We consider a setting where we collect samples from two distinct groups, denoted and . In both groups, data come in sequentially and are i.i.d. We thus have two data streams, i.i.d. and i.i.d. where we assume that ,
representing some parameterized underlying family of distributions, all assumed to have a probability density or mass function denoted byon some outcome space .
-variables (Grünwald et al., 2019; Vovk and Wang, 2021) are a tool for constructing tests that keep their Type-I error control under optional stopping and continuation. Previously, Turner et al. (2021) developed
-variables for testing equality of both groups, i.e. with null hypothesis. Here we first generalize these -variables to more general null hypotheses in which we may have . We then use these generalized -variables to construct anytime-valid confidence sequences; these provide confidence sets that remain valid under optional stopping and continuation (Darling and Robbins, 1967; Howard et al., 2021).
As in (Turner et al., 2021), we first design -variables for a single block of data , where a block is a set of data consisting of outcomes in group and outcomes in group , for some pre-specified and . An -variable
is then, by definition, any nonnegative random variablesuch that
Turner et al. (2021) first defined such an -variable for so that it would tend to have high power against a given simple alternative . Their -variable is of the following simple form (with ):
These -variables can be extended to sequences of blocks by multiplication, and can be extended to composite alternatives by sequentially learning from the data, for example via a Bayesian prior on . The and used for the -th block are allowed to depend on past data, but they must be fixed before the first observation in block occurs. For simplicity, in this note we only consider the case with and that remain fixed throughout; extension to the general case is straightforward. By a general property of -variables, at each point in time, the running product of block -variables observed so far is itself an -variable, and the random process of the products is known as a test martingale (Grünwald et al., 2019; Shafer, 2021). An -variable-based test at level is a test with, in combination with any stopping rule , reports ‘reject’ if and only if the product of -values corresponding to all blocks that were observed at the stopping time and have already been completed, is larger than . Such a test has a Type-I error probability bounded by irrespective of the stopping time that was used; see the aforementioned references for much more detailed introductions and, for example (Henzi and Ziegel, 2021), for a practical application. In case is convex, the -variable (2) has the so-called GRO-(growth-rate-optimality) property: it maximizes, over all -variables (i.e. over all nonnegative random variables satisfying (1)) the logarithmic growth rate
which implies that, under , the expected number of data points before the null can be rejected is minimized (Grünwald et al., 2019).
Below, in Theorem 1 in section 2, which generalizes Theorem 1 in Turner et al. (2021), we extend (2) to the case of general null hypotheses, , allowing for the case that the elements of have two different components, and provide a condition under which it has the GRO property. From then onwards we focus on what we call ‘the contingency table setting’ in which both streams are Bernoulli, denoting the probability of in group . For this case, Theorem 2 gives a simplified expression for the -variable and shows that the GRO property holds if is convex. Then we will extend this -variable to deal with composite and use this to define anytime-valid confidence sequences. We illustrate these through simulations. All proofs are in Appendix A.
2 General Null Hypotheses
In this section, we first construct an -variable for general null hypotheses that generalizes (2). We then instantiate the new result to the case. Our goal is thus to define an -variable for a block of data points with points in group , . For notational convenience we define, for ,
as the joint distribution ofand , so that so that we can write the null hypothesis as . Our strategy will be to first develop an –variable for a modified setting in which there is only a single outcome, falling with probability in group and in group .
To this end, for , we define , all distributions with a refering to the modified setting with just one outcome. We let be the set of all distributions on with finite support. For , we define . We set . We further define, for given alternative , , to be, if it exists, the conditional probability density satisfying
with the distribution for with . Clearly we can rephrase (4) equivalently as:
where is the KL divergence. Here we extended the conditional distributions and (corresponding to densities and ) to a joint distribution by setting (and similarly for ) and we extended . We have now constructed a modified null hypothesis of joint distributions for a single ‘group’ outcome and ‘data’ outcome . We let be the convex hull of .
