Glossary

The terms used throughout Pro Edge^™, defined plainly. For how these concepts fit together in practice, see our Methodology.

Account restrictions (limiting)

A bookmaker's risk control: cutting the maximum stake a customer may place - sometimes to a fraction of the published limit - or barring them from certain markets. The trigger is usually the quality of the betting rather than the profit itself: consistently positive closing-line value, bets that land ahead of a line movement, and concentrated activity in side markets or low-liquidity leagues are the marks of a sharp account. Restriction is indirect evidence that CLV measures real skill - the bookmakers themselves treat it as the most reliable sign of a lasting edge - and it is why serious bettors spread their activity across several high-limit books and betting exchanges.

Accumulator (acca)

Several selections combined on one slip that pays only if every leg wins. The combined price is the product of the individual prices, d = d₁ × d₂ × … × d_n - and the bookmaker's margin compounds in exactly the same way. Four legs each priced about 5% below fair carry an expected return of (1/1.05)⁴ − 1 ≈ −17.7%, against roughly −4.8% on any one of them alone. An accumulator only makes sense when every leg is positive expected value on its own and the legs are genuinely independent.

American and fractional odds

Two other ways of writing the same price as decimal odds. Fractional (UK) odds a/b state net profit per stake, so the decimal equivalent is 1 + a/b: 5/2 = 3.50 and 4/6 ≈ 1.67. American odds state, with a plus sign, the net profit on a 100 stake (+A = 1 + A/100) and, with a minus sign, the stake needed to win 100 (−A = 1 + 100/A): +150 = 2.50, −200 = 1.50, and decimal 1.91 is roughly −110. In every format the implied probability is the reciprocal of the decimal price (−110 → 52.4%). Decimal is the only format in which expected value, Kelly stakes and overround follow directly, without sign cases - which is why Pro Edge™ quotes it - but US-sports markets are quoted in American odds, so converting is part of daily work.

Arbitrage (arbing)

Simultaneously backing every outcome of an event across different bookmakers at prices that collectively guarantee a profit regardless of the result. A function of price discrepancy between books rather than predictive edge, and not the strategy this service is built around.

Asian handicap

A goal-based handicap applied to a match before kick-off to level two unevenly matched sides, removing the draw as a possible outcome (or reducing it to a partial refund on a quarter-line). Common in football markets as an alternative to the standard three-way match result.

Asian totals (Asian over/under)

The totals market with finer lines. With N goals in normal time and a line T, the over wins when N > T, is refunded when N = T and loses when N < T; the under is its mirror image. On a half line (2.5) the bet is binary, since N is a whole number; on a whole line (2.0) the exact total is a push; on a quarter line (2.25) the stake is split equally between the two neighbouring lines, 2.0 and 2.5, giving five possible settlements, from full win through half win, push and half loss to full loss. In the base model N is the sum of two independent Poisson counts, so it is itself Poisson with mean λ_home + λ_away; the fair price of a line comes from P(N > T) and P(N < T) with the probability of a refund taken out.

Backtesting

Testing a model's selections against historical prices and results to estimate how it would have performed, before any of it is risked on a live market.

Bankroll

The fund a bettor sets aside for wagering, kept separate from day-to-day finances. A unit is typically fixed as a small, constant fraction of starting bankroll and then held flat; Kelly-derived sizing instead recalculates the stake against current bankroll on every bet. Either way, sizing is defined relative to bankroll, not as a fixed cash figure, so a staking plan stays valid regardless of its size.

Bayesian updating

A method of revising a probability estimate as new evidence arrives, rather than treating the original estimate as fixed. Formally, the posterior probability of an outcome given new evidence is proportional to the prior probability of that outcome multiplied by the likelihood of the evidence under it: P(outcome | evidence) ∝ P(outcome) × P(evidence | outcome). In our pipeline, the prior is the model's early-week estimate and the evidence is late team news, so a fixture's assessment can shift as lineups are confirmed close to kick-off, rather than staying frozen at whatever it was days in advance.

Betting exchange (back and lay)

A marketplace in which bettors take each other's bets and the operator charges commission on net winnings. Backing a selection is betting that it happens; laying it is betting that it does not, taking the bookmaker's side, with a liability of the backer's stake times the price minus one: laying 10 at 3.00 risks 20 to win 10. With commission c, the effective back price is 1 + (d − 1)(1 − c), so 3.00 at 5% is worth 2.90. Prices are set by supply and demand and the back-lay spread is tight in liquid markets, which makes exchange prices among the best market estimates of margin-free probability - professional models are often calibrated to them, and closing-line value is frequently measured against them. Laying is also what makes hedging and trading out of a position possible.

