Game-Theoretic Statistics and Safe Anytime-Valid Inference

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arXiv:2210.01948v2 [math.ST] 17 Jun 2023

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Game-Theoretic Statistics and

Safe Anytime-Valid Inference

Aaditya Ramdas, Peter Grünwald, Vladimir Vovk and Glenn Shafer

Abstract. Safe anytime-valid inference (SAVI) provides measures of statisti-

cal evidence and certainty—e-processes for testing and conﬁdence sequences

for estimation—that remain valid at all stopping times, accommodating con-

tinuous monitoring and analysis of accumulating data and optional stopping

or continuation for any reason. These measures crucially rely on test martin-

gales, which are nonnegative martingales starting at one. Since a test mar-

tingale is the wealth process of a player in a betting game, SAVI centrally

employs game-theoretic intuition, language and mathematics. We summa-

rize the SAVI goals and philosophy, and report recent advances in testing

composite hypotheses and estimating functionals in nonparametric settings.

Key words and phrases: Test martingales, Ville’s inequality, universal in-

ference, reverse information projection, e-process, optional stopping, con-

ﬁdence sequence, nonparametric composite hypothesis testing.

1. INTRODUCTION

Stop when you are ahead. Increase your bet to make up

ground when you are behind. This is called martingaling

in the casino. It often succeeds in the short or medium

term, leading novice gamblers to think they can beat the

odds and day traders to think they can beat the market

(Dimitrov, Shafer and Zhang,2022). The same delusion

arises in science, where sampling until a signiﬁcant result

is obtained is an important source of irreproducibility.

The fallacy of sampling until a signiﬁcant result is ob-

tained has been discussed by statisticians at least since the

1940s, when Feller (1940) saw it happening in the study

of extra-sensory perception. Anscombe (1954) famously

called it “sampling to a foregone conclusion”, and this in-

evitability was also pointed out by Robbins (1952).

But disapproval by statisticians has hardly dented the

prevalence of the practice. In one widely publicized ex-

ample, a team of researchers apparently demonstrated

beneﬁts from “power posing” (Carney, Cuddy and Yap,

2010). The lead author later disavowed the conclusion and

identiﬁed the team’s peeking at the data as one of her rea-

sons (Carney, Fact 5):

We ran subjects in chunks and checked the ef-

fect along the way. It was something like 25

Carnegie Mellon University, Pittsburgh, USA (e-mail:

aramdas@cmu.edu). Centrum Wiskunde &Informatica,

Amsterdam, Netherlands (e-mail: pdg@cwi.nl). Royal

Holloway London, UK (e-mail: v.vovk@rhul.ac.uk). Rutgers

University, USA (e-mail: gshafer@business.rutgers.edu).

subjects run, then 10, then 7, then 5. Back then

this did not seem like p-hacking. It seemed like

saving money (assuming your eﬀect size was

big enough and p-value was the only issue).

Ten years ago, an anonymous survey of over 2000 psy-

chologists found 56% admitting to “deciding whether to

collect more data after looking to see whether the results

were signiﬁcant” (John, Loewenstein and Prelec,2012).

Bayesian inference with a prior deﬁned by a statisti-

cian’s beliefs before seeing any of the data is not aﬀected

by (planned) peeking. Problems quickly arise, however,

when default or pragmatic priors are used to test com-

posite null hypotheses. These problems are especially se-

vere for commonly used pragmatic priors that depend on

the sample size, covariates, or other aspects of the data

(De Heide and Grünwald,2021).

As emphasized by Johari et al. (2022); Howard et al.

(2021); Grünwald, De Heide and Koolen (2023); Shafer

(2021); Pace and Salvan (2020), amongst others, we need

to go beyond disapproval of peeking, and we instead

should give researchers tools to fully accommodate it.

The branch of mathematical statistics that enables this,

sequential analysis, was brilliantly launched in the 1940s

and 1950s by Wald, Anscombe, Robbins, and others. The

innovations introduced by Robbins, Darling, Siegmund

and Lai included conﬁdence sequences that are valid at

any and all times and tests of power one. But these ideas

occupied only a small niche in sequential analysis re-

search until around 2017. Since then, interest has ex-

ploded and much conceptual progress has been made in

parallel threads, which we attempt to summarize.

