State-Dependent Conformal Prediction

• State-dependent conformal prediction methods adapt prediction sets based on local covariate, temporal, or state information rather than using a fixed global rule.
• They encompass diverse approaches—such as covariate-adaptive intervals, kernel-weighted time-series, and invariant-regime calibration—to handle non-exchangeable, heteroskedastic, and temporally dependent data.
• These techniques aim for finite-sample marginal coverage and asymptotic optimality while providing actionable uncertainty quantification for dynamic and dependent data settings.

Searching arXiv for papers on state-dependent conformal prediction and related dependence-aware conformal methods. State-dependent conformal prediction denotes a family of conformal inference methods in which prediction sets, score thresholds, or uncertainty envelopes vary with the current covariate, temporal context, or dynamical state rather than being determined by a single global calibration rule. In the recent literature, this label covers several distinct constructions: covariate-adaptive prediction intervals for nonparametric regression, kernel-weighted conformal intervals for dependent time series, invariant-regime calibration for stochastic dynamical systems, region-wise perception-error bounds for neuro-symbolic verification, and policies that choose a data-dependent miscoverage level $\tilde\alpha$ on a per-example basis (Sesia et al., 2021, Lee et al., 2024, Bakhtiaridoust et al., 30 Jun 2026, Waite et al., 28 Feb 2025, Geng et al., 2 Dec 2025, Gauthier et al., 5 Oct 2025). The unifying motivation is that standard split conformal prediction assumes exchangeability and typically calibrates a single empirical quantile, whereas many scientific and engineering settings exhibit heteroskedasticity, skewness, temporal dependence, state-localized error structure, or deployment regimes that are not well represented by a generic i.i.d. calibration sample.

1. Baseline conformal formulation and the scope of “state dependence”

In standard split conformal prediction, one observes calibration data $z_i=(x_i,y_i)$, evaluates a nonconformity score $s(x,y)$, and forms the prediction set

$\mathcal{C}(X_{n+1})=\{y:s(X_{n+1},y)\le \hat q\},$

where $\hat q$ is the $\frac{\lceil (n+1)(1-\alpha)\rceil}{n}$-type empirical quantile of the calibration scores. Under exchangeability, the usual guarantee is

$$\Pr\!\left(Y_{n+1}\notin \mathcal{C}(X_{n+1})\right)\le \alpha.$$

This formulation is the common starting point for many later state-dependent variants (Snell et al., 18 Feb 2025).

The state-dependent literature modifies this baseline in several ways. In exchangeable regression, the interval itself is allowed to change with $X=x$ through an estimated conditional distribution, so that asymmetry and width reflect local properties of $Y\mid X=x$ rather than a global residual scale (Sesia et al., 2021). In dependent time series, the calibration rule is made a function of recent residual history or of a stationary invariant regime, because past and future scores are no longer exchangeable pointwise (Lee et al., 2024, Bakhtiaridoust et al., 30 Jun 2026). In verification problems, the uncertainty bound becomes piecewise in the system state, $\eta(x)=\eta_i$ on regions $z_i=(x_i,y_i)$0, so that symbolic reachability uses region-specific disturbances instead of a single worst-case bound (Waite et al., 28 Feb 2025, Geng et al., 2 Dec 2025). In adaptive-coverage formulations, even the miscoverage level ceases to be fixed: a learned policy outputs $z_i=(x_i,y_i)$1 as a function of calibration summaries and test-side information (Gauthier et al., 5 Oct 2025).

A recurrent conceptual point is that state dependence does not imply exact finite-sample conditional coverage. One line of work explicitly notes that exact finite-sample conditional coverage is impossible in general, and therefore targets finite-sample marginal validity together with approximate or asymptotic conditional validity, or with improved efficiency under structural assumptions (Sesia et al., 2021). Another line instead seeks specialized validity notions for dependent data, such as rolling one-step coverage under invariant-regime calibration or finite-sample validity under partial exchangeability for Markov sequences (Bakhtiaridoust et al., 30 Jun 2026, Basarkar et al., 28 Apr 2026).

2. Covariate-adaptive intervals in exchangeable regression

A canonical exchangeable example is "Conformal Prediction using Conditional Histograms" (Sesia et al., 2021), which introduces conformal histogram regression (CHR) for nonparametric regression. CHR is explicitly state-dependent in the sense that the prediction interval changes with the feature vector $z_i=(x_i,y_i)$2 through an estimated conditional distribution of $z_i=(x_i,y_i)$3. The paper sets three goals: finite-sample marginal coverage,

$z_i=(x_i,y_i)$4

approximate conditional coverage, and short interval length.