Binomial distribution

The distribution of the number of successes K in n independent trials with the same probability p: P(K = k) = C(n, k) p^k(1 − p)^n−k, with mean np and variance np(1 − p). Over a run of bets it describes the number of winners, and it is the mathematics underneath win rate, sample size and statistical significance. It puts a number on luck: across 100 bets at 2.00 with a true 50% chance - zero expected value - the probability of at least 55 winners is about 18.4%, so roughly one bettor in five or six with no edge at all shows a +10% ROI or better after 100 bets. Inferring an edge from a short record is therefore unsound; the binomial gives an exact test of whether a hit rate is consistent with chance.

Bivariate Poisson model

An extension of the independent Poisson model that lets the two sides' goal counts correlate (Karlis and Ntzoufras, 2003). Three independent Poisson variables X₁, X₂ and X₃ with rates λ₁, λ₂ and λ₃ give the home and away goals as X₁ + X₃ and X₂ + X₃; the shared component, with Cov = λ₃, captures what acts on the whole match - tempo, refereeing, weather, pitch. With λ₁ = 1.5, λ₂ = 1.0 and λ₃ = 0.1 - the same expected goals, 1.6 and 1.1, as an independent model - the correlation is about 0.075, the draw rises from 24.89% to 25.98% and 0-0 from 6.72% to 7.43%. Small as that looks, ignoring it misprices draw and low-total markets systematically.

Bootstrap (resampling)

Bradley Efron's method (1979) for estimating the uncertainty of a statistic from the sample itself, with no assumption about its distribution. From the n observed bets, draw n with replacement B times (typically 10,000 or more), compute the measure on each resample - ROI, CLV, Sharpe ratio, maximum drawdown - and read the 2.5th and 97.5th percentiles of the B results as a 95% interval. For 300 bets at 2.00 with 159 winners, a +6% ROI, the interval runs from about −5.3% to +17.3%: zero sits comfortably inside it. Its real value is where no closed formula exists - mixed prices, skewed high-odds portfolios, drawdown. Correlated bets, such as several in one round of fixtures, need a block bootstrap, or the uncertainty is understated.

Both-teams-to-score (BTTS)

A market betting on whether both sides score at least once during the fixture, independent of the match result or final scoreline. Read directly off the same scoreline matrix used to price totals and correct score: with A the event that the home side is shut out and B the event the away side is shut out, P(BTTS) = 1 − P(A) − P(B) + P(A ∩ B) by inclusion-exclusion, where P(A ∩ B) is simply the model's probability of a 0-0.

Bradley-Terry model

A strength model built on paired comparisons (Bradley and Terry, 1952): each team has a positive strength π, and the probability that i beats j is P(i > j) = π_i/(π_i + π_j). On a log scale, θ = ln π, that is a logistic function of θ_i − θ_j, so the Elo rating can be read as a running estimate of the same model - in Elo's base-10, 400-point form, P = 1/(1 + 10^−Δ/400). Strengths of 3 and 2 give the stronger side 60%; a 200-point Elo gap gives about 76.0%. The parameters are fitted by maximum likelihood across every result, which puts results against different opponents on one scale; extensions for draws (Rao-Kupper, Davidson) adapt it to football, and home advantage enters as a multiplier on the home side's strength.

Brier score

A scoring rule for probabilistic forecasts: the mean squared gap between each forecast probability and the outcome it predicted, scored 1 if the event happened and 0 if it didn't. Lower is better and 0 is perfect. Like log loss it rewards accuracy and honest uncertainty together, but it punishes a confident miss far more gently - log loss grows without limit as a forecast approaches certainty on the wrong side, while the worst a Brier score can be is 1. That gap is why our backtests are scored on log loss instead: on a book of real stakes, the rare confident error costs far more than a squared-error view of it suggests.

Calibration

A model is well calibrated if, among every selection it assigns a given probability, that outcome actually occurs at roughly that rate over a large sample - a 60% probability should win close to 60% of the time, not 80% or 40%. Calibration, not accuracy on any single pick, is the property a probabilistic model is judged on.

Central limit theorem (CLT)

The sum of many independent, identically distributed variables with finite variance is approximately normal whatever their own shape, with mean nμ and standard deviation σ√n. A betting record is exactly such a sum. At level stakes, 2.00 odds and a true 52% chance (+4% expected value), each bet is worth +0.04 units with a standard deviation of 2√(0.52 × 0.48) ≈ 0.999; after 1,000 bets the expected result is +40 units with a standard deviation of about 31.6, so the chance of still being behind is Φ(−40/31.6) ≈ 10.3%. Two consequences follow: expected profit grows linearly but noise only with the square root of the sample, so an edge separates from luck as the sample grows; and on small samples - especially at high odds, where the distribution is strongly skewed - the normal approximation is poor, and an exact binomial calculation or simulation is needed.

