The Topology of Statistical Verifiability

The Topology of Statistical Verifiability

Konstantin Genin \IfArrayPackageLoaded
Department of Philosophy Carnegie Mellon University Pittsburgh, Pennsylvania
Department of Philosophy Carnegie Mellon University Pittsburgh, Pennsylvania
konstantin.genin@gmail.com
   Kevin T. Kelly \IfArrayPackageLoaded
Department of Philosophy Carnegie Mellon University Pittsburgh, Pennsylvania
Department of Philosophy Carnegie Mellon University Pittsburgh, Pennsylvania
kk3n@andrew.cmu.edu
Abstract

Topological models of empirical and formal inquiry are increasingly prevalent. They have emerged in such diverse fields as domain theory [2, 17], formal learning theory [19], epistemology and philosophy of science [11, 16, 9, 10, 3], statistics [7, 8] and modal logic [18, 5]. In those applications, open sets are typically interpreted as hypotheses deductively verifiable by true propositional information that rules out relevant possibilities. However, in statistical data analysis, one routinely receives random samples logically compatible with every statistical hypothesis. We bridge the gap between propositional and statistical data by solving for the unique topology on probability measures in which the open sets are exactly the statistically verifiable hypotheses. Furthermore, we extend that result to a topological characterization of learnability in the limit from statistical data.

J. Lang (Ed.): TARK 2017 EPTCS 251, 2017, pp. The Topology of Statistical VerifiabilityLABEL:LastPage, doi:10.4204/EPTCS.251.17

The Topology of Statistical Verifiability

Konstantin Genin \IfArrayPackageLoaded
Department of Philosophy
Carnegie Mellon University
Pittsburgh, Pennsylvania
Department of Philosophy
Carnegie Mellon University
Pittsburgh, Pennsylvania
konstantin.genin@gmail.com and Kevin T. Kelly \IfArrayPackageLoaded
Department of Philosophy
Carnegie Mellon University
Pittsburgh, Pennsylvania
Department of Philosophy
Carnegie Mellon University
Pittsburgh, Pennsylvania
kk3n@andrew.cmu.edu

1 Verifiability from Propositional Information

The results in this section appear in [6], [9], and [3], but we restate them here to clarify the intended analogy between propositional and statistical verifiability. Let be a set of possible worlds, or possibilities one takes seriously, consistent with the background assumptions of inquiry. A proposition is identified with the set of worlds in which it is true, so propositions are subsets of . Let be arbitrary propositions. Logical operations correspond to set-theoretic operations in the usual way: is conjunction, is disjunction, is negation, and is deductive entailment of by . Finally, is deductively valid iff and is deductively contradictory iff .

In the propositional information setting, information states are propositions that rule out relevant possibilities. For every in , let be the set of all information states true in . It is assumed that is non-empty (at worst, one receives the trivial information ). Furthermore, it is assumed that for each in , there exists in such that . The underlying idea is that a sufficiently diligent inquirer in eventually receives information as strong as an arbitrary information state in . Since that is true of both and , there must be true information as strong as .

Example 1.1.

Let be the set of all infinite binary sequences. Each world determines an infinite sequence of observable outcomes. Let be the initial segment of of length . Let be the set of all worlds having as an initial segment. Let be the set of all for every . Think of the length of the initial segment observed as the “stage” of inquiry. There is exactly one such information state in at every stage, and is entailed by for every .

Example 1.2.

Let be the set of all real numbers. Think of the possible “stage-” information states in as the open intervals of width that contain . Then is the set of all intervals containing of width , for some natural number . It follows that for every there is a stage such that every stage- information state in entails .

Let , the set of all possible information states. It follows from the two assumptions on that is a topological basis. Therefore, the closure of topological basis under union, denoted , is a topological space. We assume that is countable, since any language in which the data are recorded is at most countably infinite. The elements of are called open sets. The complements of open sets are called closed sets. A clopen set is both open and closed. A locally closed set is the intersection of an open and a closed set. Information state verifies proposition iff entails . The interior of a proposition , denoted , is the set of all worlds , such that there is verifying . Hence, is the set of worlds in which is eventually verified by information. It is an elementary result that is open iff . The closure of , denoted , is the set of all worlds in which is compatible with all information, i.e. . The boundary of , denoted , is defined as . Every information state consistent with is consistent with both and .