The oracle target is the shortest interval with conditional mass at least $z_i=(x_i,y_i)$5: $z_i=(x_i,y_i)$6 Unlike standard residual-based split conformal intervals, this oracle interval is state-dependent and can be asymmetric, which is essential for skewed or heteroskedastic responses. CHR approximates this object by partitioning the response domain into bins $z_i=(x_i,y_i)$7, estimating the conditional bin probabilities

$z_i=(x_i,y_i)$8

and then solving a discrete shortest-interval problem over contiguous bins. In the implementation emphasized in the paper, many conditional quantiles are estimated and converted into a piecewise-constant conditional density $z_i=(x_i,y_i)$9, which is then integrated over bins to obtain $s(x,y)$0 (Sesia et al., 2021).

Conformalization is achieved through a nested family of histogram intervals $s(x,y)$1, indexed by a target mass level $s(x,y)$2, together with the conformity score

$s(x,y)$3

If $s(x,y)$4 is the empirical $s(x,y)$5 quantile of the calibration scores $s(x,y)$6, the final interval is

$s(x,y)$7

and exchangeability yields finite-sample marginal coverage without requiring $s(x,y)$8 to be accurate (Sesia et al., 2021).

The distinctive asymptotic result is that, under assumptions including i.i.d. data, consistency of the histogram estimator, continuous bounded conditional densities, unimodality of $s(x,y)$9, and a compatible estimated histogram, the CHR interval is asymptotically equivalent to the oracle shortest conditional interval. The paper states both asymptotic length optimality relative to the oracle and approximate conditional validity up to vanishing error. It also introduces a randomized boundary-bin removal rule to make intervals tighter on average while preserving the nesting structure required for conformal calibration. Empirically, CHR achieves target marginal coverage, often improves estimated conditional coverage, and produces the shortest intervals among the compared methods, especially when the conditional distribution is skewed (Sesia et al., 2021).

3. Temporal dependence, local weighting, and invariant-regime calibration

State dependence becomes structurally different when observations are sequential and non-exchangeable. "Kernel-based Optimally Weighted Conformal Prediction Intervals" (Lee et al., 2024) treats time-series prediction as a local conditional quantile estimation problem on residual histories. Given a pretrained point predictor $\mathcal{C}(X_{n+1})=\{y:s(X_{n+1},y)\le \hat q\},$0, residuals are

$\mathcal{C}(X_{n+1})=\{y:s(X_{n+1},y)\le \hat q\},$1

and the current state is a recent residual vector

$\mathcal{C}(X_{n+1})=\{y:s(X_{n+1},y)\le \hat q\},$2

The conformal interval is

$\mathcal{C}(X_{n+1})=\{y:s(X_{n+1},y)\le \hat q\},$3

where $\mathcal{C}(X_{n+1})=\{y:s(X_{n+1},y)\le \hat q\},$4 is estimated by a reweighted Nadaraya–Watson conditional quantile estimator and

$\mathcal{C}(X_{n+1})=\{y:s(X_{n+1},y)\le \hat q\},$5

The learned local weights $\mathcal{C}(X_{n+1})=\{y:s(X_{n+1},y)\le \hat q\},$6 satisfy normalization and local moment constraints and are designed to approximate oracle weights under non-exchangeability. Under strong mixing assumptions on the residual-state process, smoothness conditions, and bandwidth conditions, the paper proves asymptotic conditional coverage,

$\mathcal{C}(X_{n+1})=\{y:s(X_{n+1},y)\le \hat q\},$7

together with a finite-sample marginal coverage-gap bound involving the distance between learned and oracle weights and the maximal local weight concentration (Lee et al., 2024).

"Invariant-Measure Conformal Prediction" (Bakhtiaridoust et al., 30 Jun 2026) addresses a different dependence problem: learned stochastic dynamical systems

$\mathcal{C}(X_{n+1})=\{y:s(X_{n+1},y)\le \hat q\},$8

with predictor $\mathcal{C}(X_{n+1})=\{y:s(X_{n+1},y)\le \hat q\},$9. The paper argues that standard conformal prediction breaks because trajectory samples are temporally dependent, the state distribution changes over time, and recursive multi-step errors accumulate. Its central move is to calibrate not on arbitrary trajectory samples but on independent samples from an invariant measure $\hat q$0 of the Markov kernel $\hat q$1, satisfying