Closing-line value (CLV)

The gap, in percentage points of implied probability, between the closing price - the market's final, most efficient price before the event starts - and the price taken on a selection: CLV = closing implied probability − implied probability at the price taken. A positive figure means the market's final assessment priced the side taken as more likely to win than the odds implied when the selection was published, in other words a better price than the market ultimately settled on. Computed the same way as edge, but benchmarked against the close rather than the price at publication. Consistent positive CLV, averaged across a large sample, is the most reliable single indicator of a repeatable edge, independent of any one result.

Confidence interval

A range around a model's point estimate that reflects how certain - or uncertain - it is, rather than presenting a single probability as though it were exact. A wide interval signals that the underlying methods disagree or the available data is thin, and is itself a reason for caution rather than something to be averaged away. Strictly, frequentist methods produce confidence intervals and Bayesian methods credible intervals; the term is used here in its everyday sense to cover both.

Correct score

A market betting on the exact final scoreline of a fixture (e.g. 2-1), rather than the match result, a goal-based handicap, or a threshold like totals. The finest-grained market a scoreline matrix can price directly, and consequently the one whose probabilities are most sensitive to the low-scoring-cell adjustment a Poisson-based model applies - see Dixon-Coles correction.

Correlation

The degree to which two outcomes move together rather than independently, and the reason a multi-leg bet is rarely worth what it looks like. Combine "over 2.5 goals" with "both teams to score" in the same match and the two legs rise and fall together, so the true probability of both landing is much higher than multiplying their two prices implies - and a bookmaker prices the combination accordingly rather than at the naive product. Most of the apparent value in a long bet builder or accumulator is this effect misread: legs treated as independent when the whole point of choosing them was that they aren't.

Decimal odds

A price format, standard across Europe, in which the figure quoted is the total return per unit staked, including the original stake (decimal odds of 1.91 return £1.91 from a £1 stake, a £0.91 profit). We quote prices exclusively in decimal odds rather than fractional or American formats, since it converts most directly to implied probability.

Distribution drift

A structural shift in the statistical relationship a model was calibrated on - a rule change, a change in refereeing strictness, a shift in how efficiently a market prices a league - that silently moves the baseline the model assumes is stable. Distinct from overfitting: an overfit model was miscalibrated from the outset on the data it was trained on, while drift describes a model that was well calibrated and has since gone stale as the world it describes changed under it. Guarded against by continuously monitoring live calibration against the original backtest rather than assuming a single historical fit remains valid indefinitely; a league whose drift is severe enough to break calibration is pulled from publication until it is re-estimated.

Dixon-Coles correction

An adjustment to a Poisson-based scoreline model that corrects for the two sides' goal counts not being quite independent in practice: low-scoring draws, in particular 0-0 and 1-1, occur slightly more often than the naive product of two independent Poisson rates predicts, while 1-0 and 0-1 occur slightly less often. Introduced by Dixon and Coles (1997), the correction rescales exactly those four cells of the scoreline matrix by a factor governed by a single dependence parameter fitted from historical data, rather than discarding the Poisson framework altogether. Left uncorrected, a naive Poisson model misprices precisely the low-scoring markets - correct score and, to a lesser extent, both-teams-to-score - that depend most on getting those cells right.

Drawdown

The decline from a bankroll's peak to its subsequent trough before a new peak is reached. A normal feature of any genuinely profitable strategy - variance guarantees it will happen - so its size and duration matter more than its mere occurrence.

Dutching

Splitting a total stake T across several mutually exclusive outcomes so that any of them returns the same amount: the stake on outcome i is T(1/d_i)/Σ(1/d_j), and the return is T/Σ(1/d_j), so the combined position behaves as one bet at 1/Σ(1/d_j). The draw at 3.40 and the away win at 3.60 take 51.43 and 48.57 of a 100 stake and return 174.86 either way - a synthetic double chance at 1.7486. Dutching creates no value of its own; it is worth doing when the prices are the best available, even across different books, and the combined price beats the reciprocal of the modelled probability. Extended to every outcome with Σ(1/d_j) < 1, it becomes arbitrage.

Edge

The percentage-point gap between a model's fair probability for an outcome and the market's overround-adjusted implied probability. A positive, backtested edge is the precondition for publishing a selection.

Elo rating

A rating system that carries a side's strength as a single number and updates it after every match by the distance between the result and what the ratings had predicted: beat a much stronger opponent and the rating moves sharply, beat a much weaker one and it barely moves at all. Devised for chess and transferred to football and other sports because the update rule needs nothing but results, while the gap between two ratings maps directly onto a win probability. Its limitation is the other face of that simplicity - it sees results only, never the manner of them, so it adjusts slowly to a side whose underlying performance has shifted faster than its scorelines have.

Ensemble (model ensemble)

A forecasting approach that combines several independently built models - each capturing different signal - and reconciles their outputs into a single probability, rather than relying on any one method alone. Disagreement between the constituent models is treated as information about uncertainty, not noise to be discarded.

