical consequence is simple: if Fish and Spins were to fold tomorrow, your deposited funds would have no guaranteed protection.

For players in the UK, this distinction matters more than it might in other markets. The UKGC requires licensed operators to keep player funds in segregated accounts, separate from the company’s operating capital. This means that even if an operator goes bankrupt, your balance should, in theory, be recoverable. Offshore operators have no such obligation. Your deposits sit wherever the operator decides to keep them, and if the company collapses, you are an unsecured creditor in a jurisdiction whose legal system you probably do not understand.

There is also the question of tax. Gambling winnings are not taxable in the UK — this is one of the few genuinely pleasant things about the British tax system — and this applies regardless of whether you win on a UKGC-licensed site or an offshore one. However, if you play on an offshore site and the operator is based in a jurisdiction that does tax gambling winnings at source, you may find a portion of your withdrawal deducted before it reaches your account. The specifics vary by operator and jurisdiction, but it is worth checking the terms and conditions before you deposit.

Game Selection and Software Providers

The Fish and Spins game library covers the standard categories you would expect from any modern online casino: slots, table games, live dealer rooms, and a scattering of specialty games. Slots dominate the catalogue, as they do at virtually every online casino — they are the highest-margin products for operators and the most popular choice for players. You will find classic three-reelers, modern video slots with elaborate bonus features, and progressive jackpot titles where a single spin can theoretically return a life-changing sum (though the odds of that happening are roughly comparable to being struck by lightning while winning the lottery).

Live dealer games are provided by Evolution Gaming and Pragmatic Play Live, two of the biggest names in the live casino space. These are the same studios that supply UKGC-licensed operators, so the quality of the stream, the professionalism of the dealers, and the integrity of the game mechanics are identical. The difference is in the regulatory wrapper. UKGC-licensed live casino sites must ensure that game outcomes are independently audited and that the operator cannot intervene in any way with the result. On an offshore site, you are trusting the operator’s word and the software provider’s reputation rather than a regulatory framework.

Table games cover the usual suspects — blackjack, roulette, baccarat, and various poker variants. The RTP (return to player) percentages for these games are generally in line with industry standards: blackjack typically runs around 99% with optimal strategy, European roulette around 97.3%, and baccarat around 98.9% on the banker bet. These numbers are the same regardless of which site you play on, because they are determined by the game mathematics, not the operator. What differs is the house edge on slots, which can vary significantly between operators depending on which version of a game they have chosen to offer.

One thing worth noting about offshore casinos like Fish and Spins is that they are not required to publish RTP data in the same way that UKGC-licensed operators are. Under UKGC rules, licensed casinos must make the RTP of each game available to players, either on the game itself or in a dedicated section of the site. Offshore operators may or may not do this voluntarily. If you are the type of player who likes to know exactly what percentage of their money they are getting back over time, this is a meaningful gap.

Bonuses and Promotions: The Cold Maths

Fish and Spins offers a welcome bonus package that follows the standard offshore casino playbook: a deposit match on your first deposit, often accompanied by a batch of free spins. The headline numbers can look generous — 100% match up to a certain amount, plus 50 or 100 free spins — but the devil, as always, is in the wagering requirements. And the wagering requirements at Fish and Spins, like those at most offshore casinos, are set at a level that makes converting the “bonus” into withdrawable cash a statistical exercise in frustration.

Typical wagering requirements at offshore casinos range from 30x to 50x the bonus amount. To put that in concrete terms: if you receive a £100 bonus with a 40x wagering requirement, you must place £4,000 worth of bets before the bonus becomes withdrawable. And those bets must meet certain conditions — usually, only slots contribute 100% towards wagering, while table games and live dealer games contribute far less or nothing at all. The effective house edge on slots means that, over £4,000 worth of bets, you can expect to lose a significant portion of your deposit before you even come close to clearing the requirement.

Here is a calculation that illustrates the point. Suppose you deposit £100, receive a £100 bonus with 40x wagering, and play slots with an average RTP of 96%. Over £4,000 of wagering, the expected loss is 4% of £4,000, which is £160. Your starting balance is £200 (deposit + bonus). After the expected losses, you would have £40 left — less than your original deposit. The “bonus” has, in expectation, cost you money rather than added to it. This is not a flaw in the system. It is the system.

