CDF-First: Conditional Density Estimation with an Explicit Smoothing Bandwidth

Decisions that depend on a tail probability or a quantile need the conditional distribution, not a conditional mean. Estimators of it differ in where they obtain normalization: a closed-form family, a Jacobian determinant, a simplex over cells, or a partition function. We present CDF-First, which takes it from a boundary condition: each factor of an autoregressive CDF is a bounded monotone network pinned to $0$ and $1$ at the ends of a declared domain, and the density is read from an interval probability of half-width $\delta$. At every parameter value, and with no smoothness assumption on the target, the model is an exactly normalized conditional density, and each factor equals the model's own convolved with a uniform kernel of half-width $\delta$; the objective is proper for that smoothed conditional, not the unsmoothed one. The smoothing is therefore a stated parameter, and we test whether it is the one applied. On four synthetic targets with known ground truth it bounds the realized smoothing from above, within $5\%$, in $253$ of the $258$ fits for which the smoothing is identifiable, which realize a median of $84\%$ of it; raising $\delta$ over two decades improves accuracy on three targets without measurably degrading the fourth. Accuracy is competitive rather than superior: lowest error on three of the four synthetic targets, and the lowest mean held-out negative log-likelihood on four of seven UCI benchmarks, tied on two and second on one.