This new methodology diﬀers from traditional statisti-

cal testing in the way it quantiﬁes evidence against sta-

tistical hypotheses. The traditional approach casts doubt

on a hypothesis when a selected test statistic takes too ex-

treme a value. This leads to quantifying evidence against

the hypothesis by the p-value—the probability the hy-

pothesis assigns to the test statistic being so large. The

new methodology instead casts doubt on a hypothesis

when a selected nonnegative statistic is large relative to

its expected value. Imagining that we bought the statis-

tic for its expected value when we selected it, we call

the ratio of its realized to its expected value a betting

score and take this as a measure of our evidence. In the

case of a composite hypotheses, we use the inﬁmum of

betting scores for the multiple hypotheses and call this

an e-value. The sequential analog is an e-process—a se-

quence of e-values that monitor the accumulation of ev-

idence. E-processes permit anytime-valid inference; we

can repeatedly decide whether to collect more data based

on the current e-value without invalidating later assess-

ments, stopping whenever and for any reason whatsoever.

This anytime-validity is a form of safety. This safety may

come with a price, of course; there may (or may not) be

tradeoﬀs between safety and power; see Section B.

From a technical point of view, the new methodol-

ogy is based on the concept of a test martingale, along

with its betting interpretation. Although martingales be-

came important in probability theory more than a half-

century ago, their potential has still not been fully ex-

ploited in statistics, and the new emphasis on nonnega-

tive (super)martingales has produced a plethora of power-

ful new methods. These include conﬁdence sequences for

many functionals that can be used with multi-armed ban-

dits and new sequential tests for composite null hypothe-

ses. This responds to the need for rigorous methods in

settings that have emerged with the development of infor-

mation technology in the past half-century, including “liv-

ing meta-analysis” (Ter Schure, Grünwald and Ly,2021),

the industrial use of A/B testing (Johari et al.,2022) and

bandit experiments (Howard and Ramdas,2022).

The new methods can be most clearly presented in the

language of game-theoretic probability (Shafer and Vovk,

2001,2019). Here successive observations are Reality’s

moves in a game. Two other players move before Reality

on each round: Forecaster gives probabilities for the out-

come, and Skeptic bets by choosing a real-valued func-

tion of the outcome, paying its expected value, and receiv-

ing its realized value. If Skeptic always chooses nonneg-

ative functions, then the factor by which he multiplies his

money (the ratio of the realized to the expected value un-

der forecaster’s probabilities) is his “betting score” or “e-

value” (Shafer,2021). If he reinvests his money on each

round, the betting scores multiply, producing cumulative

betting scores that are products of the betting scores for

each round so far. Because Skeptic is a free agent, the

option of stopping or continuing or even switching to a

diﬀerent experiment on the next round is intrinsic to the

game, and the cumulative betting score or e-value quanti-

ﬁes the evidence against the Forecaster (and his probabil-

ities): Skeptic refutes the odds by making money betting

at those odds; more money is more evidence that the odds

do not reﬂect reality.

Betting games often ﬁt statistical practice better than

measure-theoretic probability models. In particular, they

accommodate fully the opportunistic behavior that we

want to allow. George Barnard, in his review of Wald’s

book on sequential analysis (Barnard,1947), called for

embedding statisticians in the sequential decision-making

of experimental scientists, in which each batch of obser-

vations is followed by deliberation about whether to stop

or to continue, perhaps with a modiﬁed experiment. The

use of a prespeciﬁed stopping time, which prescribes con-

tinuing only until a certain data-dependent condition is

met, obscures or erases this sequential deliberation, pre-

tending that all the decisions ﬂow from a stopping strategy

adopted in advance. Barnard’s suggestion is better cap-

tured by our game-theoretic framework, where a single

stopping rule is replaced by notions of evidence that re-

main valid at any stopping time not speciﬁed in advance.

Because most readers will be unfamiliar with game-

theoretic probability as developed by Shafer and Vovk

(2001,2019), we use the relatively familiar apparatus of

measure theory (ﬁltrations, stopping times, martingales,

etc.) and new concepts deﬁned within that apparatus (e-

values, e-processes, etc.). Frequently, however, we return

to the betting story, where our martingales are wealth pro-

cesses for Skeptic.

1.1 Basic terminology

We begin with a sample space Ωequipped with a ﬁl-

tration F≡(Ft)t≥0(an increasing nested sequence of σ-

ﬁelds), and a set Πof probability distributions on (Ω,F).