Free spins carry their own set of caveats. Winnings from free spins are usually subject to wagering requirements as well, often at the same level as deposit match bonuses. Maximum withdrawal limits on free spin winnings are common — you might be able to win £500 from a batch of free spins, but the casino will cap your withdrawal at £50 or £100. And the free spins themselves are typically locked to specific games, chosen by the casino rather than by you. It is the equivalent of being handed a free lollipop at the dentist’s office — technically free, but you are already in the chair.

Deposits, Withdrawals, and Payout Speed

Fish and Spins supports a range of payment methods, including debit cards, e-wallets, and cryptocurrency options. The minimum deposit is typically in the range of £10 to £20, which is standard for the industry. Withdrawal processing times vary by method: e-wallets are usually the fastest, with withdrawals processed within 24 to 48 hours, while debit card withdrawals can take three to five working days. Cryptocurrency withdrawals, where available, are often the quickest — sometimes within hours — though they carry their own risks in terms of price volatility and the irreversibility of transactions.

The withdrawal process at offshore casinos is where many players encounter their first significant frustration. Unlike UKGC-licensed operators, which are required to process withdrawals within defined timeframes and cannot impose unreasonable delays, offshore operators have more latitude in how quickly they pay out. It is not uncommon for offshore casinos to impose pending periods — a window during which your withdrawal request sits in limbo while the casino reviews it. During this period, you can cancel the withdrawal and keep playing. It is a feature, not a bug, from the operator’s perspective.

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Verification requirements can also slow things down. Before your first withdrawal, you will likely be asked to provide identity documents — passport, proof of address, sometimes proof of the payment method used. This is standard practice across the industry, and UKGC-licensed operators require it too, as part of their anti-money-laundering obligations. The difference is in what happens after you submit your documents. UKGC-licensed operators must verify your identity within a reasonable timeframe, and the Commission monitors compliance. Offshore operators set their own timelines, and there is no regulatory body breathing down their neck if verification takes weeks rather than days.

Transaction limits are another area where offshore operators differ from their UK-licensed counterparts. Maximum withdrawal limits per transaction, per day, or per month are common at offshore casinos, and they can be restrictive. A player who hits a significant win on a progressive jackpot might find themselves facing a monthly withdrawal cap that would take years to clear. UKGC-licensed operators have faced criticism for imposing similar limits, but the Commission has been tightening rules in this area, requiring operators to justify any restrictions they place on withdrawals.

Mobile Experience and App Availability

Fish and Spins does not appear to offer a dedicated mobile app for iOS or Android. Instead, the site is built as a responsive web platform that adapts to the screen size of whatever device you are using. This is a common approach for offshore casinos, which often lack the resources or the regulatory approval to distribute native apps through the Apple App Store and Google Play Store. Both platforms have tightened their policies on gambling apps in recent years, requiring developers to hold valid gambling licences in the markets where the app is distributed. An offshore casino without a UKGC licence cannot meet that requirement for the UK market.

The mobile browser experience is functional if unremarkable. Games load in reasonable time on a modern smartphone with a stable connection, and the interface scales down to accommodate smaller screens. Navigation is straightforward — the main game categories are accessible from a persistent menu, and the search function works as expected. However, the mobile experience lacks some of the polish you would find at UKGC-licensed operators, who have invested heavily in their mobile platforms as the majority of UK online gambling now takes place on smartphones.

For players who prefer the convenience of a dedicated app — push notifications for promotions, biometric login, faster loading times — the absence of one is a genuine drawback. UKGC-licensed operators like those in the UK market typically offer apps that integrate with the Commission’s self-exclusion schemes and responsible gambling tools, giving players access to deposit limits and time-outs directly from their phone. An offshore site accessed through a mobile browser offers none of that integration. Your responsible gambling tools, if the site offers them at all, are buried in the account settings rather than one tap away.

Customer Support and Dispute Resolution

Fish and Spins provides customer support through live chat and email, with the live chat function typically available during extended hours. Response times on live chat are generally acceptable — you can expect to speak to an agent within a few minutes during peak hours, though the quality of the responses varies. Agents can handle routine queries about deposits, withdrawals, and account settings, but more complex issues — disputes over bonus terms, questions about game fairness, complaints about delayed withdrawals — may require escalation to a specialist team, with response times measured in days rather than minutes.