Expected goals (xG)

A measure of chance quality: the probability that a given shot is scored, estimated from the circumstances of the chance rather than what actually happened to it - distance, angle, body part, the pass that created it, the defensive pressure on the shooter. Summed across a match it gives each side an expected-goals figure, which describes the performance far more stably than the scoreline does, since a deflection or a fine save swings the score while barely moving xG. Its value is precisely that it regresses less: a side running well ahead of its xG over a short run is usually finishing luckily rather than sustainably - see regression to the mean - and a goal-rate model is better off pricing that gap than chasing the results it produced.

Expected value (EV)

The probability-weighted average profit or loss of a bet - what it would return on average if repeated many times at the same probability and price. A positive-EV bet is profitable on average over a large enough sample, even though any single instance of it can still lose.

Fair odds

The price with no margin in it, exactly matching an outcome's estimated probability p: d_fair = 1/p, so 58.4% gives 1/0.584 ≈ 1.71. Any available price above the fair odds is a positive expected value bet on that estimate; the gap between the two, in probability terms, is the edge.

Favourite-longshot bias

A well-documented regularity: longshots are priced worse than their true chances warrant, so a disproportionate share of the bookmaker's margin sits on them. Removing the overround proportionally therefore overstates the longshots' probabilities and understates the favourites' - the error the power method and the Shin model are designed to correct.

Flat staking

Sizing every bet at the same fixed unit, regardless of the size of its edge. Simpler to verify and more behaviourally robust than edge-proportional methods, at the cost of some theoretical long-run growth - see Kelly criterion for the alternative, and our Methodology for why we publish this way.

Fractional Kelly

Staking a fixed fraction c of the Kelly criterion stake, f = c × f*, typically with c between 0.25 and 0.5. Full Kelly is optimal only when the probability is known exactly; with estimation error, overstaking is asymmetrically dangerous, because the growth rate falls to zero at twice the Kelly stake. For a small edge, fractional Kelly keeps about c(2 − c) of full Kelly's long-run growth while the variance of the bankroll falls with c²: half Kelly delivers about 75% of the growth with a quarter of the variance. At 2.00 odds and a 55% chance, full Kelly stakes 10% of bankroll and half Kelly 5%, with expected log growth per bet of 0.00501 and 0.00375. Pro Edge™ publishes flat 1-unit stakes rather than Kelly sizing; members who size by Kelly do so at their own discretion.

Gambler's fallacy

The mistaken belief that in a sequence of independent events past outcomes change the next one - that after five losing bets a winner is "due". For independent events the probability is the same on every trial. Raising stakes to win losses back, as the martingale does, increases the risk of ruin without improving the expected value at all.

Gradient-boosted trees

A machine-learning technique that builds a sequence of shallow decision trees, each one trained on the error the trees before it still leave unexplained - a form of gradient descent carried out over functions rather than parameters, typically scored on log loss when the target is a probability. Regularised by shrinking each tree's contribution and capping its depth, which trades a slower fit for one far less prone to memorising noise. Well suited to capturing non-linear interactions between variables - fixture congestion compounding with travel distance, for example - that simpler additive models like Poisson regression miss unless the interaction is specified by hand.

Hedging

Placing a bet on the opposing outcome of an existing position to lock in a profit or cap a loss before an event finishes, usually after a significant price move. A risk-management decision independent of whether the original selection still represents value.

Home advantage

The extra performance of the side playing at home, from crowd support, no travel, familiar surroundings and - research suggests, to a large degree - refereeing decisions. In Poisson-based models it enters as a multiplier: log λ_home = μ + γ + attack_home − defence_away, where γ is the home-advantage parameter; γ = 0.25 raises the home side's expected goals by e^0.25 ≈ 1.284, or 28.4%. Its size varies by sport, league and era, has declined over the long run, and the behind-closed-doors matches of 2020-21 confirmed that crowds contribute materially. Serious models therefore estimate it per league and time-weight it rather than fixing it; where the market still prices the folk version of home advantage, the value can lie with the away side.

Implied probability

The probability of an outcome implied by its price: for decimal odds, the reciprocal of the odds (a price of 2.00 implies 50%). Summed across every outcome of a market, implied probabilities exceed 100% by exactly the bookmaker's overround, which is why they are adjusted before being compared with a modelled probability.

In-play (live) betting

Betting a market after the event has started, on prices that update continuously as the state of play changes. Distinct from the pre-match markets every Pro Edge^™ selection is published into: an in-play price already reflects information no pre-kick-off model can hold - the current score, a red card, an injury - and trading it profitably depends on latency and a different modelling approach altogether, neither of which we claim. What a member chooses to do with an existing position once the event is under way is entirely their own decision; see hedging.

