Online conformal prediction must balance fast adaptation to distribution shift against stable coverage: feedback-driven methods react quickly but become volatile, while strongly discounted Bayesian methods lag and inflate intervals at tight coverage. We introduce \textbf{State-Adaptive Bayesian Conformal Prediction (SA-BCP)}, which forms the predictive quantile as a gated convex combination of long-term temporal inertia and local spatial evidence from a kernel density estimate, controlled by a single interpretable evidence threshold $K$. We establish three results: (i) asymptotic marginal validity of the resulting intervals; (ii) a closed-form expression for the MSE-optimal threshold, $K^*_{\mathrm{MSE}}=α(1-α)/M^{\mathcal{T}}$, trading the coverage-indicator (Bernoulli) variance against the temporal structural bias $M^{\mathcal{T}}$; and (iii) a rolling-origin procedure for selecting $K$ online -- consistent under stationarity, with $O(\sqrt{T\log N})$ regret against the best fixed $K$ and, for a segmented variant, a sublinear dynamic-regret bound under bounded drift. Across four financial-volatility and weather datasets, three target coverage levels, and eight baselines (including the strongest recent conditional-quantile methods, SPCI and KOWCPI), SA-BCP attains at-or-above-nominal coverage in most settings while producing substantially sharper intervals -- up to roughly $3\times$ lower Winkler score than discounted Bayesian CP at the tightest coverage -- and a coverage-matched audit confirms these efficiency gains are not an artifact of under-coverage. We disclose one principal limitation: a volatility-specialized conformal-GARCH competitor remains more efficient on its home volatility-base series, though it does not transfer across domains.

翻译：在线共形预测需要在快速适应分布漂移与稳定覆盖率之间取得平衡：基于反馈的方法反应迅速但波动剧烈，而强折扣的贝叶斯方法在严格覆盖率下会滞后且区间膨胀。我们提出**状态自适应贝叶斯共形预测（SA-BCP）**，该方法通过一个可解释的单证据阈值$K$驱动的门控凸组合，将长期时间惯性与核密度估计的局部空间证据结合，形成预测分位数。我们建立了三个结论：（i）所得区间的渐近边际有效性；（ii）均方误差最优阈值的闭式表达式$K^*_{\mathrm{MSE}}=α(1-α)/M^{\mathcal{T}}$，该表达式权衡了覆盖指标（伯努利）方差与时间结构偏差$M^{\mathcal{T}}$；（iii）一种在线选择$K$的滚动原点程序——在平稳性假设下具有一致性，相对于最优固定$K$的遗憾界为$O(\sqrt{T\log N})$，且对于分段变体，在有界漂移下具有次线性动态遗憾界。在四个金融波动率和天气数据集、三个目标覆盖率水平以及八个基线方法（包括最强的近期条件分位数方法SPCI和KOWCPI）中，SA-BCP在大多数设置下达到或超过名义覆盖率，同时产生显著更窄的区间——在最严格覆盖率下，其Winkler得分比折扣贝叶斯共形预测低约3倍。覆盖匹配审计证实，这些效率提升并非欠覆盖的假象。我们揭示了一个主要局限性：一种专为波动率设计的共形GARCH竞争方法在其原生波动率基序列上仍保持更高效率，尽管它无法跨领域迁移。