A method is a function from information states to propositions. Method is infallible iff its output is always true, i.e. iff for all .111This is equivalent to requiring that the method’s conclusions are deductively entailed by the data, i.e. that for all information states . For this reason, infallible methods are deductive and vice-versa. Suppose that one desires to arrive at true belief concerning some proposition without exposing oneself to the possibility of error. A verifier for is an infallible method that converges to belief in iff is true. That is, is a verifier for iff

  1. is infallible and

  2. iff there is such that for all entailing .

Say that is verifiable iff there exists a verifier for . Say that is refutable iff its complement is verifiable, and that is decidable iff is both verifiable and refutable. For example, if you are observing a computation by an unknown program, it is verifiable that that the program will halt at some point, but it is not verifiable that it will never halt. In the setting of Example 1.1, it is verifiable that a zero will be observed at some stage, but not that it will be observed at every stage. Verifiability is fundamentally a topological concept.

Theorem 1.1.

Proposition is verifiable iff is open.

Theorem 1.1 implies that if is not open, then there is in general no error-avoiding method that arrives at true belief in . Every method that converges to true belief in worlds in which is never verified must leap beyond the information available, and expose itself to error thereby.

The infallibility requirement is too strict to allow for inductive learning that draws conclusions beyond the information provided. The following success criterion is less demanding. A limiting verifier for is a method that converges to true belief in iff is true. That is, is a limiting verifier for iff it satisfies V2. Say that is limiting verifiable iff there exists a limiting verifier of . In the setting of Example 1.1, no method verifies the constantly-zero hypothesis , but that hypothesis is verified in the limit by the method that conjectures as long as it is not refuted by information. The following is a topological characterization of the propositions that are verifiable in the limit:

Theorem 1.2.

Proposition is limiting verifiable iff is a countable union of locally closed sets. If is metrizable, then is limiting verifiable iff is a countable union of closed sets.

Finally, an empirical problem is a countable partition of the worlds in into a set of answers. For , write for the answer true in . A method is a solution to iff it converges, on increasing information, to the true answer in , i.e. iff for every , there exists such that for all entailing . A problem is solvable iff it has a solution.

Theorem 1.3.

Problem is solvable iff every answer is a countable union of locally closed sets.

Theorems like 1.1, 1.2, and 1.3 constitute an exact correspondence between topology and learnability.

2 Verifiability from Statistical Information

There is a seeming gulf between propositional information and statistical samples. Propositional information literally rules out relevant possibilities. In sharp contrast, a random sample is often logically compatible with every possible probability distribution. We sidestep that fundamental difficulty by solving for the unique topology in which the open sets are precisely the statistically verifiable propositions, which provides an exact, statistical analogue of Theorem 1.1.

2.1 Samples and Worlds

A sample space is a set of possible random samples equipped with a topology generated by a basis . The worlds in assign probabilities to every set in , the Borel -algebra generated by the topology on . The topology on the sample space reflects what is verifiable about the sample itself. As in the purely propositional setting, it is verifiable that sample lands in iff is open, and it is decidable whether sample falls into region iff is clopen. For example, suppose that region is the closed interval , and suppose that the sample happens to land right on the end-point of . Suppose, furthermore, that given enough time and computational power, the sample can be specified to arbitrary, finite precision. But no finite degree of precision: ; ; ; suffices to determine that is truly in . But the mere possibility of a sample hitting the boundary of does not matter statistically, if the chance of obtaining such a sample is zero. A Borel set for which is said to be almost surely clopen (decidable) in .222A set that is almost surely clopen in is sometimes called a continuity set of . Borel set is almost surely clopen iff it is almost surely clopen in every in , and a collection of Borel sets is almost surely clopen iff every element of is almost surely clopen.

Example 2.1.

Consider the outcome of a single coin flip. The set of possible outcomes is . Since every outcome is decidable, the appropriate topology on the sample space is , the discrete topology on . Let be the set of all probability measures assigning a bias to the coin. Since every element of is clopen, every element is also almost surely clopen.

Example 2.2.