We assume that some distribution P∈Πgoverns our data

X≡(X1,X2,...). The variables X1,X2,... need not be in-

dependent and identically distributed (iid) under P. We

use Xtas a shorthand for X1,...,Xt.

When we say we are testing P, we mean that we are

testing the null hypothesis H0that P∈P. When we say

we are testing Pagainst Q, we mean that the alternative

hypothesis H1is that P∈Q. Typically, Pand Qare either

non-intersecting or nested subsets of Π. We always use

boldface P,Qfor sets of distributions, and a normal P,Q

for a single distribution.

A sequence of random variables Y≡(Yt)t≥0is called a

process if it is adapted to F—i.e., if Ytis measurable with

respect to Ftfor every t. Often Ft:=σ(Xt), with F0being

trivial (F=∅,Ω}), and in this case Ytbeing measurable

with respect to Ftmeans that Ytis a measurable function

GAME-THEORETIC STATISTICS AND SAFE ANYTIME-VALID INFERENCE 3

of X1,...,Xt. But Fis sometimes a coarser ﬁltration (we

discard information, see e.g. Section 4.1.2) or a richer one

(we add external randomization).1Yis called predictable

if Ytis measurable with respect to Ft−1.

A stopping time (or rule) τis a nonnegative integer val-

ued random variable such that {τ≤t} ∈ Ftfor each t≥0.

In words: we know at each time whether the rule is telling

us to stop or keep going. Denote by Tthe set of all stop-

ping times, including ones that may never stop.

1.2 A terse technical summary of the paper

We give a short technical summary below, foreshadow-

ing topics to be deﬁned and discussed in more depth later.

The ﬁeld of safe anytime-valid inference (SAVI) aims

to develop measures of statistical evidence and cer-

tainty that remain valid at arbitrary stopping times (pos-

sibly unknown in advance), accommodating continuous

monitoring and analysis of accumulating data and op-

tional stopping or continuation for any reason. There is

a strong sense in which admissible SAVI methods —

power-one tests, conﬁdence sequences, anytime-valid p-

values and e-processes — must rely centrally on nonneg-

ative martingales (Ramdas et al.,2020). Nonnegative (su-

per)martingales are endowed with a strong and direct con-

nection to gambling: every nonnegative supermartingale

corresponds to a wealth process in some game, and vice

versa (every “fair/legal” gambling strategy to test the null

hypothesis results in a wealth process that is a nonnega-

tive supermartingale).

These facts give rise to the following central principle

in game-theoretic statistics: “testing by betting”. In order

to test a null hypothesis Pagainst an alternative Q, we

set up a game such that (a) if the null is true, meaning

P∈P, then no betting strategy can reliably make money

(any gambler’s wealth is a nonnegative supermartingale),

and (b) if the null is false, meaning P∈Q, it is possible

to bet smartly to make money in that game. This principle

arguably has roots dating back (at least) to Ville (1939),

and was recently discussed in depth in the point null case

by Shafer et al. (2011) and Shafer (2021) and for com-

posite nulls by Grünwald, De Heide and Koolen (2023);

Waudby-Smith and Ramdas (2023), etc.

The game works as follows. Before observing Xt∈ X,

Skeptic puts forward a bet St:X → [0,∞], which satisﬁes

EP[St(Xt)] ≤1 for every P∈P.(1)

Then, Xtis revealed. The interpretation is that at each time

t, one can buy, at the price of 1 monetary unit, a ticket

1The ﬁltration may be coarsened, as explained by Alan Turing

(c 1941, p.1): “When the whole evidence. . . is taken into account it

may be extremely diﬃcult to estimate the probability of the event,

. . . may be better to form an estimate based on a part of the evidence

...”

that will pay oﬀSt(Xt) units. One can buy as many tick-

ets as one likes. (1) simply expresses that under the null,

one does not expect to get back more than one invests in

this game. At time 1, Skeptic invests 1 monetary unit; at

each time t, she reinvests all the money she observed so

far. Skeptic’s wealth after tsteps is then clearly given by

i=1Si(Xi), which is nonnegative by deﬁnition, and eas-

ily checked to be a supermartingale under P. If Qis ap-

propriately separated from P, good betting strategies can

force the wealth in setting (b) (alternative is true) to grow

to inﬁnity exponentially fast, and we wish to maximize

the exponent. Maximizing the exponent corresponds to

maximizing the expected logarithm of the wealth; such a

“log-optimality” objective has information-theoretic roots

dating back to Kelly (1956) and Breiman (1961), but also

(implicitly or explicitly) appears in the work of Ville,

Wald, Robbins, etc.

