Several models, one forecast. The deep question is what you average. Average the probabilities and you get a linear opinion pool (a mixture). Average the quantiles and you get a Vincent average (the mean/median ensembles). They can look strikingly different — this page shows why, with the maths and live figures.
It comes down to what you think the disagreement between models means
Different models disagree. How you combine them should depend on why you think they differ (Howerton et al., 2023):
Each model is a slightly-off measurement of the same underlying predictive distribution. Averaging should sharpen the estimate, cancelling noise — like averaging repeated readings. → quantile / Vincent averaging.
The models represent genuinely different stories about the system, and we are uncertain which one holds. We should keep all their possibilities on the table. → linear opinion pool (a probability mixture).
These are two independent choices, not one. A second, separate axis is weighting — do you trust every model equally, prune the worst, or optimise the weights?
| What you average → Weights ↓ | Quantiles (Vincent) | Probabilities (LOP) |
|---|---|---|
| Equal | Mean / median ensemble | Unweighted opinion pool |
| Pruned | Filtered mean ensemble | Filtered opinion pool |
| Optimised | Inverse-WIS, QRA | CRPS-tuned opinion pool |
Average the forecasts horizontally — line up the quantiles and average them
Take each model's value at a given quantile level p — say every model's 90th percentile — and average those numbers. Do it for every level and you have the ensemble's quantiles. Writing Qi for model i's quantile function:
This is the Vincent average (Stella Vincent, 1912). Two flavours appear in the course:
Average the forecasts vertically — average the probabilities at each value
Instead of averaging the quantiles, average the distribution functions themselves. At every value x, take each model's cumulative probability Fi(x) and average them. The word "linear" refers to this linear combination of the distributions:
The clearest way to picture it is as a sampling recipe — exactly what lopensemble does with the posterior samples:
Because each model keeps its own samples, a pool is at least as wide as its members and can be multi-modal. The law of total variance makes the cost precise:
The first term is the models' own uncertainty; the second is an extra chunk of variance the pool adds simply because the models disagree. Quantile averaging has no such term — that single equation is the whole difference between the two methods.
Two component forecasts, combined both ways at once
Two models, A and B, each a Normal. The opinion pool averages them vertically; the Vincent average averages them horizontally. Pull the two means apart and watch the pool split into two humps while the Vincent average stays a single sharp peak — even though both share the exact same mean.
Top: densities. Bottom: the same forecasts as CDFs — the pool (purple) is the curves averaged ↕ vertically; the Vincent average (green) is the curves averaged ↔ horizontally.
The pool's variance is the models' average spread plus their disagreement. Drag the means together and the orange (between-model) part vanishes — only then do the two methods nearly coincide.
Trust the models equally, prune the worst, or optimise
Weighting is orthogonal to what you average: you can weight either a quantile average or an opinion pool. The course covers three schemes:
lopensemble does).Below: three component models pooled with weights you control. Watch how re-weighting slides the opinion pool between its humps, while the Vincent average just glides along — and how an "optimised" winner-take-all weighting collapses both onto a single model.
| Quantile averaging (Vincent) | Linear opinion pool | |
|---|---|---|
| You average… | Quantiles (horizontally ↔) | Probabilities (vertically ↕) |
| Result is a… | Single re-centred distribution | Mixture of the components |
| Spread vs members | Average of their spreads — stays sharp | Average spread + disagreement — wider |
| Can be multi-modal? | No (shape preserved) | Yes, if models disagree |
| Built from | Quantile forecasts | Samples (a mixture draw) |
| Assumes models are… | Noisy estimates of one truth | Rival candidates for the truth |
| In the course | Mean / median ensemble, inverse-WIS, QRA | lopensemble mixture |
| Watch out for | Over-confidence if models genuinely disagree | Under-confidence / fat tails; multi-modality may be unrealistic |