Time series II
Conditional coverage and sharpness
The first page sorted the marginal guarantees — a question now settled, generously. This page is about what a forecaster actually wants: coverage conditional on the visible past, with intervals near the conditional oracle’s width. Here is what each route delivers, and the accounting problem nobody has done.
Two facts frame every attempt. First, exact distribution-free conditional coverage is impossible in finite samples (Lei & Wasserman 2014; Barber et al. 2021) — dependence does not soften that, so every route below is an approximation plus, ideally, a measurement of the residual gap. Second, there is no distribution-free width theorem anywhere in this literature, for a structural reason: the interval’s width is a quantile of whatever scores you supply, so with no assumption on the model behind the scores there is nothing to prove. Coverage theorems are assumption-free; sharpness theorems are always model-quality theorems in disguise.
Route 1: condition through the score
Make the score itself carry the state: volatility-normalized residuals |y − f̂(x)| / σ̂(past), quantile scores with lagged covariates, SPCI’s fitted conditional quantile of recent residuals, HopCPT’s attention over similar past regimes, TQA’s per-step quantile adjustment. This is a sensible default, and the worked example shows both its power and its ceiling: the conditional quality is exactly the conditional quality of the wrapped model. Raw quantile regression ties conformalized quantile regression on conditional metrics, while the conformal layer contributes the marginal certificate on top. The certificate never becomes conditional; it just stops being violated so visibly, and by Barber & Pananjady’s accounting the marginal certificate survives the dependence for any of these scores.
Route 2: reweight calibration toward the present state
Localized conformal prediction (Guan 2023) weights calibration points by similarity to the current one; Hore & Barber (2025) show randomized local weights buy a genuine local guarantee; Gibbs, Cherian & Candès (2025) deliver exact coverage simultaneously over a finite-dimensional class of covariate shifts, chosen in advance. These are the strongest conditional statements actually on offer. The price: the conditioning class is fixed a priori and necessarily coarse — “calm vs stormy” you can have; “conditional on this exact history” you cannot, by the impossibility results. And the time-series versions of these guarantees mostly lack the dependence accounting that the marginal case now enjoys: reweighting toward similar past states while the calibration set is itself serially dependent is largely unpriced territory.
Route 3: make width the objective
Bellman conformal inference (Yang, Candès & Lei 2024) treats interval length as the thing to minimize, subject to long-run coverage, and solves the resulting stochastic control problem by dynamic programming over multi-step forecasts. It fixes the known degeneracy of the online game: adaptive methods can satisfy their guarantee with intervals oscillating between empty and infinite. Empirically it produces much shorter intervals. No distributional width guarantee is claimed, and by the structural point above, none could be without model assumptions. The consistency-based width results (EnbPI, SPCI converging to oracle width) live in the same trade: assume the model learns the truth, and sharpness follows from the assumption.
Route 4: measure the gap that remains
Whatever route you take, the deliverable should include a measurement of what marginality left on the table. The classical tool is the PIT (Diebold, Gunther & Tay 1998); the level-free versions are the distance covariance between the conformal rank and the visible past (the Feynman–Wigner note, runnable via the guardrails on the code page), the width of the realized-coverage fan (the fan note), and classify-the-miss diagnostics (ERT). The accounting frame that unifies them: the coverage–sharpness gap of a marginally calibrated predictor is the mutual information between its conformal rank and the side information ignored (Marginally Useful) — and for a time series the ignored side information is the recent past itself.
The price, and what remains open
Barber & Pananjady priced the coverage cost of dependence: at most minτ {τ/(n+1) + 2β(τ)}, tight, two-sided, vanishing as the calibration grows. The sharpness cost of marginality is priced in the two-prices note, for the single-shape family: the per-step log-score regret of the marginal predictive against the past-conditional oracle equals the entropy-rate gap ΔH = I(S₀; past). That is the information-gap identity applied to the past. For Gaussian scores it exponentiates into width: oracle intervals are e−ΔH narrower at matched coverage, √(1−φ²) for AR(1). The asymmetry is the point: the coverage tax vanishes in n, the sharpness rent never does. What remains open is the general accounting — beyond the single-shape family, with estimation costs for the feasible conditional model included, and a distribution-free-modulo-mixing width statement rather than an oracle comparison.