Consider the outcome of a continuous measurement. Then the sample space is the set of real numbers. Let the basis of the sample space topology be the usual interval basis on the reals. That captures the intuition that it is verifiable that the sample landed in some open interval, but it is not verifiable that it landed exactly on the boundary of an open interval. There are no nontrivial decidable (clopen) propositions in that topology. However, in typical statistical applications, contains only probability measures that assign zero probability to the boundary of an arbitrary open interval. Therefore, every open interval is almost surely decidable, i.e. .

Product spaces represent the outcomes of repeated sampling. Let be an index set, possibly infinite. Let be sample spaces, each with basis . Define the product of the as follows: let be the Cartesian product of the ; let be the product topology, i.e. the topology in which the open sets are unions of Cartesian products , where each is an element of , and all but finitely many are equal to . When is finite, the products of basis elements in are the intended basis for . Let be the -algebra generated by . Let be a probability measure on , the Borel -algebra generated by the . The product measure is the unique measure on such that, for each expressible as a Cartesian product of , where all but finitely many of the are equal to , . Let denote the -fold product of with itself.

2.2 Statistical Tests

A statistical method is a measurable function from random samples to propositions over .333The -algebra on the range of the method is assumed to be the power set. A test of a statistical hypothesis is a statistical method . Call the acceptance region, and the rejection region of the test.444The acceptance region is , rather than , because failing to reject licenses only the trivial inference . The power of test is the worst-case probability that it rejects truly, i.e. . The significance level of a test is the worst-case probability that it rejects falsely, i.e. .

A test is feasible in iff its acceptance region is almost surely decidable in . Say that a test is feasible iff it is feasible in every world in . More generally, say that a method is feasible iff the preimage of every element of its range is almost surely decidable in every world in . Tests that are not feasible in are impossible to implement — as described above, if the acceptance region is not almost surely clopen in , then with non-zero probability, the sample lands on the boundary of the acceptance region, where one cannot decide whether to accept or reject. If one were to draw a conclusion at some finite stage, that conclusion might be reversed in light of further computation. Tests are supposed to solve inductive problems, not to generate new ones. Therefore we consider only feasible methods in the following development.

2.3 The Weak Topology

A sequence of measures converges weakly to , written , iff for every almost surely clopen in . It is immediate that iff for every -feasible test , . It follows that no feasible test of achieves power strictly greater than its significance level. Furthermore, every feasible method that correctly infers with high chance in , exposes itself to a high chance of error in “nearby” . It is a standard fact that one can topologize in such a way that weak convergence is exactly convergence in the topology: the usual sub-basis is given by sets of the form , where is almost surely clopen in .555Recall that a sequence converges to in a topology iff for every open set containing , there is such that for all . If a topology is first countable, converge to in the topology iff is in the topological closure of the . That topology is called the weak topology. If is second countable and metrizable, then the weak topology on is also second countable and metrizable, e.g. by the Prokhorov metric [4, Theorem 6.8]. When is countable and almost surely clopen, the weak topology is generated in a particularly natural way.666That condition is satisfied, for example, in the standard case in which the worlds in are Borel measures on , and all measures are absolutely continuous with respect to Lebesgue measure, i.e. when all measures have probability density functions, which includes normal, chi-square, exponential, Poisson, and beta distributions. It is also satisfied for discrete distributions like the binomial, for which the topology on the sample space is the discrete (power set) topology, so every acceptance zone is clopen and, hence, feasible. Naturally, it is satisfied in the particular cases of Examples 2.1 and 2.2.

Lemma 2.1.

Suppose that is a countable, almost surely clopen basis for . Let be the algebra generated by . Then the collection for and is a countable sub-basis for the weak topology.

That sub-basis for the weak topology has two fundamental advantages over the standard sub-basis. First, its closure under finite intersection is evidently a countable basis. Second, it is easy to show that the sub-basis elements are statistically verifiable. The following observations are easy consequences of the Lemma. In the setting of Example 2.1, the set of all for , assigning open intervals of biases for the coin, forms a sub-basis for the weak topology on . In fact, it forms a basis. If is the world in which the bias of the coin is exactly and is the world in which the bias is exactly , then the converge to in the weak topology.

