E-values measure evidence through a null expectation bound. Build a coin-toss e-process, apply Ville’s inequality and distinguish fixed-time validity from safe stopping.
E-values and always-valid inference
An e-value is an observed nonnegative evidence score whose expected value is at most one under the null hypothesis. Always-valid inference uses evidence procedures that preserve their error guarantee when the researcher chooses when to stop after seeing the accumulating data. For a quant monitoring a live research experiment, this makes the difference between a valid sequential decision and repeatedly checking a fixed-sample test until it happens to reject.
The sequential object is an e-process. A collection of individually valid fixed-time e-values does not automatically have its stopping-time guarantee. This article concerns statistical evidence; the similarly named epidemiological E-value measures sensitivity to unmeasured confounding and is a different concept.
One score, then a process
Write E for the random score before observing the data and e for its realised value. For a null family P0, fixed-time validity means
E≥0,EP[E]≤1for every P∈P0.
Markov's inequality then gives PP(E≥1/α)≤α. A score at least 20 therefore supports a level-5% rejection for that valid test. An e-value is not constrained to lie between zero and one, and a larger value points against the null. Vovk and Wang develop this expectation-based definition and its combination rules.
Let Fn record the information available after observation n. An e-process (En) is nonnegative, observable from that information, and satisfies EP[Eτ]≤1 at every finite stopping timeτ, for every null distribution P. A stopping time decides whether to stop using information available now; choosing a favourable historical endpoint with hindsight is different.
A nonnegative supermartingale starting at one is a sufficient construction:
E0=1,EP[En∣Fn−1]≤En−1.
The general e-process class is broader; an e-process need not itself be a supermartingale. The null must be specified fully enough to verify the required guarantee, including nuisance parameters in a composite null.
Eight heads cross a boundary that six heads do not
Consider independent coin tosses. The null is a fair coin, p0=0.5, and the alternative is fixed in advance at p1=0.75. Heads is coded as one and tails as zero. This is a deliberately simple probability model, not an assumption about financial returns.
The likelihood ratio of the alternative to the null, after hn heads and tn tails, is
Every head multiplies the score by 1.5; every tail halves it. Under the null, the next multiplier has conditional mean
0.5(1.5)+0.5(0.5)=1.
Consequently (En) is a nonnegative null martingale, hence an e-process. The calculation uses the next toss being fair conditional on the past, rather than merely a 50% unconditional head frequency.
Observations on the illustrated path
Current score
Has the path crossed 20?
Six consecutive heads
11.390625
No
Eight consecutive heads
25.62890625
Yes
Eight heads, then one tail
12.814453125
Yes, at observation 8
Constructed path of eight heads then a tail: E starts at 1, reaches 11.390625 after six heads and 25.62890625 after eight, crossing 20. The tail lowers E to 12.814453125, while the always-valid p-value stays approximately 0.039018.
Original constructed coin path under a fixed 0.75 alternative against a fair-coin null. Current evidence can fall; the running maximum records the earlier crossing. Figure values are rounded.
Ville's inequality provides the simultaneous guarantee:
PH0(∃n≥1:En≥20)≤0.05.
This bounds the chance of ever crossing, not the chance of crossing at a single preselected time. The SAVI paper, §§2.4–2.5, states the e-process guarantee and Ville's bound.
Under the null, the particular path of eight initial heads has probability 2−8=1/256=0.00390625. That is not the total probability of eventual rejection: other paths cross later. For example, one initial tail followed by ten heads gives 0.5×1.510=28.83251953125. The 5% bound covers all crossing paths together; it does not assert equality to 5%.
Turning the running maximum into an always-valid p-value
Define
pnav=min{1,max0≤k≤nEk1}.
Since E0=1, the outer minimum is redundant here. For every u∈(0,1), the event that this p-value ever falls to u is the event that the e-process reaches 1/u. Ville's inequality therefore gives
PP(∃n:pnav≤u)≤u.
This is a p-value process that remains valid when inspected and stopped using the available information. Johari, Pekelis and Walsh, §4, formalise this stopping-time interpretation of always-valid inference.
After eight heads, p8av=1/25.62890625≈0.039018. If a tail arrives next, the current score halves and its reciprocal rises to approximately 0.078037. The running-maximum construction keeps p9av≈0.039018. These are different summaries: current accumulated score versus the smallest rejection level attained so far. The running maximum itself is generally not an e-process.
Neither summary is a posterior probability that the coin is fair. In this simple two-model example, the likelihood ratio can update prior odds to posterior odds, but prior odds must still be supplied. It is not a probability of replication or a measure of economic profit.
Multiplication needs a conditional check
To build a product En=∏i=1nBi, with E0=1, require nonnegative factors satisfying
EP[Bi∣Fi−1]≤1
under every null distribution. Pulling the already observed product outside the conditional expectation proves the supermartingale inequality. The Conditional Expectation lesson develops the tower and pull-out properties used here; its full teaching requires membership, with a public preview.
Marginal validity alone does not justify multiplying scores. For example, let B=2 on a single fair head and zero on a tail. Then E[B]=1, but multiplying the same result twice gives E[B2]=2. Reusing evidence has broken validity. Independent valid factors suffice, but independence is stronger than the conditional condition needed above; Vovk and Wang, §4, make this distinction.
For research monitoring, the guarantee follows the specified data model and evidence construction. Refitting the alternative to all observations and reporting an unadjusted maximised likelihood ratio is not the fixed-alternative calculation above. Adaptation can be valid when bets use only past information and retain conditional validity. Serially dependent returns, overlapping observations or a changing null require a construction that covers those features. Monitoring one valid process also does not automatically resolve multiple hypothesis testing across many strategies.