$\hat q$2

Calibration pairs $\hat q$3 are sampled with $\hat q$4 and $\hat q$5, calibration scores are

$\hat q$6

and the one-step prediction set is

$\hat q$7

The key theorem states that if $\hat q$8 is invariant and the calibration states and test initial state are sampled independently from $\hat q$9, then for every fixed time step $\frac{\lceil (n+1)(1-\alpha)\rceil}{n}$0,

$\frac{\lceil (n+1)(1-\alpha)\rceil}{n}$1

This is a rolling one-step guarantee: the same calibrated set remains valid at every fixed time as long as the system remains in the invariant regime. For recursive forecasting, the paper assumes $\frac{\lceil (n+1)(1-\alpha)\rceil}{n}$2 is $\frac{\lceil (n+1)(1-\alpha)\rceil}{n}$3-Lipschitz and derives the error recursion

$\frac{\lceil (n+1)(1-\alpha)\rceil}{n}$4

leading to horizon-dependent tubes with factor

$\frac{\lceil (n+1)(1-\alpha)\rceil}{n}$5

The corresponding multi-step guarantee is

$\frac{\lceil (n+1)(1-\alpha)\rceil}{n}$6

The resulting uncertainty tubes are explicitly pre-deployment: $\frac{\lceil (n+1)(1-\alpha)\rceil}{n}$7 is fixed in advance from invariant calibration data, so no future residuals are needed before bounds are produced. The paper identifies rolling one-step prediction, receding-horizon forecasting, self-triggered control, and fault detection or observer design as settings where this matters. In its nonlinear rotational stochastic benchmark, invariant-measure calibration achieves rolling one-step coverage about $\frac{\lceil (n+1)(1-\alpha)\rceil}{n}$8 with small radius, whereas independent non-invariant and dependent non-invariant calibration degrade coverage, and the naive one-step baseline undercovers badly at longer horizons (Bakhtiaridoust et al., 30 Jun 2026).

These two works embody different dependence-aware meanings of state dependence. KOWCPI conditions on a local residual-history state and seeks asymptotic conditional validity under strong mixing, while imCP aligns calibration to the stationary operating law of the Markov process and restores exchangeability of one-step scores at each fixed time in rolling deployment (Lee et al., 2024, Bakhtiaridoust et al., 30 Jun 2026).

4. Markov-structured future-sequence prediction

"Conflict Forecasting via Conformal Prediction for Markov Processes" (Basarkar et al., 28 Apr 2026) applies conformal prediction to a discrete-state Markov sequence $\frac{\lceil (n+1)(1-\alpha)\rceil}{n}$9, with the target being a future state-sequence

$$\Pr\!\left(Y_{n+1}\notin \mathcal{C}(X_{n+1})\right)\le \alpha.$$0

In the conflict application, the four states are peacetime, escalation, war, and deescalation. The paper’s central methodological point is that pointwise exchangeability fails for Markov chains, so the replacement symmetry is partial exchangeability of full sequences. Two sequences of equal length are partially exchangeable if they have the same initial state and the same transition counts; under that condition they have the same probability (Basarkar et al., 28 Apr 2026).

The conformal construction is candidate-specific. For a proposed future sequence, the observed and proposed path is augmented to length $$\Pr\!\left(Y_{n+1}\notin \mathcal{C}(X_{n+1})\right)\le \alpha.$$1, a transition matrix is estimated on the augmented sequence, and the sequence is decomposed into permutable $$\Pr\!\left(Y_{n+1}\notin \mathcal{C}(X_{n+1})\right)\le \alpha.$$2-blocks, where $$\Pr\!\left(Y_{n+1}\notin \mathcal{C}(X_{n+1})\right)\le \alpha.$$3 is the last state of the proposed future sequence. The nonconformity score $$\Pr\!\left(Y_{n+1}\notin \mathcal{C}(X_{n+1})\right)\le \alpha.$$4 is defined for each allowed block permutation $$\Pr\!\left(Y_{n+1}\notin \mathcal{C}(X_{n+1})\right)\le \alpha.$$5, and a randomized conformal p-value $$\Pr\!\left(Y_{n+1}\notin \mathcal{C}(X_{n+1})\right)\le \alpha.$$6 is computed from the permutation scores. The prediction set is

$$\Pr\!\left(Y_{n+1}\notin \mathcal{C}(X_{n+1})\right)\le \alpha.$$7

State dependence enters here through the transition counts, the candidate future path itself, and the final state $$\Pr\!\left(Y_{n+1}\notin \mathcal{C}(X_{n+1})\right)\le \alpha.$$8, which determines the block structure (Basarkar et al., 28 Apr 2026).

