post-hoc confidence sequences via e-processes

Let $\mathcal{P}$ be some set of distributions, and ${\mathcal{P}}_{\theta }\subset \mathcal{P}$ be those distributions which share some parameter $\theta$: eg the set of distributions on $[0,1]$ with mean $\theta$. We receive observations from $P\in {\mathcal{P}}_{{\theta }^{\ast }}$ for some ${\theta }^{\ast }$ and want to form a post-hoc CS (see post-hoc valid confidence sequences) for ${\theta }^{\ast }$.

Let $(E_t^{\theta }{)}_{t\ge 1}$ be an e-process for ${\mathcal{P}}_{\theta }$:

$$\underset{P\in {\mathcal{P}}_{\theta }}{\sup }\mathbb{E}[E_{\tau }^{\theta }]\le 1\text{for all stopping times }\tau .$$

For any $\alpha \in (0,1)$, let $C_t(\alpha )=\{\varphi :E_t^{\varphi }<1/\alpha \}$. (This is precisely how we’d built a CI or CS with an e-process or an e-value at a fixed time $t$). Then, $\{(C_t(\alpha ){)}_{t\ge 1}:\alpha \in (0,1)\}$ is immediately post-hoc valid: For any stopping time $\tau$,

$$\underset{P\in {\mathcal{P}}_{{\theta }^{\ast }}}{\sup }{\mathbb{E}}_P\left[\underset{\alpha \in (0,1)}{\sup }\frac{1({\theta }^{\ast }\notin C_{\tau }(\alpha ))}{\alpha }\right]=\underset{P\in {\mathcal{P}}_{{\theta }^{\ast }}}{\sup }{\mathbb{E}}_P\left[\underset{\alpha \in (0,1)}{\sup }\frac{1(E_{\tau }^{{\theta }^{\ast }}\ge 1/\alpha )}{\alpha }\right]\le \underset{P\in {\mathcal{P}}_{{\theta }^{\ast }}}{\sup }{\mathbb{E}}_P[E_{\tau }^{{\theta }^{\ast }}]\le 1,$$

where the second inequality follows from basic case analysis on the indicator.

Note that e-processes accomplish this for free. We didn’t have to add additional assumptions. Therefore procedures which are already working with e-processes (such as estimating means by betting) are immediately post-hoc valid in this sense.