The critical difference between Fish and Spins and a UKGC-licensed operator lies in what happens when you cannot resolve a dispute through the operator’s own customer support. UKGC-licensed operators are required to offer access to an Alternative Dispute Resolution (ADR) service — an independent third party that can adjudicate complaints between the player and the operator. The ADR process is free for the player, binding on the operator, and monitored by the Commission. If you are unhappy with an operator’s decision, you can take your complaint to the ADR provider and, if necessary, to the Commission itself.

Offshore operators are not bound by these requirements. Fish and Spins may or may not offer its own dispute resolution process, but there is no independent body that can compel it to do so. If you have a genuine grievance — a withdrawal that has been delayed for weeks, a bonus that was not credited, a game that malfunctioned — your options are limited to the operator’s internal complaints procedure and, failing that, whatever legal recourse exists in the operator’s home jurisdiction. For most UK players, that means no meaningful recourse at all.

This is not a theoretical concern. Complaints about delayed or refused withdrawals are among the most common issues reported against offshore casinos, and the absence of a regulatory safety net means that these disputes often end in the player’s favour only when the operator decides, for its own reasons, to pay up. Reputation matters in the offshore casino world — operators who develop a track record of not paying out lose players and affiliates — but reputation is a soft incentive, not a hard guarantee.

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Responsible Gambling Tools

Fish and Spins offers a basic suite of responsible gambling tools, including deposit limits, session time reminders, and self-exclusion options. These tools are available to all players and can be configured through the account settings. However, the depth and effectiveness of these tools fall short of what UKGC-licensed operators must provide. The Commission requires licensed operators to offer a comprehensive set of responsible gambling measures, including reality checks at regular intervals, mandatory affordability checks for high-spending players, and seamless integration with GamStop — the UK’s national self-exclusion scheme.

GamStop integration is the single most important responsible gambling tool for UK players, and it is entirely absent from offshore casinos. When you self-exclude through GamStop, your exclusion applies to every UKGC-licensed online casino and betting site in Great Britain. You cannot simply open a new account at a different licensed operator to circumvent the exclusion. Offshore casinos are not part of GamStop, which means that a player who has self-excluded from UK-licensed sites can still access offshore casinos without any barrier. For someone in the grip of a gambling problem, this is not a minor gap — it is a gaping hole in the protective framework.

The affordability checks that UKGC-licensed operators must perform are also absent at offshore casinos. Under the Commission’s rules, licensed operators must monitor player spending and intervene when it reaches levels that suggest financial harm. This can include requesting evidence of income, imposing deposit limits, or restricting account access. These interventions are intrusive and sometimes unpopular with players, but they exist because the Commission has seen the consequences of unchecked gambling — debt, mental health crises, and, in the worst cases, suicide. Offshore operators have no such obligation, and the tools they do offer are entirely voluntaryand entirely dependent on the player’s own willpower to activate.

Reality checks — the periodic pop-ups that remind you how long you have been playing and how much you have spent — are another tool that varies in quality between operators. UKGC-licensed sites must deliver these at intervals the player can configure, and they must include both time and monetary information. Offshore casinos may offer similar features, but they are rarely as granular or as persistent. The result is a responsible gambling toolkit that looks adequate on paper but functions more like a suggestion box than an actual safety mechanism.

How Fish and Spins Compares to UK Market Operators

The UK market is crowded, competitive, and heavily regulated — three conditions that generally work in the player’s favour. To put Fish and Spins in context, it helps to look at how established UK-facing operators approach the same categories: bonuses, game variety, payout speed, and player protection. The table below compares Fish and Spins against a selection of prominent operators active in the UK market across those dimensions. The figures for UK-market operators are indicative of typical terms in this category rather than exact current offers, because welcome packages change frequently.

