Notes on Sum-Tests and Independence Tests

Abstract

We study statistical sum-tests and independence tests, in particular for computably enumerable semimeasures on a discrete domain. Among other things, we prove that for universal semimeasures every 0 1 -sum-test is bounded, but unbounded 0 1 -sum-tests exist, and we study to what extent the latter can be universal. For universal semimeasures, in the unary case of sum-test we leave open whether universal 0 1 -sum-tests exist, whereas in the binary case of independence tests we prove that they do not exist.

Nonapproximability of the normalized information distance

“…In [2] it is shown that the function E is upper semicomputable, D(x, y) = E(x, y) + O (log E(x, y)), the function E is a metric (more precisely, that it satisfies the metric (in)equalities up to a constant), and that E is minimal (up to a constant) among all upper semicomputable distance functions D satisfying the mild normalization conditions y: y =x 2 −D (x,y) 1 and…”

“…(Here and elsewhere in this paper "log" denotes the binary logarithm.) It should be mentioned that the minimality property was relaxed from the D functions being metrics [2] to symmetric distances [10] to the present form [11] without serious proof changes. The normalized information distance (NID) e is defined by…”

m-sophistication

“…As a corollary of Proposition 3 it follows that: By [7] every lower semicomputable sumtest for m is bounded by a constant, which implies that if d was lower semicomputable, than d + 0, and thus only the constant e = 0 is allowed.…”

“…x BB(k)}. It is a very slow growing function, dominated by any unbounded non-decreasing function [7].…”

“…The question rises whether for the lower semicomputable semimeasure unbounded sumtests exists in some computability class. It turns out that for the hypotheses of independence there are no unbounded computable and lower semicomputable sumtests, but there are upper semicomputable sumtests of maximal magnitude l(x) [7].…”