3 Statistical Verifiability

In Section 1, proposition was said to be verifiable iff there is an infallible method that converges on increasing information to iff is true. That condition implies that there is a method that achieves every bound on chance of error, and converges to iff is true.777If for every your chance of error is less than , then your chance of error is zero: you are almost surely infallible. In statistical settings, one cannot insist on such a high standard of infallibility. Instead, say that is verifiable in chance iff for every bound on error, there is a method that achieves it, and that converges in probability to iff is true. The reversal of quantifiers expresses the fundamental difference between statistical and propositional verifiability and, hence, between statistical and propositional information. Say that a family of feasible tests of is an -verifier in chance of iff for all :

  1. , for all and

  2. , for all .

Say that is -verifiable in chance iff there is an -verifier in chance of . Say that is verifiable in chance iff is -verifiable in chance for every .

The preceding definition only bounds the chance of error at each sample size. One might strengthen SV1 to the requirement that the overall chance of error be bounded, when is false. Furthermore, one might also strengthen SV2 by requiring almost sure convergence to , rather than mere convergence in probability in every measure in . Say that a family of feasible tests of is an almost sure -verifier of iff

  1. for all and

  2. for all .

Say that is almost surely -verifiable iff there is an almost sure -verifer of . Say that is almost surely verifiable iff is almost surely -verifiable, for every . Clearly, if is almost surely verifiable, then is verifiable in chance.

We now weaken the preceding two criteria of statistical verifiability to arrive at statistical notions of limiting verifiability. Say that a family of feasible methods is a limiting verifier in chance of iff

Say that is limiting verifiable in chance iff there is a limiting verifier in chance of .

As before, there is an almost sure version of that success criterion. Say that a family of feasible methods is a limiting almost sure verifier of iff

Say that is limiting a.s. verifiable iff there is a limiting a.s. verifier of .

Finally, there is a natural statistical analogue of solvability. Recall that an empirical problem is a countable partition of the worlds in into a set of answers. Say that a family of feasible methods is a solution in chance to iff for every , . Say that is solvable in chance iff there exists a solution in chance to . A family of feasible methods is an almost sure solution to iff for every , . Furthermore, say that is almost surely solvable iff there exists an almost sure solution to .

4 Results

Theorem 4.1 states that, for sample spaces with countable, almost surely clopen bases, verifiability in chance and almost sure verifiability are equivalent to being open in the weak topology. As promised in the introduction, that fundamental result lifts the topological perspective to inferential statistics.

Theorem 4.1.

Suppose that is a set of Borel measures on , a metrizable sample space with countable, almost surely clopen basis . Then the following are equivalent:

  1. is -verifiable in chance for some ;

  2. is almost surely verifiable;

  3. is open in the weak topology.

For an elementary application of the Theorem, consider, in the setting of Example 2.1, the sharp hypothesis that the bias of the coin is exactly . That hypothesis is almost surely refutable, but it is not almost surely verifiable. Since a topological space is determined uniquely by its open sets, Theorem 4.1 implies that the weak topology is the unique topology that characterizes statistical verifiability under the weak conditions stated in the antecedent of the theorem. Thus, under those conditions, the weak topology is not merely a convenient formal tool—it is the topology of statistical information.

Here is the promised statistical analogue of Theorem 1.2.

Theorem 4.2.

Suppose that is a set of Borel measures on , a metrizable sample space with countable, almost surely clopen basis . Then the following are equivalent:

  1. is limiting verifiable in chance;

  2. is limiting almost surely verifiable;

  3. is a countable union of closed sets in the weak topology.

Finally, there is a natural statistical analogue of Theorem 1.3.

Theorem 4.3.

Suppose that is a set of Borel measures on , a metrizable sample space with countable, almost surely clopen basis . Then the following are equivalent:

  1. is solvable in chance;

  2. is almost surely solvable;

  3. partitions into countable unions of closed sets in the weak topology.888A similar result is proven in [7, Theorem 2] under different conditions. Dembo and Peres do not require their methods to be feasible, so Theorem 4.2 does not straightforwardly generalize their result. It is not difficult to reprove Theorem 4.2 without that requirement to obtain a generalization of the result in [7].