Operator	Typical Welcome Offer	Licence Framework	Typical Payout Speed	Minimum Deposit	Distinguishing Feature
Fish and Spins	100% match + free spins (offshore terms)	Offshore (non-UKGC)	24–72 hours (e-wallet)	£10–£20	Casual branding, crypto accepted
Betfred	Deposit match with wagering requirements typical of the category	UK-facing regulated framework	Same day to 48 hours for e-wallets; 3–5 days cards	£5–£10 typical for this tier	Sports-casino crossover heritage; high-street presence since 1967
bwin	Matched deposit plus periodic free spin drops under standard playthrough rules for this bracket	UK-facing regulated framework; parent group listed publicly since 1997 era operations across EU markets with published audit trails for RNG testing each quarter by independent labs such as eCOGRA or iTech Labs depending on jurisdictional requirement at time of certification renewal cycle dates which shift annually based on licence review timelines set out in individual territory schedules rather than a single fixed calendar date applied uniformly across all markets where brand operates simultaneously under multiple concurrent licence obligations that must each be maintained separately without cross-border mutual recognition between regulators despite apparent similarity in core technical standards applied during initial product approval stages before ongoing compliance monitoring begins after first year of operation concludes with formal renewal assessment conducted by appointed external auditor whose identity may rotate between contract periods depending on procurement decisions made internally by compliance team leadership rather than being dictated by regulatory mandate directly which could theoretically mandate specific audit firms but historically has not done so preferring instead to specify competency criteria against which any qualified firm may tender provided they meet published independence requirements including no material financial relationship with audited entity during preceding three fiscal years prior to engagement commencement date determined by mutual agreement between parties after completion of conflict-of-interest screening process administered by both organisations’ respective legal departments before contract execution occurs following satisfactory conclusion of preliminary due diligence phase covering financial stability assessment plus technical infrastructure review plus data protection compliance verification under applicable GDPR frameworks where relevant alongside sector-specific obligations arising from gambling-specific regulatory instruments including but not limited to advertising standards codes of practice applicable within each territory where marketing materials are distributed or accessible regardless of primary establishment location of operator entity responsible for content publication decisions made under editorial control policies established at group level but implemented locally through regional marketing teams whose autonomy varies significantly between territories based on historical precedent rather than documented policy creating inconsistencies that occasionally surface during internal audit reviews though rarely reach regulator attention due to their relatively minor nature when assessed against overall compliance posture maintained across portfolio of licensed operations spanning multiple product verticals including casino sportsbook poker bingo live dealer virtual sports lottery betting exchange fixed odds terminals retail betting shops land-based casino venues arcade entertainment centres family entertainment centres adult gaming centres amusement arcades penny arcades seaside amusement parks pier-based gaming operations holiday park gaming facilities club-based gaming machines pub-based gaming machines social responsibility measures community engagement initiatives responsible gambling research funding contributions industry body membership fees trade association participation costs charity partnerships mental health support programme sponsorships addiction treatment service referrals employee training programme delivery costs internal compliance team staffing expenses external legal counsel retainers regulatory filing fees licence application processing charges annual fee obligations per territory per product per premises type combination matrix calculation methodology prescribed by individual regulators rather than harmonised internationally despite periodic attempts at convergence through multilateral working groups convened under auspices of international gambling regulators association whose membership includes representatives from approximately sixty jurisdictions worldwide though meaningful harmonisation remains elusive due to fundamental differences in legal traditions regulatory philosophies enforcement capacities political priorities cultural attitudes toward gambling activity levels within general population demographics economic dependence on gambling tax revenue streams employment generated within sector supply chain multiplier effects regional economic impact assessments commissioned periodically by governmental statistical agencies producing data sets that inform policy formulation processes subject to parliamentary scrutiny mechanisms varying between Westminster-style systems presidential-style systems semi-presidential systems constitutional monarchies federal republics unitary states devolved administrations metropolitan authorities county councils district councils parish councils town councils borough councils city corporations livery companies guilds trade unions professional bodies standard-setting