Combining Several Forecasts Into a Better One

If you have several estimates of the same probability — from different methods, different people, or different models — combining them beats picking one, almost always. The details of how you combine them matter less than the fact that you do.

Why averaging works

Individual estimates contain signal plus error. If the errors are at least partly independent, averaging cancels some of the error while keeping the signal.

The condition is independence. Five estimates derived from the same source are one estimate counted five times, and averaging them produces false confidence rather than accuracy. Where crowd wisdom runs out.

Which average

Arithmetic mean of probabilities. Simple, robust, hard to beat. Start here.

Median. More resistant to one wild estimate. Useful when your inputs include something unreliable.

Mean of log-odds. Convert each probability to log-odds, average, convert back. This treats the distance from 50% to 60% as smaller than the distance from 89% to 99%, which is the right geometry for probabilities. It also naturally produces a more extreme result than the arithmetic mean, which is usually an improvement.

Weighted by track record. Better in principle, and it requires enough data on each source to estimate the weights, which is the same small-sample problem in a new place. Equal weights are often the safer choice.

Extremising

A consistent finding from forecasting research: the simple average of a group of forecasters is systematically underconfident, and pushing it away from 50% improves the score.

The reason is that each forecaster hedges against their own uncertainty. When many hedged estimates agree, the agreement itself is evidence, and the average does not capture it.

A modest transformation toward the extreme — via log-odds, or by applying a small exponent — usually improves accuracy. Overdoing it is harmful, and the right amount depends on how correlated your sources are. Correlated sources deserve less extremising, because their agreement is not independent evidence.

Aggregating across time

Your own forecasts at different moments are an ensemble too, and a recency-weighted average of them is often better than your latest one, because the latest one is the most exposed to whatever you read this morning.

If you find your combined-over-time estimate consistently beats your latest, you are overreacting to news. That is a specific, fixable diagnosis. Updating on news.

A market price is an aggregate

Worth stating plainly: the price is already a weighted combination of many participants' views, weighted by conviction and capital. That makes it a strong default input to any ensemble of your own.

Two cautions. It is not independent of the other forecasts you are using, because those forecasters probably looked at it too. And its weighting is by capital, not by accuracy, so it inherits the constraints in where crowd wisdom runs out.

Combining across venues

Two venues pricing the same event give you two aggregates from partly different populations. A liquidity-weighted combination of them is often better than either alone.

This is a much more defensible use of cross-venue data than treating the difference as an opportunity. The difference is mostly explained by structure and costs; the combination is genuinely more informative than either side. Why venues disagree and what a gap is worth.

The practical recipe

Take your own estimate, the market price, and any other independent estimates you trust. Average in log-odds. Extremise slightly if your sources are genuinely independent. Write down the result and the inputs.

Then score both the combination and your own individual estimate over time. If the combination wins — it usually does — you have learned something useful about how much weight your own view deserves.

Next

The mechanics category begins with what a spread costs you.