The paper contrasts this conformal procedure with a likelihood-based baseline that sorts all future sequences by their Markov-model probability and retains the highest-probability-mass set reaching level $$\Pr\!\left(Y_{n+1}\notin \mathcal{C}(X_{n+1})\right)\le \alpha.$$9. That baseline is efficient if the model is correct, but it has no model-robust finite-sample validity. In simulation with a known four-state Markov generator, conformal prediction is well calibrated across forecast horizons $X=x$0 and target levels from $X=x$1 to $X=x$2. In real conflict data, the conformal method remains close to nominal coverage across horizons, whereas the likelihood method often overcovers at lower target levels and may fail at target coverage $X=x$3 when a transition never appeared in calibration data. The paper also emphasizes a practical limitation: when calibration data are dominated by a single state, the $X=x$4-block structure can become inefficient or unintuitive. A proposed workaround appends an artificial extra time point $X=x$5, but this changes the meaning of the prediction set by conditioning on a return to peacetime at the extra future time (Basarkar et al., 28 Apr 2026).

5. State-dependent bounds for verification and control

A separate branch of the literature uses state-dependent conformal prediction as a statistical interface for symbolic verification of autonomous systems with neural perception. "State-Dependent Conformal Perception Bounds for Neuro-Symbolic Verification of Autonomous Systems" (Waite et al., 28 Feb 2025) studies systems of the form

$X=x$6

and seeks high-confidence reachable sets satisfying

$X=x$7

The paper argues that previous CP-based verification methods are conservative because they use either global bounds or time-dependent bounds, whereas perception error is often heteroskedastic in the state space, not just in time. It therefore partitions the state space into $X=x$8 disjoint regions,

$X=x$9

and defines a piecewise-constant bound

$Y\mid X=x$0

For each region, the calibration score on trajectory $Y\mid X=x$1 is the maximum perception error over the segment lying inside that region,

$Y\mid X=x$2

together with a sentinel $Y\mid X=x$3. Region-wise conformal quantiles are then computed by

$Y\mid X=x$4

and a union-bound argument yields the global guarantee

$Y\mid X=x$5

Two gradient-free partition optimization methods are proposed: a genetic algorithm and simulated annealing. The objective can be either Experience Loss or Experience Time-Decay Loss, with the latter using $Y\mid X=x$6 to penalize earlier errors more strongly. In the Mountain Car case study, the baseline time-based CP has verification time $Y\mid X=x$7 s and max reachable set size $Y\mid X=x$8, whereas the best state-based configuration, GA + ETDL with $Y\mid X=x$9, has verification time $\eta(x)=\eta_i$0 s and max reachable set size $\eta(x)=\eta_i$1, about a $\eta(x)=\eta_i$2 reduction in max reachable set size (Waite et al., 28 Feb 2025).

"Statistical-Symbolic Verification of Perception-Based Autonomous Systems using State-Dependent Conformal Prediction" (Geng et al., 2 Dec 2025) develops the same state-space-partitioning idea further. It allocates region-specific confidence levels $\eta(x)=\eta_i$3 satisfying $\eta(x)=\eta_i$4, defines

$\eta(x)=\eta_i$5

and proves the trajectory-wide bound

$\eta(x)=\eta_i$6

If worst-case reachability is then performed under the disturbance bounds $\eta(x)=\eta_i$7, the resulting reachable sets satisfy

$\eta(x)=\eta_i$8

The paper optimizes both the partition $\eta(x)=\eta_i$9 and the confidence allocation $z_i=(x_i,y_i)$00 with a genetic algorithm using a weighted loss

$z_i=(x_i,y_i)$01

which emphasizes frequently visited regions and earlier time steps (Geng et al., 2 Dec 2025).

Because region-dependent disturbances increase hybrid-system branching, the paper also introduces a branch-merging reachability algorithm based on clustering Taylor-model branches and computing union enclosures: $z_i=(x_i,y_i)$02 The evaluation reports reduced conservatism compared with a time-based conformal baseline. In Mountain Car, the best state-based method has reachable set size $z_i=(x_i,y_i)$03 versus $z_i=(x_i,y_i)$04 for the best time-based baseline, with test coverage $z_i=(x_i,y_i)$05 versus $z_i=(x_i,y_i)$06. In the autonomous racing car case study, the best state-based method has reachable set size $z_i=(x_i,y_i)$07 versus $z_i=(x_i,y_i)$08 for the best time-based baseline, while achieving higher verified safe distances from the wall and empirical coverage often between $z_i=(x_i,y_i)$09 and $z_i=(x_i,y_i)$10 (Geng et al., 2 Dec 2025).