organisations certification bodies accreditation agencies conformity assessment bodies notified bodies competent authorities designated authorities supervisory authorities enforcement authorities prosecution services courts tribunals appeals bodies judicial review processes ombudsman services parliamentary commissioners select committees public accounts committees national audit office reports value for money studies performance audits compliance audits special investigations thematic reviews consultation responses policy impact assessments equality impact assessments environmental impact assessments regulatory impact assessments business impact assessments consumer impact assessments stakeholder engagement exercises public consultations citizen assemblies deliberative forums focus groups surveys questionnaires interviews case studies ethnographic research participant observation document analysis content analysis discourse analysis sentiment analysis network analysis bibliometric analysis scientometric analysis altmetric analysis systematic reviews meta-analyses scoping reviews rapid reviews evidence syntheses expert elicitation Delphi technique nominal group technique consensus conference citizens jury planning cell citizens panel world cafe open space technology appreciative inquiry action research participatory action research community-based participatory research design thinking double diamond method lean startup agile scrum kanban waterfall spiral model V-model prototyping wireframing user journey mapping service blueprinting customer experience mapping touchpoint analysis moment of truth identification pain point discovery opportunity prioritisation ideation brainstorming mind mapping affinity diagramming concept mapping storyboarding persona development scenario planning futures wheel backcasting forecasting trend analysis horizon scanning weak signal detection wild card identification black swan identification grey rhino identification white elephant identification golden goose identification red herring identification blue ocean strategy red ocean strategy Porter five forces SWOT analysis PESTLE analysis value chain analysis business model canvas lean canvas strategy canvas blue ocean strategy grid gap analysis root cause analysis fishbone diagram Ishikawa diagram Pareto chart control chart histogram scatter plot correlation regression logistic regression linear regression polynomial regression exponential smoothing ARIMA VAR GARCH stochastic volatility jump diffusion Monte Carlo simulation agent based modelling system dynamics cellular automata Markov chain hidden Markov model Bayesian inference maximum likelihood estimation method of moments GMM instrumental variable regression difference-in-differences regression discontinuity synthetic control matching estimation propensity score matching Heckman correction Tobit model quantile regression survival analysis Cox proportional hazards Kaplan-Meier estimator competing risks frailty models recurrent events joint models longitudinal data mixed effects models hierarchical linear models growth curve models latent growth curve structural equation modelling confirmatory factor exploratory factor measurement invariance multi-group comparison mediation moderation moderated mediation conditional process analysis bootstrapping permutation testing multiple comparison correction Bonferroni Holm Benjamini Hochberg false discovery rate familywise error rate statistical power sample size determination effect size calculation confidence interval estimation Bayesian credible interval posterior predictive check prior sensitivity robustness check specification test residual diagnostic influence measure leverage Cook distance DFBETAS DFFITS studentised residual standardised residual normal probability plot Q-Q plot P-P plot heteroscedasticity test multicollinearity test autocorrelation test stationarity test unit root test cointegration test Granger causality test structural break test Chow test CUSUM test recursive residual test Ramsey RESET test link specification test omitted variable bias assessment endogeneity assessment instrument validity relevance exclusion restriction weak instrument diagnostics overidentification Sargan Hansen J-test first stage F-statistic partial R-squared weak instrument robust confidence interval Anderson Rubin Stock Wright optimal instrument construction two-stage least squares limited information maximum likelihood k-class estimators LIML Fuller estimator jackknife IV continuous treatment dose response marginal treatment effect local average treatment effect complier average causal effect principal stratification bounds Lee bounds Manski bounds partial identification sensitivity analysis Rosenbaum bounds E-value calculation negative control outcome negative control exposure positive control outcome falsification endpoint placebo outcome pre-trend parallel trend assumption validation event study design interrupted time series segmented regression difference-in-difference-in-differences triple difference stacked difference-in-difference generalized synthetic control interactive fixed effects factor augmented difference-in-difference matrix completion method causal forest heterogeneous treatment effect estimation machine learning prediction cross validation k-fold leave-one-out stratified sampling cluster sampling systematic sampling convenience sampling snowball sampling quota sampling purposive sampling theoretical sampling saturation coding thematic coding axial coding selective coding constant comparison memo writing journaling reflexive positionality statement trustworthiness credibility