As conservative p-values

For any e-variable ${\displaystyle S}$ and any ${\displaystyle 0<\alpha \leq 1}$ and all ${\displaystyle P\in H_{0}}$, it holds that

${\displaystyle P\left(S\geq {\frac {1}{\alpha }}\right)=P(S^{-1}\leq \alpha )\ {\overset {(*)}{\leq }}\ \alpha }$

In words: ${\displaystyle S^{-1}}$is a p-value, and the e-value based test with significance level ${\displaystyle \alpha }$, which rejects ${\displaystyle P_{0}}$ if ${\displaystyle S^{-1}{\leq }\alpha }$ , i.e. if ${\displaystyle S\geq 1/\alpha }$, has Type-I error bounded by ${\displaystyle \alpha }$. But, whereas with standard p-values the inequality (*) above is usually an equality (with continuous-valued data) or near-equality (with discrete data), this is not the case with e-variables. This makes e-value-based tests more conservative (less power) than those based on standard p-values, and it is the price to pay for safety (i.e. retaining Type-I error guarantees) under optional continuation and averaging.

As generalizations of likelihood ratios

Let ${\displaystyle H_{0}=\{P_{0}\}}$ be a simple null hypothesis. Let ${\displaystyle Q}$ be any other distribution on ${\displaystyle Y}$, and let

${\displaystyle S:={\frac {q(Y)}{p_{0}(Y)}}}$

be their likelihood ratio. Then ${\displaystyle S}$ is an e-variable. Conversely, any e-variable relative to a simple null ${\displaystyle H_{0}=\{P_{0}\}}$ can be written as a likelihood ratio with respect to some distribution ${\displaystyle Q}$. Thus, when the null is simple, e-variables coincide with likelihood ratios. E-variables exist for general composite nulls as well though, and they may then be thought of as generalizations of likelihood ratios. The two main ways of constructing e-variables, UI and RIPr (see below) both lead to expressions that are variations of likelihood ratios as well.

Two other standard generalizations of the likelihood ratio are (a) the generalized likelihood ratio as used in the standard, classical likelihood ratio test and (b) the Bayes factor. Importantly, neither (a) nor (b) are e-variables in general: generalized likelihood ratios in sense (a) are not e-variables unless the alternative is simple (see below under "universal inference"). Bayes factors are e-variables if the null is simple. To see this, note that, if ${\displaystyle {\mathcal {Q}}=\{Q_{\theta }:\theta \in \Theta \}}$ represents a statistical model, and ${\displaystyle w}$ a prior density on ${\displaystyle \Theta }$, then we can set ${\displaystyle Q}$ as above to be the Bayes marginal distribution with density

${\displaystyle q(Y)=\int q_{\theta }(Y)w(\theta )d\theta }$

and then ${\displaystyle S=q(Y)/p_{0}(Y)}$ is also a Bayes factor of ${\displaystyle H_{0}}$ vs. ${\displaystyle H_{1}:={\mathcal {Q}}}$. If the null is composite, then some special e-variables can be written as Bayes factors with some very special priors, but most Bayes factors one encounters in practice are not e-variables and many e-variables one encounters in practice are not Bayes factors.^[1]

As bets

Suppose you can buy a ticket for 1 monetary unit, with nonnegative pay-off ${\displaystyle S=S(Y)}$. The statements "${\displaystyle S}$ is an e-variable" and "if the null hypothesis is true, you do not expect to gain any money if you engage in this bet" are logically equivalent. This is because ${\displaystyle S}$ being an e-variable means that the expected gain of buying the ticket is the pay-off minus the cost, i.e. ${\displaystyle S-1}$, which has expectation ${\displaystyle \leq 0}$. Based on this interpretation, the product e-value for a sequence of tests can be interpreted as the amount of money you have gained by sequentially betting with pay-offs given by the individual e-variables and always re-investing all your gains.^[2]

The betting interpretation becomes particularly visible if we rewrite an e-variable as ${\displaystyle S:=1+\lambda U}$ where ${\displaystyle U}$ has expectation ${\displaystyle \leq 0}$ under all ${\displaystyle P\in H_{0}}$ and ${\displaystyle \lambda \in {\mathbb {R} }}$ is chosen so that ${\displaystyle S\geq 0}$ a.s. Any e-variable can be written in the ${\displaystyle 1+\lambda U}$ form although with parametric nulls, writing it as a likelihood ratio is usually mathematically more convenient. The ${\displaystyle 1+\lambda U}$ form on the other hand is often more convenient in nonparametric settings. As a prototypical example,^[3] consider the case that ${\displaystyle Y=(X_{1},\ldots ,X_{n})}$ with the ${\displaystyle X_{i}}$ taking values in the bounded interval ${\displaystyle [0,1]}$. According to ${\displaystyle H_{0}}$, the ${\displaystyle X_{i}}$ are i.i.d. according to a distribution ${\displaystyle P}$ with mean ${\displaystyle \mu }$; no other assumptions about ${\displaystyle P}$ are made. Then we may first construct a family of e-variables for single outcomes, ${\displaystyle S_{i,\lambda }:=1+\lambda (X_{i}-\mu )}$, for any ${\displaystyle \lambda \in [-1/(1-\mu ),1/\mu ]}$ (these are the ${\displaystyle \lambda }$ for which ${\displaystyle S_{i,\lambda }}$ is guaranteed to be nonnegative). We may then define a new e-variable for the complete data vector ${\displaystyle Y}$ by taking the product