5 Related Work

Section 1 recapitulates foundational results in topological learning theory. Results stated in that section appear previously in [6], [9], and [3]. The theorems stated in section 4 are new, as far as we can tell. In statistical terminology, our Theorem 4.1 provides necessary and sufficient conditions for the existence of a Chernoff consistent test. Although there is extensive statistical work on pointwise consistent hypothesis testing, we are unaware of any topological result analogous to Theorem 4.1. The closest work is [15], where a topological characterization is given for consistent hypothesis testing of ergodic processes with samples from a discrete, finite alphabet. That result is incomparable with our own, because, although our work is done in the i.i.d setting, we allow samples to take values in an arbitrary, separable metric space. Furthermore, the topology employed in [15] is not the weak topology, but the topology of distributional distance. The existence of uniformly consistent tests is investigated topologically in [8], where some sufficient conditions are given. Limiting statistical solvability, or discernability, as it is known in the statistical literature, has been investigated topologically in [7] and [12]. The results of [7] are generalized to ergodic processes in [13]. Although the setting is slightly different, our Theorem 4.3 gives a simpler back-and-forth condition than the one given in [7] and is arrived at more systematically, by building on the fundamental Theorem 4.1. The weak topology is used in [7], but our Theorem 4.1 shows that the weak topology is the unique topology for which the open sets are exactly the statistically verifiable propositions. Our result shows, therefore, that the weak topology is more than just a convenient technical device.

6 Conclusion

This note lifts the topological perspective on empirical inquiry to statistics. In the deductive setting, open sets are deductively verifiable by true, propositional information. Theorem 4.1 exhibits a topology on probability measures in which the open sets are exactly the propositions statistically verifiable from random samples. In the deductive setting, learnability in the limit receives an elegant topological characterization [3, 9]. Theorems 4.2 and 4.3 provide analogous topological characterizations of learnability in the limit from statistical data. In light of those fundamental bridge results, we expect many of the streamlined insights of formal learning theory to apply literally to the concrete statistical problems that arise in statistics and machine learning. Of particular interest is the learning theoretic vindication of Ockham’s razor, developed topologically in [9], and [10].

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Appendix A Proofs and Lemmas

a.1 Deductive Verifiability

Proof of Theorem 1.1.

Right to left. Suppose that is open, and that is true in . Let if entails , and let otherwise. Since is a union of information states, there is an information state true in that entails . Therefore, . Furthermore, for any information state true in , we have that . So converges to true belief in . Furthermore, if then either , or , so avoids error in all worlds. Left to right. Suppose that is not open. Then is true in some , such that for all information true in , does not entail , i.e. there is . Suppose, for contradiction, that verifies . Then , for some true in . But, by assumption, there is . So does not avoid error in . ∎

Proof of Theorem 1.2.

Left to right. Suppose that is a limiting verifier of . Let

For each , let , and let . We claim that:

To prove the claim, iff there is such that for all information states , iff there is such that . Since , and is countable, is expressed as a countable union of locally closed sets. If the topology is metrizable, every open set — and therefore every locally closed set — can be expressed as a countable union of closed sets. Right to left. Every countable union of locally closed sets can be expressed as a disjoint union of locally closed sets [3, Proposition 3]. Let be a disjoint union, for open. Let be the least such that and , if such an exists, and let otherwise. Let if , and let otherwise. Suppose that . Let be the least integer such that . Then for , either or . For each , let be an information state true in such that , if , and let , otherwise. Let be an information state true in that entails . Finally, let . Then , for every such that . Finally, suppose that . Then for each , either or . Suppose that , for some . Then . Let be an information state true in and entailing . Then , because is either , or is some , which was chosen to be disjoint from . ∎

Proof of Theorem 1.3.

See the proof of Theorem 2 in [9]. ∎

a.2 The Statistical Setting

a.2.1 The Sample Space

The following Lemma states that is always feasible to perform logical operations (e.g. , , and ) on feasible tests.

Lemma A.1 (Lemma 6.4 [14]).

The almost surely clopen sets in , denoted , form an algebra.

Proof of Lemma a.1.

One has that , since . Moreover, is closed under complement, since . Furthermore, since , it follows that if , then . Therefore, is closed under finite union as well. ∎

Hypothesis tests are often constructed to reject if the number of samples landing in a particular region exceeds some threshold. The following lemma states that such a test is -feasible, if the region is almost surely clopen in .

Lemma A.2.