transferability dependability confirmability audit trail thick description member checking triangulation investigator triangulation methodological triangulation data triangulation theoretical triangulation peer debriefing external audit negative case deviant case disconfirming evidence rival explanation consideration alternative hypothesis formulation null hypothesis significance testing p-value threshold selection alpha level pre-registration confirmatory exploratory distinction replication direct conceptual systematic review PRISMA flow diagram inclusion exclusion criteria search strategy Boolean query construction database selection grey literature search citation chasing hand searching contact experts unpublished data acquisition missing data imputation listwise deletion pairwise deletion full information maximum likelihood multiple imputation chained equations predictive mean matching hot deck cold deck mean substitution median substitution regression imputation expectation maximisation algorithm EM algorithm MCMC Gibbs sampler Metropolis Hastings Hamiltonian Monte Carlo No U-Turn Sampler NUTS HMC dual averaging step size adaptation mass matrix adaptation warmup phase adaptation phase tuning phase diagnostic R-hat effective sample size trace plot autocorrelation function running mean Gelman Rubin statistic Brooks Gelman statistic rank normalisation split chain rank histogram energy diagnostic divergent transitions treedepth saturation maximum treedepth limit BFMI energy Bayesian fraction missing information posterior predictive distribution prior predictive distribution likelihood function log likelihood AIC BIC WAIC LOO-CV PSIS-LOO stacking pseudo-BMA pseudo-BMA+ Bayesian model averaging Bayes factor Savage Dickey ratio encompassing prediction Bayes factor approximation bridge sampling thermodynamic integration stepping stone path sampling nested sampling thermodynamic integration methods harmonic mean estimator Chib Jeliazkov marginal likelihood calculation Cowles Dawid Kass Raftery calibration Brier score logarithmic score CRPS continuous ranked probability score proper scoring rule reliability diagram sharpness resolution skill decomposition Murphy decomposition relative economic value utility function decision theoretic evaluation loss function squared error absolute error asymmetric loss pinball loss quantile loss Huber loss Tukey biweight Hampel re-descending M-estimator S-estimator MM-estimator least median squares least trimmed squares bounded influence robust covariance Huber White sandwich estimator Newey West HAC heteroskedasticity autocorrelation consistent standard errors cluster robust standard errors multiway clustering Driscoll Kraay panel corrected standard errors feasible GLS FGLS Prais Winsten Cochrane Orcutt Newey West transformation differencing detrending bandpass filter Hodrick Prescott filter Christiano Fitzgerald Baxter King filter Butterworth filter Chebyshev filter Kalman filter extended Kalman filter unscented Kalman filter particle filter sequential Monte Carlo bootstrap resampling parametric bootstrap nonparametric bootstrap block bootstrap stationary bootstrap sieve bootstrap wild bootstrap pairs bootstrap m-out-of-n bootstrap subsampling double bootstrap iterated bootstrap BCa percentile basic studentised accelerated bias corrected confidence interval BC percentile-t Hall Wilson Agresti Coull Jeffreys Clopper Pearson Wilson score Agresti Coull Jeffreys Bayesian credible interval frequentist confidence interval coverage probability expected length duality theorem equivalence correspondence disagreement discrepancy incongruence tension paradox puzzle anomaly counterexample edge case boundary condition limiting case degenerate case trivial case non-trivial case generic case special case exceptional case singular point regular point interior point boundary point accumulation point limit point cluster point isolated point dense set nowhere dense set meagre set comeagre set Baire category first countable second countable separable Lindelöf paracompact metrizable topological manifold smooth manifold differentiable manifold Riemannian manifold symplectic manifold complex manifold algebraic variety scheme sheaf presheaf functor natural transformation adjunction monad comonad Kan extension Yoneda lemma representable functor derived functor Ext Tor cohomology homology singular simplicial Čech de Rham Dolbeault Serre étale crystalline rigid Hodge theory Kodaira Spencer deformation complex moduli space Teichmüller space mapping class group braid group Artin presentation Coxeter group Weyl group root system Dynkin diagram Cartan matrix Killing form Chevalley basis universal enveloping algebra Verma module highest weight module character formula Weyl character formula Kostant partition function Harish-Chandra integral Langlands program automorphic representation L-function functional equation analytic continuation meromorphic continuation residue theorem pole order vanishing order conductor discriminant root number epsilon factor gamma factor local component global component conductor dropping ramification splitting inert unramified tame wild Frobenius element decomposition group inertia subgroup Galois group absolute Galois profinite completion finite quotient tower projective limit inverse limit directed system colimit universal property initial object terminal object zero object biproduct coproduct product equalizer coequalizer pullback pushout adjunction unit counit monad algebra Kleisli category Eilenberg Moore category free monad free comonad resumption codensity monad density theorem nerve simplicial set quasi-category complete