When does an information score really reward better information?

What makes one source of information better than another? And how can we check that a numerical score respects that comparison? In the preprint “On the Monotonicity of Information Measures,” Xiaoyu Cheng and Yonggyun Kim turn these questions into tests involving small changes to signal probabilities.

The work is mathematical, not an empirical study. Its results establish consistency conditions within specified models of information; they do not measure how people actually value news or demonstrate that more information always improves market outcomes.

Two meanings of better information

A statistical experiment, in this setting, maps each possible state of the world to probabilities of different signals. An information measure assigns a number to that entire arrangement.

The Blackwell order compares experiments by asking whether one can be produced from another by adding noise that does not depend on the underlying state. If so, the original is at least as informative. Equivalently, it supports at least as high an expected payoff in every decision problem.

That is a demanding comparison: many experiments cannot be ranked this way. The Lehmann order considers a narrower class of decision problems with an ordered structure, where higher actions become more attractive in higher states. Because it asks for agreement across fewer problems, it ranks more pairs of experiments.

The paper studies Lehmann comparisons among experiments satisfying the monotone likelihood ratio property. Roughly, higher signals must provide relatively stronger evidence for higher states. This is a substantive restriction, not a feature of every information source.

A global guarantee from local checks

The authors begin with a simple operation: randomly replace one signal with another, using the same replacement probability in every state. This adds noise and cannot improve Blackwell informativeness.

Their first main result concerns continuous information measures that are differentiable wherever all signal probabilities are positive. Such a measure respects the Blackwell order exactly when three requirements hold: relabeling signals leaves its value unchanged; splitting a signal into proportionally identical copies leaves its value unchanged; and its first-order change under every signal replacement is nonpositive.

“First-order change” means the initial rate at which the score changes as a tiny amount of noise is introduced. Crucially, the condition must hold at every relevant experiment, not just at one example. Together with the other assumptions, these local checks guarantee the ranking across all Blackwell-comparable experiments.

For the Lehmann order, the elementary changes are more targeted. A low signal can be replaced with the next higher signal only in states below a cutoff, or a high signal with the next lower signal only in states above a cutoff. These reverse signal replacements weaken the association between high states and high signals.

Under the paper’s regularity assumptions, split invariance and nonpositive first-order changes in the specified reverse-replacement directions characterize Lehmann monotonicity.

Why the proof needs carefully chosen paths

The difficult step is showing that local checks really cover every relevant global comparison. Simply averaging two experiments satisfying the likelihood-ratio restriction can violate that restriction.

The authors instead construct paths that stay within the permitted set, sometimes splitting or merging signals along the way. They use probability–probability plots, which compare cumulative signal probabilities across adjacent states. Coordinated replacements reshape one plot while leaving the others unchanged. Along these paths, the derivative conditions ensure that the information score never rises.

They also translate the tests into posterior beliefs: probabilities assigned to states after a signal has been observed, starting from a fixed prior belief.

Entropy passes, but convexity is not enough

An important application is the entropy cost, the expected reduction in Shannon entropy, a standard measure of uncertainty. On experiments, this equals mutual information between the state and signal. The authors prove that it respects the Lehmann order for every prior assigning positive probability to every state.

Other familiar mathematical properties do not automatically suffice. For posterior-separable measures, which average a fixed function of updated beliefs, convexity ensures Blackwell monotonicity but need not ensure Lehmann monotonicity. The additional comparisons impose additional restrictions.

The paper also identifies posterior-mean-ratio measures whose collective rankings characterize the Lehmann order among positive experiments satisfying the likelihood-ratio restriction. One score alone is not claimed to capture the whole order.

These are finite-state, finite-signal results in an arXiv preprint. Extensions to continuous signals and optimization over the restricted experiments remain open. The contribution is a way to verify proposed information measures—not a universal claim that information benefits everyone.