Kelly criterion

A staking formula that maximises the long-run geometric growth rate of a bankroll while keeping the risk of ruin bounded. For decimal odds d and win probability p, the full-Kelly stake is the fraction f* = (p(d − 1) − (1 − p)) / (d − 1) of bankroll; fractional variants (half-Kelly, quarter-Kelly) trade growth for lower volatility. Pro Edge^™ publishes flat 1-unit stakes rather than Kelly-sized ones, for consistency and ease of verification; a member who wants Kelly-style sizing can apply the formula to their own bankroll against our published edge, entirely at their own discretion and risk.

Law of large numbers

The theorem underpinning why expected value is a meaningful figure at all: as the number of independent repetitions of a bet increases, the average outcome converges to the true expected value with probability 1, even though no individual repetition is bound to match it. It says nothing about any single bet or short run - variance dominates there - and everything about why a genuine edge, however small, becomes visible only once realised over a sufficiently large sample.

Line movement

The change in a market's price between when it opens and when it closes, driven by new information and the weight of money on either side. A price that has already moved against a published selection by kick-off has, by definition, eroded some of the original edge.

Log loss

A scoring rule for probabilistic forecasts that penalises confident, wrong predictions far more heavily than cautious ones. For a single prediction p on an outcome that resolves as 1 (occurred) or 0 (didn't), log loss is −[y·log(p) + (1 − y)·log(1 − p)]; averaged across every prediction, it rewards both accuracy and honest uncertainty. A model that is well calibrated and avoids overconfidence scores better on log loss than one that is merely accurate on average, which is why we use it - rather than plain accuracy - to score every backtest.

Logarithmic growth rate

The long-run geometric growth of a bankroll per bet: G(f) = p ln(1 + f(d − 1)) + (1 − p) ln(1 − f), where f is the fraction of bankroll staked. After n bets the bankroll multiplies by roughly e^nG, so long-run wealth is governed by G, not by expected value - and the Kelly criterion is simply the stake that maximises it. At 2.00 odds and a 55% chance, the 10% Kelly stake gives G = 0.00501, about 150 times the bankroll after 1,000 bets in theory; half Kelly gives G = 0.00375, about 43 times; double Kelly, 20%, gives G ≈ −0.00014 - a shrinking bankroll from positive-expectation bets. That is the paradox of overbetting: too large a stake turns a genuine edge into a loss.

Logistic regression

The standard model for the probability of a two-way outcome: the log-odds of the probability are a linear function of the explanatory variables, ln(p/(1 − p)) = β₀ + β₁x₁ + …, so p = 1/(1 + e^−z) for the linear predictor z, always inside (0, 1). The coefficients are fitted by maximum likelihood and read directly: e^β is the odds ratio for a one-unit change. With β₀ = 0.2 for home advantage and β₁ = 0.004 per Elo point, a 100-point gap gives z = 0.6 and p ≈ 64.6%, the 100 points multiplying the odds of winning by e^0.4 ≈ 1.49. Its strengths are interpretability and well-calibrated output; its limit is that non-linear interactions must be specified by hand, which is why it often sits beside gradient-boosted trees in an ensemble as the baseline.

Market efficiency

The degree to which a price already reflects all available public information. A perfectly efficient market leaves no edge to find; the case for a quantitative approach rests on markets being efficient enough that the closing line is a meaningful benchmark, yet not so efficient that a disciplined, backtested edge cannot survive contact with them.

Markov chain

A model of a process that moves between states where the next state depends only on the current one, not on the route taken to reach it. It suits anything in sport that is naturally a sequence of states - a tennis match as a chain of point, game and set scores, or a football match as a chain of scorelines - because the full distribution of final outcomes follows from a single transition rule. Where Monte Carlo simulation reaches the same answer by re-running the scenario thousands of times, a Markov chain can often reach it exactly, at the cost of requiring the process to be genuinely memoryless - an assumption worth testing rather than assuming, since momentum and fatigue are exactly the kind of history it discards.

Martingale (stake doubling)

A progression that doubles the stake after every loss, so the first win recovers all previous losses plus the original stake. It cannot work: by the optional stopping theorem, with finite capital and stake limits no staking scheme changes the expected value of the bets - negative-expectation bets cannot be combined into a positive-expectation sequence. It trades frequent small wins for a rare, catastrophic loss: from a 1-unit base, ten straight losses cost 1,023 units and demand a next stake of 1,024. At even chances the probability of at least one run of ten losses in 1,000 bets is about 38.5%, and still about 28.5% with a 52% chance of winning each bet; bookmakers' stake limits rule it out in practice too. In probability theory a martingale is also the stochastic process describing a fair game, which is where the system got its name.

Maximum likelihood estimation (MLE)

The standard way to fit a statistical model's parameters: choose the values under which the observed data are most probable, θ̂ = arg max Σ ln f(x_i | θ). In a Poisson model, the estimate of a side's scoring rate is its goals per game: 16 goals in 10 matches gives λ̂ = 1.6, with a standard error of √(λ/n) = 0.4 and a 95% interval of roughly 0.82 to 2.38 - ten games in, the estimate is still very loose. Dixon-Coles and other goal-based strength models fit every side's attack and defence and the home advantage together, in one time-weighted likelihood. MLE is consistent and efficient on large samples but overfits small ones, so in practice it is combined with regularisation or a Bayesian prior that shrinks the estimates.