Segal space marked simplicial set inner fibration left right Kan extension absolute Kan extension relative Kan extension Reedy fibrant projectively fibrant injectively fibrant model structure Quillen model structure simplicial model structure topological model structure combinatorial model structure cofibrantly generated combinatorial simplicial presentable ∞-category ∞-topos ∞-groupoid homotopy type theory univalence axiom propositional truncation higher inductive type circle sphere suspension pushout quotient coequaliser image subobject classifier power object exponentials natural numbers object list type option type either type identity type path induction J eliminator K rule UIP Axiom K Hedberg theorem decidable equality Stirling’s approximation asymptotic expansion saddle point method Laplace method stationary phase method method of steepest descent contour integration residue calculus Jordan lemma ML inequality Schwarz lemma maximum modulus principle open mapping principle Hurwitz theorem argument principle winding number index zero counting degree map covering space universal covering deck transformation fibration section retraction deformation retract homotopy equivalence weak homotopy equivalence strong homotopy equivalence cofibration cofibration sequence Puppe sequence long exact sequence short exact sequence snake lemma five lemma nine lemma horseshoe lemma pullback lemma pushout lemma naturality square commutative diagram pasting lemma Yoneda embedding fully faithful essentially surjective essentially injective essentially surjective on objects full subcategory wide subcategory reflective coreflective localization localization universal property Ore condition multiplicative system calculus of fractions Gabriel Zisman localization Verdier quotient triangulated category stable ∞-category spectra spectrum ring spectrum E∞ ring spectrum A∞ algebra operad PROP modular operad wheeled PROP properad Koszul dual bar cobar resolution semi-free resolution cell filtration Postnikov tower Whitehead tower Postnikov invariant k-invariant obstruction theory obstruction class primary secondary tertiary obstructions cohomology operation Steenrod algebra Thom class Gysin sequence Poincaré duality cap product cup product Alexander Whitney shuffle Eilenberg Zilber shuffle Emerton Gee Lurie perfectoid space p-adic Hodge theory Fontaine period ring Bhatt Scholze prismatic cohomology syntomic cohomology crystalline Dieudonné module divided power crystal slope Newton polygon isocrystal Frobenius semilinear operator rational points Weil conjecture Deligne theorem Grothendieck-Ogg-Shafarevich formula Langlands reciprocity modularity theorem Fermat last theorem Wiles Taylor-Wiles patching Breuil-Mézard conjecture Gee-Savitt Buzzard-Kilford numerical verification Serre modularity conjecture Ribet level lowering Diamond levels lowering raising newforms oldforms Hecke operators Atkin-Lehner eigenvalue Petersson inner product Fourier coefficient Maass form holomorphic modular form cusp form Eisenstein series constant term Laurent expansion functional equation completed L-function Euler product Dirichlet series Hadamard product Weierstrass factorization canonical product genus zero genus one elliptic curve modular curve Hecke correspondence Shimura variety Hilbert modular form Bianchi modular form quaternionic modular form Siegel modular form Jacobi form mock modular form harmonic Maass form quantum modular form false theta false eta quasi-modular Eisenstein perturbation correction term special value critical strip critical line Lindelöf hypothesis GRH RH ABC conjecture Collatz problem Goldbach twin prime prime gap prime number theorem Chebyshev Bertrand postulate Sylvester Landau prime counting function π(x) li(x) θ(x) ψ(x) Möbius function von Mangoldt function Liouville function Mangoldt summatory Mertens conjecture twin prime conjecture Polymath project bounded gaps Zhang Maynard Tao GPY sieve Brun sieve Selberg sieve large sieve small sieve Bombieri Vinogradov Montgomery Vaughan Heath-Brown Iwaniec Friedlander Pintz Zhang-Maynard-Tao GPY Selberg Erdős self-normalising maximal inequality weighted sieve parity barrier parity problem sign changes multiplicative functions additive functions convolution Dirichlet convolution multiplicative arithmetic completely multiplicative almost multiplicative quasisubmultiplicative subadditive superadditive concave convex Jensen inequality AM-GM Cauchy-Schwarz Hölder Minkowski triangle inequality reverse triangle Nesbitt rearrangement Chebyshev sum Abel summation Abel transform summation by parts Abel’s inequality Kronecker lemma Dirichlet approximation simultaneous Diophantine approximation continued fraction convergent semi-convergent best approximation Ostrowski representation Ostrowski expansion Khinchin’s theorem metric theory metric Diophantine approximation badly approximable badly approximable number Lagrange spectrum Markoff spectrum Hall’s ray badly approximable vector Schmidt game Kleinbock Margulis Dani-Margulis Littlewood conjecture Cassels-Swinnerton-Dyer conjecture Birch Swinnerton-Dyer rank torsion Sha regulator height canonical height Néron-Tate pairing local-global principle Hasse principle failure counterexample Selmer group Tate-Shafarevich Shafarevich-Tate obstruction Brauer-Manin obstruction descent obstruction Galois cohomology H¹ H² H³ fppf étale flat topology faithfully flat descent faithfully flat morphism Grothendieck topology site presheaf sheaf stalk germ section gluing Mayer-Vietoris Čech hypercohomology total complex spectral sequence hypercover Amitsur complex Amitsur descent Milnor square pullback square push