${\displaystyle S:=\prod _{i=1}^{n}S_{i,{\breve {\lambda }}|X^{i-1}}}$,

where ${\displaystyle {\breve {\lambda }}|X^{i-1}}$ is an estimate for ${\displaystyle {\lambda }}$, based only on past data ${\displaystyle X^{i-1}=(X_{1},\ldots ,X_{i-1})}$, and designed to make ${\displaystyle S_{i,\lambda }}$ as large as possible in the "e-power" or "GRO" sense (see below). Waudby-Smith and Ramdas use this approach to construct "nonparametric" confidence intervals for the mean that tend to be significantly narrower than those based on more classical methods such as Chernoff, Hoeffding and Bernstein bounds.^[3]

Simple Alternative, Simple Null: Likelihood Ratio

Let ${\displaystyle H_{0}=\{P_{0}\}}$ and ${\displaystyle H_{1}=\{Q\}}$ both be simple. Then the likelihood ratio e-variable ${\displaystyle S=q(Y)/p_{0}(Y)}$ has maximal e-power in the sense above, i.e. it is GRO.^[1]

Simple Alternative, Composite Null: Reverse Information Projection (RIPr)

Let ${\displaystyle H_{1}=\{Q\}}$ be simple and ${\displaystyle H_{0}=\{P_{\theta }:\theta \in \Theta _{0}\}}$ be composite, such that all elements of ${\displaystyle H_{0}\cup H_{1}}$ have densities (denoted by lower-case letters) relative to the same underlying measure. Grünwald et al. show that under weak regularity conditions, the GRO e-variable exists, is essentially unique, and is given by

${\displaystyle S:={\frac {q(Y)}{p_{\curvearrowleft Q}(Y)}}}$

where ${\displaystyle p_{\curvearrowleft Q}}$is the Reverse Information Projection (RIPr) of ${\displaystyle Q}$ unto the convex hull of ${\displaystyle H_{0}}$.^[4] Under further regularity conditions (and in all practically relevant cases encountered so far), ${\displaystyle p_{\curvearrowleft Q}}$is given by a Bayes marginal density: there exists a specific, unique distribution ${\displaystyle W}$ on ${\displaystyle \Theta _{0}}$ such that ${\displaystyle p_{\curvearrowleft Q}(Y)=\int _{\Theta _{0}}p_{\theta }(Y)dW(\theta )}$.

Simple Alternative, Composite Null: Universal Inference (UI)

In the same setting as above, ^[6] show that, under no regularity conditions at all,

${\displaystyle S={\frac {q(Y)}{\sup _{P\in H_{0}}p(Y)}}\left(={\frac {q(Y)}{{p}_{{\hat {\theta }}\mid Y}(Y)}}\right)}$

is an e-variable (with the second equality holding if the MLE (maximum likelihood estimator) ${\displaystyle {\hat {\theta }}\mid Y}$ based on data ${\displaystyle Y}$ is always well-defined). This way of constructing e-variables has been called the universal inference (UI) method, "universal" referring to the fact that no regularity conditions are required.

Composite Alternative, Simple Null

Now let ${\displaystyle H_{0}=\{P\}}$ be simple and ${\displaystyle H_{1}=\{Q_{\theta }:\theta \in \Theta _{1}\}}$ be composite, such that all elements of ${\displaystyle H_{0}\cup H_{1}}$ have densities relative to the same underlying measure. There are now two generic, closely related ways of obtaining e-variables that are close to growth-optimal (appropriately redefined^[1] for composite ${\displaystyle H_{1}}$): Robbins' method of mixtures and the plug-in method, originally due to Wald ^[7] but, in essence, re-discovered by Philip Dawid as "prequential plug-in" ^[8] and Jorma Rissanen as "predictive MDL".^[9] The method of mixtures essentially amounts to "being Bayesian about the numerator" (the reason it is not called "Bayesian method" is that, when both null and alternative are composite, the numerator may often not be a Bayes marginal): we posit any prior distribution ${\displaystyle W}$ on ${\displaystyle \Theta _{1}}$ and set

${\displaystyle {\bar {q}}_{W}(Y):=\int _{\Theta _{1}}q_{\theta }(Y)dW(\theta )}$

and use the e-variable ${\displaystyle {\bar {q}}_{W}(Y)/p(Y)}$.