Suppose that is almost surely clopen in . Then:

is almost surely clopen in , for , and .

Proof of Lemma a.2.

Let enumerate all -element subsets of . Then

where if , and otherwise. Since the almost surely clopen sets in form an algebra, it suffices to show that is an almost surely clopen set in . Argue by induction on . If , then is either or , which are both almost surely clopen sets in . For the inductive step, note that . By the induction hypothesis, . ∎

a.2.2 The Weak Topology

Billingsley [4] proves the following result about the product space:

Lemma A.3 (Theorem 2.8).

If is metrizable and second-countable, then iff and .

Lemma A.3 entails that the product map is sequentially continuous, and therefore, continuous, for all natural .999Continuity is relative to the weak topologies on and . Recall that a function is sequentially continuous if whenever a sequence converges to a limit , the sequence converges to . In first-countable spaces, sequential continuity is equivalent to continuity. Billingsley [4] also proves the following useful lemma:

Lemma A.4 (Theorem 2.2).

Suppose that is a -system101010 is a -system iff whenever . and that every open set is a countable union of sets. If for every in , then .

The following is a consequence of Lemma A.4.

Lemma A.5.

Suppose that is a countable, almost surely clopen basis for . Then the collection for and is a countable sub-basis for the weak topology.

Proof of Lemma a.5.

It is sufficient to show that iff the converge to in the topology generated by the sub-basis. Left to right. Suppose . Let be open in the topology generated by the sub-basis. Suppose lies in . Then there is a basic open set:

such that . Since is feasible for , for each . Therefore, there exists such that for all . Letting , it follows that for all . Therefore, the converge to in the topology generated by the sub-basis. Right to left. Suppose that the converge to in the topology generated by the sub-basis. Note that since is an algebra, the collection for , and generates the same topology.111111Notice that Let enumerate the elements of . Let be rationals lying in . Let denote the sub-basis element . Let be a surjective function from to . Let . By assumption, for every , there is such that the lie in , for all . So , for every . By Lemma A.4, . ∎

Proof of Lemma 2.1.

Immediate corollary of Lemma A.5. ∎

a.3 Statistical Verifiability

First, a useful lemma.

Lemma A.6.

The almost surely verifiable propositions are closed under finite conjunctions, and countable disjunctions.

Proof of Lemma a.6.

Suppose that are a.s. verifiable. Let . Let be such that is an a.s. -verifier for . Let if , for . By Lemma A.1, is feasible, for each . Suppose that . Then:

Suppose that . Without loss of generality, suppose . Then:

To show that the a.s. verifiable propositions are closed under countable union, suppose that are a.s. verifiable. For , let be an a.s. -verifier for with . Let if for some , and let otherwise. By Lemma A.1, is feasible for each . Suppose that . Then there exists such that . Furthermore:

Suppose that . Then:

Proof of Theorem 4.1.

1 implies 3. Suppose, for contradiction, that is not open, but that is an -verifier in chance for . Let . Then there is a sequence of in such that . Since is a verifier for , there is a sample size such that . By Lemma A.3, . So there is a such that . Contradiction.

3 implies 2. By Lemmas 2.1, A.6, it is sufficient to show that every element of the sub-basis:

is a.s. verifiable. Let , and let , for some . Define the indicator random variable by if , otherwise . Letting , it follows from Hoeffding’s inequality that:

Let if , and let otherwise. By Lemmas A.2, is feasible for all . If , then and:

Furthermore, if , then since , by the strong law of large numbers, and , we have that . Therefore, , as required.

2 implies 1. Immediate from the definitions. ∎

Proof of Theorem 4.2.

1 entails 3. Suppose that is a limiting verifier in chance of . For every , let:

Lemma A.3 and the feasibility of the entail that and are both open in the weak topology. Let . We claim that:

Observe that iff there is , such that for all iff Therefore, proposition is a countable union of locally closed sets. Since the weak topology is metrizable, every open set — and therefore every locally closed set — can be expressed as a countable union of closed sets.

3 entails 2. Suppose that is a countable union of closed sets. By theorem 4.1, for each , there exists an a.s. statistical -verifier of its complement . Let , where is the least integer in such that , if such a exists. Otherwise, let . Suppose that , and that is the least integer such that . Then:

Since each