Empirical-Bernstein confidence-sequence interval update

Inputs are real bounds $a\le b$, a nominal error level $\delta \in (0,1)$, an observation stream $(X_t{)}_{t\ge 1}$ with $X_t\in [a,b]$ almost surely and $X_t$ measurable with respect to ${\mathcal{F}}_t$, a forecast stream $({\widehat{X}}_t{)}_{t\ge 1}$ with ${\widehat{X}}_t\in [a,b]$ almost surely and ${\widehat{X}}_t$ measurable with respect to ${\mathcal{F}}_{t-1}$, and, when $c=b-a>0$, an evaluation procedure for a Borel one-sided sub-exponential uniform boundary $u_{\delta \mathrm{/}2,c}$ with crossing probability $\delta \mathrm{/}2$ and scale $c$. The output is one interval $C_t$ at each positive integer time $t$.

1. If $c=0$, return $C_t=[a,a]$ at each time.

2. If $c>0$, initialize ${\bar{X}}_0=0$ and $V_0=0$. After receiving forecast ${\widehat{X}}_t$ and then observation $X_t$, update ${\bar{X}}_t=(1-\frac{1}{t}){\bar{X}}_{t-1}+\frac{X_t}{t},V_t=V_{t-1}+(X_t-{\widehat{X}}_t{)}^2\mathrm{.}$ Return $C_t=[{\bar{X}}_t-\frac{u_{\delta \mathrm{/}2,c}(V_t)}{t},{\bar{X}}_t+\frac{u_{\delta \mathrm{/}2,c}(V_t)}{t}]\mathrm{.}$

The procedure uses observations only through time $t$. The accumulated quantity $V_t$ is the observed sum of squared forecast errors; it is not a predictable conditional-variance process.

Filtration and filtered probability space: ${\mathcal{F}}_t$ records information available at integer time $t$.

Predictable discrete time process: the forecast at time $t$ is measurable with respect to ${\mathcal{F}}_{t-1}$ before the observation at time $t$.

One-sided sub-exponential uniform boundary: the supplied boundary has the stated scale and crossing-probability guarantee.

Confidence sequence for a time-varying target: the output intervals are considered for simultaneous coverage of a target process.

A Howard et al., Theorem 4, §4.1