The satisfying (5) is commonly called the reverse information projection of onto . Li (1999) shows that always exists, though in some cases it may represent a sub-distribution (integrating to strictly less than one); see (Grünwald et al., 2019, Theorem 1) who, building on Li’s work, established a general relation between reverse information projection and –variables. Part 1 of that theorem establishes that if the minimum in (4) (or (5)) is achieved by some then and, for all ,
This expresses that is an -variable for our modified problem, in which within a single block we observe a single outcome in group , with chosen with probability . If were to interpret the –variable of the modified problem as in (6) as a likelihood ratio for a single outcome, its corresponding likelihood ratio for a single block of data in our original problem with outcomes in group would look like:
The following theorem expresses that this ‘extension’ of the -variable in the modified problem gives us an -variable in our original problem:
In the case that is not convex and closed, we do not have a simple expression for in general, and we may have to find it numerically by minimizing (4). In the table (Bernoulli ) case though, there are interesting for which the corresponding is convex, and we shall now see that this leads to major simplifications.
2.1 General Convex for the contingency table
In this subsection and the next, refers to the model again. We now let be any closed convex subset of that contains a point in the interior of . Again, note that the corresponding need not be convex; still, , the null hypothesis for the modified problem as defined above, must be convex if is convex, and this will allow us to design -variables for such . Let with in the interior of , and let
stand for the KL divergence between and restricted to a single block (note that in the previous subsection, KL divergence was defined for a single outcome ). The following result makes crucial use of Theorem 1:
is uniquely achieved by some . If , then . Otherwise, lies on the boundary of , but not on the boundary of . The –variable (7) is given by the distribution that puts all its mass on , i.e.
is an -variable. Moreover, this is the -GRO -variable relative to .
We can extend this -variable to the case of a composite by learning the true from the data (Turner et al., 2021). We thus replace, for each , for the block consisting of points in group and points in group , the ‘true’ for
by an estimatebased on the previous data blocks. The -variable corresponding to blocks of data then becomes
where, for ,
can be an arbitrary estimator (function from to )
is defined to achieve
. No matter what estimator we choose, (10) gives us an -variable. In Section 3, as in (Turner et al., 2021), we implement this estimator by fixing a prior and using the Bayes posterior mean, .
Let us now illustrate Theorem 2 for two choices of .
with linear boundary
First, we let , for , stand for any straight line through : . This can be extended to and similarly to . For example, we could take to be the solid line in Figure 1(a) (which would correspond to ), or the whole area underneath the line () including the line itself, or the whole area above it including the line itself ().
Now consider a that has nonempty intersection with the interior of and that is separated from the point alternative , i.e. . Simple differentiation gives that the minimum is achieved by the unique satisfying:
which can now be plugged into the -variable (9) if the alternative is the simple alternative, or otherwise into its sequential form (10). In the basic case in which }, the solution to (11) reduces to the familiar from Turner et al. (2021).
If lies above the line , then by Theorem 2, must lie on . Theorem 2 then gives that it must be achieved by the satisfying (11). Similarly, if lies below the line , then is again achieved by the satisfying (11).
with log odds ratio boundary
Similarly, we can consider that correspond to a given log odds effect size . That is, we now take
For example, we could now take to be the area under the curve (including the curve boundary itself) in Figure 1(b), which would correspond to . Now let and point alternative be such that and is separated from , i.e. . Let . As Figure 1(b) suggests, is convex. Theorem 2 now tells us that is achieved by . Plugging these into (9) thus gives us an -variable. can easily be determined numerically. Similarly, if , is convex and closed and if is separated from , the minimizing KL divergence on gives an -variable relative to .