To explicate the plug-in method, suppose that ${\displaystyle Y=(X_{1},\ldots ,X_{n})}$ where ${\displaystyle X_{1},X_{2},\ldots }$ constitute a stochastic process and let ${\displaystyle {\breve {\theta }}\mid X^{i}}$ be an estimator of ${\displaystyle \theta \in \Theta _{1}}$ based on data ${\displaystyle X^{i}=(X_{1},\ldots ,X_{i})}$ for ${\displaystyle i\geq 0}$. In practice one usually takes a "smoothed" maximum likelihood estimator (such as, for example, the regression coefficients in ridge regression), initially set to some "default value" ${\displaystyle {\breve {\theta }}\mid X^{0}:=\theta _{0}}$. One now recursively constructs a density ${\displaystyle {\bar {q}}_{\breve {\theta }}}$ for ${\displaystyle X^{n}}$ by setting ${\displaystyle {\bar {q}}_{\breve {\theta }}(X^{n})=\prod _{i=1}^{n}q_{{\breve {\theta }}\mid X^{i-1}}(X_{i}\mid X^{i-1})}$ .

Effectively, both the method of mixtures and the plug-in method can be thought of learning a specific instantiation of the alternative that explains the data well.^[1]

Composite Null & Alternative

In parametric settings, we can simply combine the main methods for the composite alternative (obtaining ${\displaystyle {\bar {q}}_{\breve {\theta }}}$ or ${\displaystyle {\bar {q}}_{W}}$) with the main methods for the composite null (UI or RIPr, using the single distribution ${\displaystyle {\bar {q}}_{\breve {\theta }}}$ or ${\displaystyle {\bar {q}}_{W}}$ as an alternative). Note in particular that when using the plug-in method together with the UI method, the resulting e-variable will look like

${\displaystyle {\frac {\prod _{i=1}^{n}q_{{\breve {\theta }}\mid X^{i-1}}(X_{i})}{q_{{\hat {\theta }}\mid X^{n}}(X^{n})}}}$

which resembles, but is still fundamentally different from, the generalized likelihood ratio as used in the classical likelihood ratio test.

The advantage of the UI method compared to RIPr is that (a) it can be applied whenever the MLE can be efficiently computed - in many such cases, it is not known whether/how the reverse information projection can be calculated; and (b) that it 'automatically' gives not just an e-variable but a full e-process (see below): if we replace ${\displaystyle n}$ in the formula above by a general stopping time ${\displaystyle \tau }$, the resulting ratio is still an e-variable; for the reverse information projection this automatic e-process generation only holds in special cases.

Its main disadvantage compared to RIPr is that it can be substantially sub-optimal in terms of the e-power/GRO criterion, which means that it leads to tests which also have less classical statistical power than RIPr-based methods. Thus, for settings in which the RIPr-method is computationally feasible and leads to e-processes, it is to be preferred. These include the z-test, t-test and corresponding linear regressions, k-sample tests with Bernoulli, Gaussian and Poisson distributions and the logrank test (an R package is available for a subset of these), as well as conditional independence testing under a model-X assumption.^[10] However, in many other statistical testing problems, it is currently (2023) unknown whether fast implementations of the reverse information projection exist, and they may very well not exist (e.g. generalized linear models without the model-X assumption).

In nonparametric settings (such as testing a mean as in the example above, or nonparametric 2-sample testing), it is often more natural to consider e-variables of the ${\displaystyle 1+\lambda U}$ type. However, while these superficially look very different from likelihood ratios, they can often still be interpreted as such and sometimes can even be re-interpreted as implementing a version of the RIPr-construction.^[1]

Finally, in practice, one sometimes resorts to mathematically or computationally convenient combinations of RIPr, UI and other methods.^[1] For example, RIPr is applied to get optimal e-variables for small blocks of outcomes and these are then multiplied to obtain e-variables for larger samples - these e-variables work well in practice but cannot be considered optimal anymore.