For testing a point null Pagainst a point alternative Q,

the log-optimal bet is simply given by the likelihood ratio

St=dQt/dPt, where Ptand Qtare the conditional prob-

abilities for the tth observation given the past, under P

and Qrespectively. Thus the realized likelihood ratio of

Qagainst Pis precisely the optimal wealth of a gambler

betting against P. This central fact provides much intu-

ition for extensions and generalizations.

For composite alternatives Q, the Skeptic often hedges

their bets by not betting all their money on a single Q∈Q,

instead spreading their investment over Qusing a mixture

(“prior”) distribution R. To illustrate using the toy case in

which Qis countable and Rhas mass function r,r(q)=a

would mean that a fraction aof Skeptic’s money is in-

vested in q— importantly in general Rneither has a fre-

quentist (‘drawing from an urn’) nor a Bayesian (‘belief’)

interpretation here. This method of mixtures plays a cen-

tral role in this paper: an instance of Laplace’s method

for approximating a maximum by an integral, it appears

directly within our anytime-valid context in Ville (1939)

and Robbins (1970), and in broader sequential contexts in

Wald (1947a) and Cover (1974), among many others.

The most interesting questions in this area involve com-

posite (and often nonparametrically speciﬁed) nulls P.

Indeed, there really was no general theory for dealing

with composite nulls until 2017 — when, almost out of

the blue, several generic proposals for dealing with com-

posite nulls appeared. Arguably, it is this development

which caused the aforementioned explosion of interest in

the area — suddenly there was an indication that eventu-

ally almost any interesting statistical testing or estimation

problem could be converted into an anytime-valid version

with a gambling interpretation.

For such composite P, a fascinating phenomenon some-

times presents itself: for some “extremely rich” nulls P,

the game described above is hopelessly restraining: the

constraint (1) is too stringent, and the only functions St

that satisfy it are either constant or decreasing (meaning

that they cannot increase under any alternative). This hap-

pens, for example, when testing exchangeability or testing

log-concavity; see Section 5.5 for references and details.

Luckily, generalizing the above game protocol resusci-

tates the approach. There appear to be two diﬀerent types

of generalized games: (a) one can restrict the amount of

information available to the Skeptic by introducing a third

player (an “Intermediary”) who throws away some infor-

mation revealed by Reality (mathematically, Skeptic op-

erates in a shrunk ﬁltration), (b) one can instead make

the Skeptic play many games in parallel, each against a

diﬀerent subset of P, with the Skeptic’s net wealth be-

ing their worst wealth across all the parallel games. In

the ﬁrst case, Skeptic’s wealth may remain a nonnega-

tive (super)martingale, but in the second case, their wealth

is an e-process (under the null, their wealth is upper-

bounded by a diﬀerent supermartingale in each game,

and thus is bounded by one at any stopping time). While

these solutions may seem almost magical at ﬁrst glance,

they both yield fruit for the same problem mentioned

above of testing exchangeability: approach (a) is used

in Vovk, Gammerman and Shafer (2022, Part III) and ap-

proach (b) in Ramdas et al. (2022). The latter work, along

with Ruf et al. (2022), together show the centrality of e-

processes in game-theoretic statistics: e-processes exist

for many Pfor which nonnegative (super)martingales do

not.

When Pand Qhave a common reference measure,

meaning that likelihood-ratio based methods are still in

play, two key ideas stand out: universal inference (Wasserman, Ramdas and Balakrishnan,

2020), and the reverse information projection (Grünwald, De Heide and Koolen,

2023). The former always yields an e-process, but lat-

ter always results in an e-value which can be multiplied

across batches of data to yield a supermartingale. But

sometimes the latter also directly yields an e-process (and

when it does, it dominates universal inference).

When Pand Qdo not have any common reference mea-

sures — and thus likelihood-ratio based methods may not

make any sense at the outset — the design of nonneg-

ative (super)martingales or e-processes occupies center

stage. Sometimes, the nonparametric deﬁnition of Pdi-

rectly yields a natural game, like when testing if a “sub-

Gaussian” mean is positive (Darling and Robbins,1967).

Other times, one must design new games in possibly

shrunk ﬁltrations, which may not be obvious at the outset,

like in two-sample testing (Shekhar and Ramdas,2021).

