Fixed-geometry continuous ensembles

fit_continuous_ensemble combines a finite list of existing ContinuousTree geometries. It recomputes every member's posterior on the same predictors, responses, normalization, and hyperparameters. Each member retains its coordinate-slice continuity and its selected main and pair terms. The ensemble mean is also continuous because its weights do not depend on the query row.

root = fit_continuous_tree(X, y; max_splits=0)
split = fit_continuous_tree(X, y; max_splits=1)
ensemble = fit_continuous_ensemble(X, y, [root, split];
    logprior=[log(1.0), log(1.0)])
mean_prediction = predict(ensemble, Xnew)

Supply each distinct terminal-box partition and pair-term basis once. A duplicate raises an error even when the same boxes were reached through a different split order, because repeating it would silently increase its prior mass. Candidate normalization must agree with that computed from X. The fitted ensemble copies its geometry, so later changes to a source tree do not change it. logprior gives relative log prior mass; -Inf excludes a member. At least one member must have finite posterior log mass. The default gives equal prior mass to all supplied members.

The model weights are proportional to exp(logprior[k] + evidence[k]), normalized with a log-sum-exp shift. Every evidence uses the member's conjugate linear-coefficient and inverse-gamma model. It conditions on the provided tree and pair-term basis. The result is exact conjugate averaging in the normalized model, treating normalization as fixed. By default, fitting estimates response center and scale from fitting responses. Fixing the candidate space alone therefore does not establish prior-predictive calibration for that procedure. If candidates were selected or tuned using those same responses, this is a restricted adaptive approximation: the weights do not account for the selection process or for omitted geometries.

Use response_normalization=(center=..., scale=...) to specify one common response transform for every refitted component. For example, identity normalization for one output uses (center=0.0, scale=1.0). Multiple outputs require vectors with one center and positive scale per output. Values are copied. The supplied geometries' response transforms do not affect refitting.

The prior uses these normalized response units. Choose the transform, geometry space, and hyperparameters independently of the fitting responses for a fixed conditional Bayesian model. A common response transform contributes the same likelihood Jacobian to every geometry, so it cancels in model weights. The stored comparison scores omit common constants and are not complete log evidences in original response units.

predict returns the weighted conditional mean. predict! writes that mean after reading all predictor rows, so its output may overlap X. Both accept the batch_size keyword used by continuous trees.

predictive returns location, variance, within_variance, between_variance, weights, and components. Each component has its own Student-t location, scale2, and dof, in the same target rank as a single continuous tree's predictive result. The mixture is generally not a Student-t distribution. Its variance adds the weighted conditional variances and the weighted squared differences between component means and the mixture mean. If a member with positive weight has dof <= 2, its conditional and mixture variance are infinite. These predictions condition on the supplied model space and fitted normalization.

For a scalar response, a query row can be converted to a Distributions.jl mixture for density or interval calculations. Distributions.jl is needed by this example, not by LinearTrees.jl at runtime.

using Distributions

posterior = predictive(ensemble, Xnew)
row = 1
members = [component.location[row] +
    sqrt(component.scale2[row]) * TDist(component.dof)
    for component in posterior.components]
mixture = MixtureModel(members, Categorical(posterior.weights))
logdensity = logpdf(mixture, ynew[row])
interval = [quantile(mixture, p) for p in (0.05, 0.95)]

Held-out log density and interval coverage should be assessed separately from training evidence, using the same candidate-generation policy intended for future data. The current API does not generate or enumerate candidate trees.

Check latent and observation intervals separately. Observation noise can hide mean misspecification: near-nominal observation coverage does not imply that credible intervals cover the latent function. Check several nominal levels, since one level can appear calibrated by coincidence. Aggregate coverage also need not hold at each predictor value. The benchmark protocol in bench/VALIDATION.md separates a fixed-transform prior-predictive check from the default procedure that estimates response normalization.