Across these verification papers, state dependence means that uncertainty is localized in the state space actually visited by the closed-loop system. The statistical calibration step and the symbolic reachability step remain distinct: conformal prediction supplies region-indexed disturbance bounds, and verification propagates those bounds through the known dynamics and controller (Waite et al., 28 Feb 2025, Geng et al., 2 Dec 2025).

6. Adaptive coverage policies, Bayesian reinterpretations, and recurring limitations

"Adaptive Coverage Policies in Conformal Prediction" (Gauthier et al., 5 Oct 2025) shifts state dependence from set geometry to the coverage level itself. Instead of fixing $z_i=(x_i,y_i)$11, it allows a data-dependent miscoverage level

$z_i=(x_i,y_i)$12

formalized by a policy

$z_i=(x_i,y_i)$13

which maps the sum of calibration scores and a test statistic to a miscoverage level. The policy is implemented as a neural network $z_i=(x_i,y_i)$14 trained by leave-one-out pseudo-episodes on the calibration set. For pseudo-episode $z_i=(x_i,y_i)$15, the training objective is

$z_i=(x_i,y_i)$16

which trades off prediction-set size against miscoverage. The validity mechanism is the use of e-values and post-hoc conformal inference, yielding

$z_i=(x_i,y_i)$17

In experiments on CIFAR-10, the learned adaptive policy produces smaller average set size than a fixed-e-value baseline, while preserving the post-hoc validity mechanism (Gauthier et al., 5 Oct 2025).

"Conformal Prediction as Bayesian Quadrature" (Snell et al., 18 Feb 2025) does not present a fully developed covariate-conditional or state-dependent conformal method, but it is relevant because it recasts conformal-style uncertainty quantification as a Bayesian decision problem indexed by a control parameter $z_i=(x_i,y_i)$18. The paper interprets split conformal and conformal risk control through loss functions and Bayes risk, then places a prior on the loss quantile function $z_i=(x_i,y_i)$19 rather than directly on a model parameter $z_i=(x_i,y_i)$20. Using the order-statistic spacing fact

$z_i=(x_i,y_i)$21

it derives a stochastic upper bound

$z_i=(x_i,y_i)$22

for posterior expected loss. Taking expectations recovers conformal risk control, and split conformal appears as a special case. The paper explicitly notes that its framework is compatible with letting $z_i=(x_i,y_i)$23 depend on input features or context, but it does not prove conditional coverage guarantees, nor does it provide a localized or weighted exchangeability argument. The extension to state-specific loss modeling is therefore conceptual rather than established (Snell et al., 18 Feb 2025).

Several recurring limitations organize the field. First, exact finite-sample conditional coverage is impossible in general, so methods such as CHR target finite-sample marginal validity together with asymptotic conditional behavior or oracle efficiency (Sesia et al., 2021). Second, when exchangeability fails, validity must be rebuilt from additional structure: strong mixing for kernel-weighted time-series methods, invariant measures for rolling dynamical-system guarantees, or partial exchangeability for Markov-sequence prediction (Lee et al., 2024, Bakhtiaridoust et al., 30 Jun 2026, Basarkar et al., 28 Apr 2026). Third, region-wise or horizon-wise decompositions often rely on union bounds, which preserve formal guarantees but can introduce conservatism; this is explicit in the verification literature and in multi-step propagation results for stochastic dynamics (Waite et al., 28 Feb 2025, Geng et al., 2 Dec 2025, Bakhtiaridoust et al., 30 Jun 2026). Fourth, Bayesian or adaptive reinterpretations widen the design space but do not by themselves solve the conditional-coverage problem: the Bayesian-quadrature formulation still assumes i.i.d. deployment data matching calibration data and bounded losses, while adaptive-coverage policies preserve validity through e-values rather than through covariate-conditional calibration in the classical sense (Snell et al., 18 Feb 2025, Gauthier et al., 5 Oct 2025).

Taken together, these works show that “state-dependent conformal prediction” is not a single method but a research program for replacing one-size-fits-all calibration by calibration rules indexed by covariates, residual-history states, invariant operating regimes, state-space regions, or example-specific coverage policies. The technical challenge in each case is the same: to exploit heterogeneity or dependence without discarding the finite-sample or asymptotic validity properties that make conformal inference attractive in the first place.