The entire discussion above was centered on testing by

betting, because this typically forms the technical heart

of other problems that are not cast explicitly as testing.

For example, appropriate duality concepts and inversions

allow us to translate many of these results into those for

estimation of appropriate functionals using conﬁdence se-

quences (Section 5). Both e-processes and conﬁdence se-

quences can in turn be extended to other problems like

change detection (Section 5.7), model selection, etc.

In fact, our investigations reveal a curious phenomenon:

at the heart of many (and plausibly, all) nonparametric

testing and estimation problems is a “hidden” game (of-

ten not unique: the same Pand Qmay be associated

with diﬀerent ﬁltrations and betting strategies that are e-

processses under Pand make money under Q). Further,

explicitly bringing out such games (and betting well in

them) can yield powerful new methodology as well as

new theoretical insights (Howard et al.,2020)

A full understanding of when and why this happens

is open, but we provide one hint here. Likelihood ratios

have been at the center of statistics for nearly a century.

Nonnegative (super)martingales and e-processes are sim-

ply nonparametric, composite generalizations of likeli-

hood ratios, and these have been found to exist in dozens

of problems where one cannot even begin to talk about

likelihood-ratio based methods. Thus, these tools give

us a way to work implicitly with likelihood ratios, even

when there appears to be no explicit way to do so. Given

the power (and sometimes optimality) of likelihood-ratio

based tests in parametric settings, we perhaps get a hint of

the power of our game-theoretic approaches in composite

(often nonparametric) settings.

The rest of this paper will formally deﬁne the key con-

cepts, and provide technical details of the aforementioned

methods and phenomena in diﬀerent problem settings.

2. CENTRAL CONCEPTS

In the sequel, we leave measurability assumptions and

other measure-theoretic details implicit so far as possible.

2.1 E-values

An e-variable for Pis a nonnegative random variable E

such that EP[E]≤1 for all P∈P. Its realized value, after

observing the data, is an e-value.2Often we call Eitself

an e-value, blurring the distinction between the random

variable and its realized value. (The term “p-value” is also

often used for both random variables and their values.)

When EP[E]=1, we call the e-value Eaunit bet

against P. This name evokes a story in which expected

values are prices of payoﬀ: the Forecaster predicts that

X∼P, and in order to bet against them, a Skeptic could

buy one unit of E, for the price of 1, delivering the Skep-

tic a payoﬀof E(X). Mathematically, a unit bet against P

is simply3a likelihood ratio dQ/dP for some alternative

Q. This is elementary when we use probability densities:

2Observe that we use boldface Efor expectation and normal Efor

e-values. The “e” in e-value stands both for “evidence” (because it

quantiﬁes statistical evidence against the null) and for “expectation”

(because its central property is its expectation).

3In some sense, statisticians have always been using e-values (and

test martingales), because likelihood ratios are the most important ex-

ample of e-values (and test martingales). But this direct analog only

holds when testing a single distribution P. The power and utility of e-

values, test (super)martingales and e-processes are truly realized only

dealing with a composite (and sometimes nonparametric) P.

GAME-THEORETIC STATISTICS AND SAFE ANYTIME-VALID INFERENCE 5

•EP[E]=1 can be written as RE(x)p(x)dx=1, so

that q:=E×pis a density, and E=q/p.

•If qand pare Q’s and P’s densities, then EP[q/p]=

Rp(x)q(x)

p(x)dx =1.

We use e-values when data are treated as a batch.

Their dynamic counterparts are test martingales and e-

processes, introduced next.

2.2 Test (Super)Martingales

A process Mis a martingale for Pif

(2) EP[Mt|Ft−1]=Mt−1

for all t≥1. Mis a supermartingale for Pif it satisﬁes (2)

with “=” relaxed to “≤”. A (super)martingale is called a

test (super)martingale if it is nonnegative and M0=1.

Game-theoretically, a test martingale for Pis the wealth

process of a gambler who bets against P. If Mis a test

martingale for P, then EP[Mt]=1 for any t≥0, and

thus each Mtis itself a unit bet against P; it is the fac-

tor by which Mmultiplies its money from time 0 to time

t. Similarly, the optional stopping theorem implies that

for any stopping time τ— even potentially inﬁnite —

EP[Mτ]≤1, and thus each Mτis also an e-value for P.