Middling

Placing bets on opposite sides of a line that has moved between the two wagers, structured so that a result landing inside the gap between the two lines wins both bets rather than merely cancelling one out. Distinct from arbitrage, which locks in a guaranteed profit at the moment both bets are placed; a middle only pays the bonus outcome if the result happens to land in the gap, and otherwise settles as one win and one loss, exactly like an ordinary hedge.

Modelled probability

The probability our models assign to an outcome, calculated independently of - and before seeing - the market price.

Moneyline

A market betting purely on which side wins a fixture, with no handicap or goal line attached - the most direct expression of a model's probability for the match result itself.

Monte Carlo simulation

Re-running a modelled scenario thousands of times with inputs drawn at random from their estimated distributions, to build a distribution of outcomes rather than a single point estimate - a common technique for translating goal-rate or scoring-rate models into full scoreline probabilities.

Net yield

Net profit expressed as a percentage of total stakes - profit divided by total turnover, not by bankroll or any other base. Under flat, unit-sized staking the figure is numerically identical to return on investment, which is why the two are often used loosely as synonyms; they can diverge once stake sizes vary, which is why we state which one a given figure represents rather than treating them as interchangeable.

Normalisation

Adjusting raw data onto a common, comparable scale before it is fed into a model - correcting for confirmed lineup changes, fixture congestion, or known structural biases such as home advantage, so the model isn't comparing distorted figures.

Overdispersion and the negative binomial

Overdispersion is a count whose variance exceeds its mean, where the Poisson distribution has them equal. In football, goals show mild overdispersion; corners, cards and player statistics usually show a lot, because the intensity of the match itself varies. The negative binomial - a Poisson mixed over a gamma-distributed rate - is then the right model, with mean μ and variance μ + μ²/r for a dispersion parameter r, becoming the Poisson as r grows without limit. A match type averaging 2.6 goals with a variance of 3.1 gives r = μ²/(σ² − μ) ≈ 13.52, and the probability of a goalless match rises from the Poisson's 7.43% to 9.27%. Ignoring overdispersion underprices the extremes - very low and very high counts - and so misprices the outer lines of totals, corner and card markets.

Overfitting

A model that has learned the noise in its historical sample rather than the genuine signal, producing a backtest that looks excellent and live performance that doesn't match it. Guarded against by testing on data the model has never seen and discarding any selection whose edge doesn't survive that out-of-sample bar.

Overround (or "vig" / "juice")

A bookmaker's built-in margin: the amount by which the sum of the implied probabilities across a market's outcomes exceeds 100%. Normalising those probabilities back to a sum of 100% recovers the market's overround-adjusted (fair) prices - the figures our modelled probabilities are compared against.

p-value and z-test

The p-value is the probability, if the null hypothesis is true - for betting, if the true ROI is zero - of a result at least as extreme as the one observed. The z-test measures the observed mean in standard errors: z = ROI/SE, with SE = σ/√n for σ the standard deviation of one bet's result. 500 bets at 2.00 with a 53% hit rate return +6%; σ = 2√(0.53 × 0.47) ≈ 0.998, SE ≈ 0.0446, z ≈ 1.34 and the one-sided p-value is about 0.09 - not significant at the usual 5% level, since roughly one bettor in eleven with no edge would do at least as well. A p-value is not the probability that a strategy is profitable, nor a measure of the edge's size; and when many strategies, leagues or parameters are tested at once, chance "discoveries" multiply, so a correction (Bonferroni, or false-discovery-rate control) is needed.

Player props

A market on one player's output in a fixture - shots, assists, cards, points, rebounds - rather than on the match result or a team total. Priced from player-level rather than team-level distributions, which makes them unusually sensitive to minutes and role: a price that looked sound in the morning can be void of meaning by kick-off if the player is rested, moved out of position, or withdrawn from the squad.

Poisson regression

A statistical model for count-based outcomes - goals scored, for instance - that estimates the rate λ at which an event occurs as a function of explanatory variables, such as each side's attacking and defensive strength: the probability of exactly k goals is P(X = k) = e^−λλ^k/k!. The workhorse for scoring-based markets like totals and correct score, though two sides' goal counts are not quite independent in practice - low-scoring draws occur slightly more often than the naive product of two Poisson rates predicts, an effect the Dixon-Coles correction adjusts for directly.

