Math & ModelsUpdated 22 Jul 20262 min read
Log Loss
Log loss answers a narrow question well: when your model said a number, how surprised should it have been by what happened?
log loss = −(1 ÷ n) × Σ ln(probability assigned to the actual outcome)
Say 70% for a home win that lands and you score −ln(0.70) = 0.357. Say 30% for the same result and you score −ln(0.30) = 1.204. Lower is better; perfect foresight scores zero.
| # | Forecast probability | Result | Brier contribution |
|---|---|---|---|
| 1 | 0.0625 | ||
| 2 | 0.1600 | ||
| 3 | 0.3025 | ||
| 4 | 0.1600 | ||
| 5 | 0.4900 | ||
| 6 | 0.0400 |
Base-rate baseline 0.2500 — lower is better.
Base-rate baseline 0.6931.
6 forecasts, of which 50% happened. Your probabilities beat simply predicting that base rate every time.
A handful of forecasts cannot separate skill from luck whatever the score says. Treat anything under a few hundred as a sanity check, not a verdict.
Why the logarithm#
Because it makes confidence expensive.
Under the Brier score, being 99% certain about something that fails costs (0.99 − 0)² ≈ 0.98 — bad, but bounded. Under log loss it costs −ln(0.01) = 4.6, roughly thirteen times the penalty of a lukewarm 60% miss. Claim certainty and be wrong often enough and log loss will bury you, which is exactly the behaviour you want from a metric that guards a staking decision. A confidently wrong 95% is the single most expensive thing a betting model can produce, because it is the forecast you stake most on.
Taken to the limit, a 0% forecast for something that happens is infinitely wrong. Real implementations clip probabilities marginally away from 0 and 1 so a single row cannot dominate the average — ours included.
Both metrics, on purpose#
Log loss and Brier are both proper scoring rules: each is minimised by reporting what you genuinely believe, so neither can be gamed by hedging everything toward 50%. That property is what makes them usable at all.
Where they disagree is temperament. Brier is forgiving and stable; log loss is unforgiving about overconfidence. A model that is well calibrated in the middle of the range but occasionally too sure at the extremes will look respectable on Brier and poor on log loss — and that gap is diagnostic, not noise. It is telling you precisely where the model is overreaching.
What a score cannot tell you#
Two limits worth holding onto.
A good score is not a profit. Scoring measures whether your probabilities are honest, not whether they are better than the price you can bet at. A model can be beautifully calibrated and still lose money against a sharper market — which is why closing line value is the metric that answers the betting question, and why our own Performance page reports both.
A small sample tells you nothing. A few dozen forecasts cannot separate skill from luck at any confidence worth acting on. See sample size for how many you actually need — the honest answer is more than almost anyone has.
Score your own forecasts with the Brier and log loss calculator, and read how our models work for what goes into the numbers being scored.
Related
- The Brier Score: How We Grade Our Own Predictions
- Calibration: The Only Promise a Prediction Can Keep
- Sample Size: How Many Bets Before You Know You Have an Edge
- How Our Models Work: Poisson, Dixon-Coles and Elo
- Expected Value in Betting: The Number That Decides Everything
- The Poisson Distribution: Turning Expected Goals Into Scorelines
- Brier Score & Log Loss Calculator — Grade Your Own Forecasts
- Closing Line Value Calculator — Grade a Bet Without Waiting for the Result
- Fair Odds Calculator — Remove the Bookmaker Margin from Any 1X2 Book
Frequently asked questions
What is log loss?
Log loss is the average negative logarithm of the probability a model gave to what actually happened. Forecast 70% for an event that occurs and you score −ln(0.70) = 0.357. Lower is better, and a perfect forecaster scores zero.
What is a good log loss score in football?
It depends entirely on how predictable the fixtures are, so the number is only meaningful against a baseline. Compare it with the log loss you would get by predicting the base rate every time — if you cannot beat that, the forecasts carry no information.
What is the difference between log loss and Brier score?
Both reward honest probabilities, but log loss is logarithmic and Brier is quadratic. A confident miss costs a bounded amount under Brier and an enormous amount under log loss, which is why log loss is preferred when confident errors are expensive.
Why is log loss infinite for a wrong certainty?
Because the logarithm of zero is undefined. Assigning 0% to something that happens is infinitely wrong — a model that claims certainty has no way to be forgiven. In practice probabilities are clipped slightly away from 0 and 1 so one row cannot swallow the whole sample.