The correspondence between unit bets against Pand

likelihood ratios with Pas the denominator extends to a

related correspondence for test martingales for P. If Qis

absolutely continuous with respect to P, we can write

(3) q(Xt)

p(Xt)=q(X1)

p(X1)

q(X2|X1)

p(X2|X1)··· q(Xt|Xt−1)

p(Xt|Xt−1),

where Xt:=(X1,...,Xt), p(Xt) is P’s density for Xt, and

q(Xt) is Q’s density for Xt. Denote the sequence deﬁned

by (3) as M; then Mis a test martingale for P, and

Mt=

i=1

Bi=q(Xt)

p(Xt),(4)

where Bt:=q(Xt|Xt−1)

p(Xt|Xt−1).(5)

Note that each Btis a unit bet against P, conditional on

Ft−1; we call BtM’s unit bet on round t.

Test martingales for Pare always of the form (4). So

choosing a test martingale for Pcomes down to choosing

an alternative Q. In applications, constructing a test mar-

tingale for Pusually amounts to constructing the numer-

ator q(Xt|Xt−1) in (5); see Section 3.2. Test supermartin-

gales can also be decomposed in the style of (4), where

the Btare single-round e-values (i.e. deﬁned as function

on a single outcome Xt) conditional on Ft−1.

Test martingales become more interesting objects in the

composite setting.

2.3 Composite Test (Super)Martingales

A process Mis a test (super)martingale for Pif it

is a test (super)martingale for every P∈P. Such com-

posite test (super)martingales are important in this pa-

per. Composite test martingales also decompose as in (4):

for every P∈P, there is a Qthat is absolutely contin-

uous with respect to Pand satisﬁes Mt=q(Xt)/p(Xt);

see Ramdas et al. (2020, Proposition 4). In other words,

composite test martingales are simultaneous likelihood

ratios.

Trivially, the constant process Mt=1 is a test martin-

gale for any P, and a decreasing process is a test super-

martingale for any P. We call a test (super)martingale

nontrivial if it is not always a constant (or decreasing)

process. In particular, we would like test martingales for P

that increase to inﬁnity under the alternative Q. But there

may be no nontrivial test martingales if Pis too large.

In this case there may still be nontrivial test supermartin-

gales (Section 5.1), but even these may not exist (Sec-

tion 5.5). For this reason, we also need e-processes.

2.4 E-processes

A family (MP)P∈Pis a test martingale family if MPis

always a test martingale for P. A nonnegative process E

is called an e-process for Pif there is a test martingale

family (MP)P∈Psuch that

(6) Et≤MP

tfor every P∈P,t≥0.

This type of deﬁnition was used by Howard et al. (2020),

who used the name “sub-ψprocess”. In parallel, Grünwald, De Heide and Koolen

(2023) implicitly deﬁned an e-process for P, also without

using the name “e-process”, as a nonnegative process E

such that

E[Eτ]≤1 for every τ∈ T ,P∈P.

In words, Emust be an e-value at any stopping time.

Ramdas et al. (2020) proved that the two deﬁnitions are

equivalent and that if Pis “locally dominated”, then ad-

missible4e-processes (see Section 8.2.3) must satisfy

(7) Et=inf

P∈PMP

for some test martingale family (MP)P∈P. (Technically, the

inf above is an “essential inﬁmum”.)

Whereas a test martingale for Pis the wealth process

of a gambler who bets against P, an e-process for Pre-

ports the minimum wealth across many simultaneous bet-

ting games, one against each P∈P, all with the same out-

comes X1,X2,... (Ramdas et al.,2022, Section 5.4).

4An e-process E≡(Et)t≥1for Pis inadmissible if there exists an-

other e-process E′for Psuch that E′≥E(E′

t≥Etalmost surely P,

for all P∈Pand all t≥1), and E′

t>Etwith positive probability under

some P∈Pand some t≥1; Eis admissible if it is not inadmissible.

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arXiv:2210.01948v2[math.ST]17Jun2023SubmittedtoStatisticalScienceGame-TheoreticStatisticsandSafeAnytime-ValidInferenceAadityaRamdas,PeterGrünwald,VladimirVovkandGlennShaferAbstract.Safeanytime-validinference(SAVI)providesmeasuresofstatisti-calevidenceandcertainty—e-processesfortestingandconﬁdenceseq...

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Game-Theoretic Statistics and Safe Anytime-Valid Inference

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