3 Anytime-Valid Confidence for the case
We will now use the -variables defined above to construct anytime-valid confidence sequences. Let be a notion of effect size as before. A -anytime-valid (AV) confidence sequence (Darling and Robbins, 1967; Howard et al., 2021) is a sequence of random (i.e. determined by data) subsets of , with being a function of the first data blocks , such that for all ,
We first consider the case in which for all values that can take, is a convex set. Fix a prior on . Based on (10) we can make an exact (nonasymptotic) AV confidence sequence
by Ville’s inequality (Grünwald et al., 2019; Turner et al., 2021). Here the are not necessarily intervals, but, potentially loosing some information, we can make a AV confidence sequence consisting of intervals by defining to be the smallest interval containing . We can also turn any confidence sequences into an alternative AV confidence sequence with sets that are always a subset of by taking the running intersection
In this form, the confidence sequences can be interpreted as the set of ’s that have not yet been rejected in a setting in which, for each null hypothesis we stop and reject as soon as the corresponding -variable exceeds . The running intersection can also be applied to the intervals .
To simplify calculations, it is useful to take a prior under which and
have independent beta distributions with parameters. We can, if we want, infuse some prior knowledge or hopes by setting these parameters to certain values — our confidence sequences will be valid irrespective of our choice (Howard et al., 2021). In case no such knowledge can be formulated (as in the simulations below), we advocate the prior, which, among all priors of the simple form asymptotically achieves the REGROW criterion (a criterion related to minimax log-loss regret, see (Grünwald et al., 2019)), i.e for the case we set to an independent beta prior on and with as was empirically found to be the ‘best’ value (Turner et al., 2021).
Log Odds Ratio Effect Size
The situation is slightly trickier if we take the log odds ratio as effect size, for is then not convex. Without convexity, Theorem 2 cannot be used and hence the validity of AV confidence sequences as constructed above breaks down. We can get nonasymptotic anytime-valid confidence sequences after all as follows. First, we consider a one-sided AV confidence sequence for the submodel of positive effect sizes , defining
where we note that is convex (since ) and also contains with . This confidence sequence can give a lower bound on . Analogously, we consider a one-sided AV confidence sequence for the submodel , defining
and derive an upper bound on . By Theorem 2, both sequences and are AV confidence sequences for the submodels with and respectively. Defining , we find, for with ,
and analogously for with . We have thus arrived at a confidence sequence that works for all , positive or negative.
In this section some numerical examples of confidence sequences for the two types of effect sizes are given. All simulations were run with code available in our software package (Turner et al., 2022).
Figure 2 shows running intersections of confidence sequences with as the additive effect size for simulations for various distributions and stream lengths. It appears for the linear boundary on is an interval, corresponding to the ‘beam’ of bounded by the lines and with being values such that . Figure B.1 in the Appendix illustrates that the running intersection indeed improves the confidence sequence, albeit slightly.
Log odds ratio boundary
If the ML estimate based on lies in the upper left corner as in Figure 3(a), the confidence sets we get at time have a one-sided shape such as the shaded region, or the shaded region in Figure 3(c), if the ML parameters lie in the lower right corner. Again, we can improve these confidence sequences by taking the running intersection; running intersections over time are illustrated in Figures 3(b) and 3(d).
We have shown how -variables for data streams can be extended to general null hypotheses and non-asymptotic always-valid confidence sequences. We specifically implemented the confidence sequences for the contingency tables setting; the resulting confidence sequences are efficiently computed and show quick convergence in simulations. For estimating absolute differences between proportions in two groups, to our knowledge, such exact confidence sequences did not yet exist. For the log odds ratio we could also have used the sequential probability ratio (SPR) in Wald’s SPR test (Wald, 1945) test, which can be re-interpreted as a (product of) -variables (Grünwald et al., 2019). However, the SPR does not satisfy the GRO property making it sub-optimal (see also (Adams, 2020)); moreover, as should be clear from the development, our method for constructing confidence sequences can be implemented for any effect size notion with convex rejection sets and , not just the log odds ratio. A main goal for future work is to use Theorem 2 to provide such sequences for sequential two-sample settings that go beyond the table.