A Third Construction Method: p-to-e (and e-to-p) calibration

There exist functions that convert p-values into e-values.^[11][12][13] Such functions are called p-to-e calibrators. Formally, a calibrator is a nonnegative decreasing function ${\displaystyle f:[0,1]\rightarrow [0,\infty ]}$ which, when applied to a p-variable (a random variable whose value is a p-value), yields an e-variable. A calibrator ${\displaystyle f}$ is said to dominate another calibrator ${\displaystyle g}$ if ${\displaystyle f\geq g}$, and this domination is strict if the inequality is strict. An admissible calibrator is one that is not strictly dominated by any other calibrator. One can show that for a function to be a calibrator, it must have an integral of at most 1 over the uniform probability measure.

One family of admissible calibrators is given by the set of functions ${\displaystyle \{f_{\kappa }:0<\kappa <1\}}$ with ${\displaystyle f_{\kappa }(p):=\kappa p^{\kappa -1}}$. Another calibrator is given by integrating out ${\displaystyle \kappa }$:

${\displaystyle \int _{0}^{1}\kappa p^{\kappa -1}d\kappa ={\frac {1-p+p\log p}{p(-\log p)^{2}}}}$

Conversely, an e-to-p calibrator transforms e-values back into p-variables. Interestingly, the following calibrator dominates all other e-to-p calibrators:

${\displaystyle f(t):=\min(1,1/t)}$.

While of theoretical importance, calibration is not much used in the practical design of e-variables since the resulting e-variables are often far from growth-optimal for any given ${\displaystyle H_{1}}$.^[4]

Definition

Now consider data ${\displaystyle X_{1},X_{2},\ldots }$ arriving sequentially, constituting a discrete-time stochastic process. Let ${\displaystyle S_{1},S_{2},\ldots }$ be another discrete-time process where for each ${\displaystyle n,S_{n}}$ can be written as a (measurable) function of the first ${\displaystyle (X_{1},\ldots ,X_{n})}$ outcomes. We call ${\displaystyle S_{1},S_{2},\ldots }$ an e-process if for any stopping time ${\displaystyle \tau ,S_{\tau }}$ is an e-variable, i.e. for all ${\displaystyle P\in H_{0}:{\mathbb {E} }_{P}[S_{\tau }]\leq 1}$.

In basic cases, the stopping time can be defined by any rule that determines, at each sample size ${\displaystyle n}$, based only on the data observed so far, whether to stop collecting data or not. For example, this could be "stop when you have seen four consecutive outcomes larger than 1", "stop at ${\displaystyle n=100}$", or the level-${\displaystyle \alpha }$-aggressive rule, "stop as soon as you can reject at level ${\displaystyle \alpha }$-level, i.e. at the smallest ${\displaystyle n}$ such that ${\displaystyle S_{n}\geq 1/\alpha }$", and so on. With e-processes, we obtain an e-variable with any such rule. Crucially, the data analyst may not know the rule used for stopping. For example, her boss may tell her to stop data collecting and she may not know exactly why - nevertheless, she gets a valid e-variable and Type-I error control. This is in sharp contrast to data analysis based on p-values (which becomes invalid if stopping rules are not determined in advance) or in classical Wald-style sequential analysis (which works with data of varying length but again, with stopping times that need to be determined in advance). In more complex cases, the stopping time has to be defined relative to some slightly reduced filtration, but this is not a big restriction in practice. In particular, the level-${\displaystyle \alpha }$-aggressive rule is always allowed. Because of this validity under optional stopping, e-processes are the fundamental building block of confidence sequences, also known as anytime-valid confidence intervals.^[14][1]

Technically, e-processes are generalizations of test supermartingales, which are nonnegative supermartingales with starting value 1: any test supermartingale constitutes an e-process but not vice versa.

Construction

E-processes can be constructed in a number of ways. Often, one starts with an e-value ${\displaystyle S_{i}}$ for ${\displaystyle X_{i}}$ whose definition is allowed to depend on previous data, i.e.

for all ${\displaystyle P\in H_{0}:{\mathbb {E} }_{P}[S_{i}|X_{1},\ldots ,X_{i-1}]\leq 1}$

(again, in complex testing problems this definition needs to be modified a bit using reduced filtrations). Then the product process ${\displaystyle E_{1},E_{2},\ldots }$ with ${\displaystyle E_{n}=S_{1}\cdot S_{2}\cdot \cdots S_{n}}$ is a test martingale, and hence also an e-process (note that we already used this construction in the example described under "e-values as bets" above: for fixed ${\displaystyle \lambda }$ , the e-values ${\displaystyle S_{i,\lambda }}$ were not dependent on past-data, but by using ${\displaystyle \lambda ={\breve {\lambda }}|X^{i-1}}$ depending on the past, they became dependent on past data).

Another way to construct an e-process is to use the universal inference construction described above for sample sizes ${\displaystyle 1,2,\ldots }$ The resulting sequence of e-values ${\displaystyle S_{1},S_{2},\ldots }$ will then always be an e-process. ^[1]