Power method

A way of removing the overround that raises every implied probability to a common power: p_i = π_i^k, with k > 1 the unique value for which the results sum to one. Because powering shrinks small probabilities proportionally more, more of the margin is taken from the longshots - handling the favourite-longshot bias with one parameter, and keeping every probability inside (0, 1). On a 2.10 / 3.40 / 3.60 market, k ≈ 1.0461 gives 46.02%, 27.80% and 26.18%, against 45.43%, 28.06% and 26.50% when the margin is removed proportionally. It usually lands close to the Shin model, and is simpler on many-outcome markets - correct score, scorers, outrights - because it needs no model of the market.

Public money

The aggregate weight of recreational betting, as distinct from sharp money. It gathers predictably - on favourites, home sides, well-supported clubs and overs - and it moves a price far less readily than its volume suggests, because a bookmaker is content to hold that action rather than shorten against it. Which makes the exception informative: a line drifting against heavy public backing usually means better-informed money has taken the other side.

Push

A bet that neither wins nor loses because the result lands exactly on the handicap or total line, returning the stake in full. Common on whole-number lines (e.g. a 2-goal handicap or a 2.0 total) rather than the quarter- or half-lines that avoid the possibility entirely.

Pythagorean expectation

Bill James's sabermetric relation estimating a team's expected winning percentage from points scored (PF) and conceded (PA): W = PF^x/(PF^x + PA^x). The exponent depends on how widely scores spread in the sport: about 1.83 in baseball, about 2 in ice hockey, about 2.37 in American football, and 13.9 to 16.5 in basketball, where many points per game make the margin relatively stable. An MLB side scoring 800 runs and conceding 700 projects to 56.1%, about 90.8 wins over 162 games; an NBA side scoring 115 and conceding 110 a game projects to about 65.1% with x = 14. Points differential predicts the future better than the win-loss record, so a team whose record runs well ahead of its Pythagorean expectation - typically on close wins - tends to fall back, by regression to the mean, and the market is often slow to price it.

Ranked probability score (RPS)

A scoring rule for forecasts over ordered outcomes (Epstein, 1969), the standard in the literature for football's home-draw-away. For r ordered outcomes, RPS = 1/(r − 1) × Σ_i=1…r−1(Σ_j≤i(p_j − e_j))², where p is the forecast probability and e is 1 for the outcome that happened and 0 otherwise; it runs from 0 to 1, lower is better. Comparing cumulative distributions makes it respect the order of the outcomes: a 55% / 25% / 20% forecast scores 0.121 on a home win, 0.171 on a draw and 0.471 on an away win, so a near miss costs less than a far one, where the Brier score summed over the outcomes gives 0.305, 0.905 and 1.005 and barely separates the draw from the away win. It is strictly proper: its expected value is lowest when the forecaster reports their true belief.

Regression to the mean

The tendency for unusually good or bad results to be followed by results closer to the underlying average, simply because extreme outcomes are partly luck. Mistaking a hot or cold streak for a genuine change in ability is one of the most common analytical errors in sports betting.

Return on investment (ROI)

Net profit as a percentage of the capital committed to generate it. In flat-staking contexts such as ours, ROI and net yield are calculated the same way and return the same figure; the distinction becomes material under variable or compounding stake sizes, where capital committed and total turnover are no longer equal.

Risk of ruin

The probability that a staking strategy depletes a bankroll to a level it cannot recover from, given a stated edge and bet sizing. Kelly-derived sizing keeps risk of ruin mathematically bounded by shrinking stakes as a bankroll falls; flat, unit-based staking - our own published approach - instead manages it through a conservative, fixed unit size and the discipline of never increasing it after a loss.

Sample size

The number of independent observations - selections, in our case - a result is based on. A short run of wins or losses is dominated by variance and says little on its own; only over a large enough sample does an edge's true size become statistically visible.

Sharp money

Wagers placed by bettors with a demonstrated history of beating the closing line, whose action tends to move market prices as bookmakers adjust around it.

Sharpe ratio

William F. Sharpe's risk-adjusted return: excess return divided by its standard deviation, S = μ/σ. For betting, μ is the average return per unit staked (the ROI) and σ the standard deviation of one bet's result; over n independent bets it scales with √n. At 2.00 odds, +4% ROI and a 52% hit rate, σ ≈ 0.999 and the per-bet Sharpe ratio is about 0.040 - about 1.27 over 1,000 bets a year. It lets strategies at different prices be compared: at the same ROI a high-odds strategy has more variance, a lower Sharpe ratio, and needs more bets for the same confidence. It ties directly to the z-test, z = S√n, so an annualised figure also says how far a year's result stands from chance. On skewed, fat-tailed distributions it misleads; there the Sortino ratio, which counts only downside risk, or the maximum drawdown is the necessary complement.

Shin model

Hyun Song Shin's market-microstructure model (1992-93) for removing the overround. It assumes a share z of the money comes from bettors with inside information, so the bookmaker loads a disproportionate margin onto longshots, where informed money does relatively more damage. From the implied probabilities π_i = 1/d_i, summing to S, the true probabilities are p_i = [√(z² + 4(1 − z)π_i²/S) − z] / [2(1 − z)], with z solved numerically so that they sum to one. On a 2.10 / 3.40 / 3.60 market z ≈ 0.0241, giving 45.87%, 27.87% and 26.26% against the proportional 45.43%, 28.06% and 26.50%: the favourite gains, the longshots lose. Because it derives the favourite-longshot bias from a model, it gives more accurate probabilities than proportional normalisation in empirical studies, and is a professional default for expected value and CLV.

Skellam distribution

The distribution of the difference between two independent Poisson variables - in goal models, the goal difference. With home goals Poisson(λ_h) and away goals Poisson(λ_a), D = home − away has P(D = k) = e^−(λh+λa)(λ_h/λ_a)^k/2I_|k|(2√(λ_hλ_a)), where I is the modified Bessel function of the first kind; its mean is λ_h − λ_a and its variance λ_h + λ_a. The match result and Asian handicap lines price directly from it: with λ_h = 1.6 and λ_a = 1.1, P(D > 0) ≈ 48.96%, P(D = 0) ≈ 24.89% and P(D < 0) ≈ 26.15%, and the two-goal home win a −1.5 handicap needs is about 25.52%. Its independence assumption tends to understate draws and low scores, which the Dixon-Coles correction and the bivariate Poisson model address.

Statistical significance

A measure of how unlikely a result would be to occur by chance alone if there were, in fact, no real edge. We stay silent on a fixture rather than publish a selection whose modelled edge hasn't cleared this bar.

Steam move

A sudden, sharp price movement across multiple bookmakers simultaneously, typically triggered by coordinated sharp money. A steam move after a selection has been published is a real-time signal that the market is converging toward - or away from - the price that was taken.

Survivorship bias

The distortion that creeps into a backtest when it only includes data that "survived" to be available for analysis - discontinued leagues, delisted teams, or restated historical odds - while silently excluding the cases that didn't. Backtests built on incomplete historical data routinely overstate how well a strategy would actually have performed.

Time decay

Weighting older matches less in a model's estimates, because a side's strength changes over time with squad turnover, injuries and managerial changes. The usual form is exponential: an observation t time units old gets weight w(t) = e^−ξt, multiplying its term in the log-likelihood, with a half-life of ln 2/ξ. Dixon and Coles used ξ = 0.0065 per half-week - a half-life of about 53 weeks, so a match a year old carries roughly half the weight of last week's. Choosing ξ trades bias against variance: too small keeps stale information, too large leaves too few effective observations and inflates the uncertainty. It should be set by walk-forward validation, optimising predictive accuracy (log loss or RPS), separately by sport and market.

Totals (Over/Under)

A market betting on whether a combined statistic in a fixture - most commonly total goals - finishes over or under a stated threshold (e.g. Over 2.5 goals), rather than on the match result itself.

Transaction costs

The frictions that erode a theoretical edge once it meets a live market - reduced limits, wider prices on smaller books, or a price moving before a stake can be placed. A backtest that ignores transaction costs overstates real-world performance, which is why ours are factored in before a model clears the bar for publication.

Unit (u)

A fixed proportion of bankroll used as the base measure for stake sizing (e.g. "1u" or "1.5u") - capped at 1/20th (5%) of bankroll and often smaller, then held fixed for a minimum of 30 days rather than recalculated bet to bet - so performance can be compared independently of any individual bankroll size.

Value bet

A bet whose odds are longer than the fair odds implied by the outcome's true probability - the bookmaker's implied probability understates the actual likelihood, so the bet carries positive expected value. Synonymous with a positive-edge selection.

Variance

The natural spread of outcomes around a statistical average. A genuinely profitable model can, and will, still produce losing streaks; variance alone does not indicate a broken model.

Void bet

A bet cancelled and the stake returned in full, typically because a fixture was postponed, abandoned, or didn't meet the conditions stated when the bet was placed. Distinct from a loss or a push: no result is graded either way.

Walk-forward validation

Testing a model the way it will actually be used: fit it on everything up to a cut-off date, score it on the period that follows, roll the cut-off forward and repeat - so every evaluation lands on fixtures the model genuinely could not have seen. The usual alternative, k-fold cross-validation, shuffles the sample at random and is quietly wrong for betting data: a randomly drawn fold trains the model on matches played after the ones it is then scored against, leaking information no live model could ever have had. The symptom is the familiar one - a backtest that looks excellent and live performance that doesn't match it, see overfitting - and walk-forward testing is the specific discipline behind the out-of-sample bar a selection has to clear here. Rolling the window forward also exposes distribution drift: a model that decays as the cut-off advances is telling you the baseline it was fitted to has moved.

Win rate

The proportion of settled selections that finish as winners. On its own it says nothing about the price taken on each selection, so two runs with an identical win rate can represent a genuine edge or a steady loss, depending on the odds - see closing-line value for the metric that accounts for this.