Signal and Noise: Fisher Information as a Measure of Communicative Substance

Some communicators say a great deal while revealing very little. Their messages are syntactically complete, responsive in tone, and superficially cooperative, yet after reading dozens of them you know scarcely more about their actual position than when you started. This is not accidental. It is a deliberate strategy — obfuscation through volume, the production of text that appears to engage while systematically withholding substance. Fisher information, a concept from mathematical statistics that measures how much an observation tells you about an unknown parameter, provides a formal framework for detecting and quantifying this pattern.

The Mathematical Foundation

Fisher information measures the amount of information that an observable random variable carries about an unknown parameter of the distribution that generated it. High Fisher information means the observation tightly constrains the parameter — the data is informative. Low Fisher information means the observation is only weakly related to the underlying parameter — the data tells you little despite being plentiful. Formally, Fisher information is the expected value of the squared score function, or equivalently, the negative expected value of the second derivative of the log-likelihood. It quantifies the curvature of the likelihood surface: sharp peaks mean high information, flat surfaces mean low information.

The Cramér-Rao bound connects Fisher information to the best possible estimation precision: no unbiased estimator can have variance lower than the reciprocal of the Fisher information. In communication analysis, this translates to a fundamental limit on how precisely a party's true position can be recovered from their statements. Low Fisher information means the party's communications are inherently uninformative — no amount of analytical effort can extract a clear position from text that does not contain one.

Application to Communication Analysis

The "unknown parameter" in communication analysis is the party's true position — their actual knowledge, intentions, and account of events. Each successive statement is an observation. Fisher information per statement measures how much that statement narrows the uncertainty about the party's position. The implementation computes the reduction in entropy of the inferred position distribution after incorporating each new statement.

Honest communicators generate high Fisher information per statement. Each message reduces uncertainty about their position in at least one dimension. After ten statements, the posterior distribution over their likely position is sharply peaked — you have a clear picture of what they know and what they claim. The information accumulates constructively because each new statement is consistent with and reinforces the constraints imposed by previous ones.

Manipulative communicators generate low Fisher information. Their statements are designed to appear responsive while minimising the actual information conveyed. After ten statements from a skilled obfuscator, the posterior distribution may be barely more concentrated than it was after one. The statements do not contradict each other — that would be detectable by simpler means — they simply fail to add up to anything. Each message acknowledges the topic without committing to specifics, references context without providing detail, and responds to questions without answering them.

The Obfuscation Metric

The ratio of word count to Fisher information produces a direct measure of communicative efficiency. Honest communicators maintain a relatively constant ratio throughout a conversation — they say what they mean proportionate to the words they use. Their information density may vary with topic complexity but does not systematically decline.

Manipulative communicators show a characteristic pattern: the word-to-information ratio increases over time, particularly when pressed on specific topics. They produce more text while conveying less substance. This escalation is diagnostic because producing genuinely high-information content requires committing to specific claims — exactly what manipulators avoid. The harder they are pressed, the more words they generate per unit of actual information, and this inflation of the ratio is measurable and reproducible.

Cumulative Information Yield

Plotting cumulative Fisher information against message count produces a growth curve that distinguishes communication styles. Honest communicators produce a curve that rises steeply at first — early messages are maximally informative — then gradually flattens as the position becomes well-established and new messages add diminishing marginal information. This is the natural shape of informative communication: rapid initial disclosure followed by refinement.

Manipulative communicators produce a nearly flat cumulative curve. The total information about their position barely increases regardless of how many statements they produce. This flatness is the quantitative signature of stonewalling dressed up as cooperation — the appearance of engagement without the substance. In legal contexts, where parties have obligations of candour and disclosure, a demonstrably flat Fisher information curve constitutes measurable evidence that those obligations are being structurally evaded.

Dimensional Selectivity

Fisher information analysis can be decomposed by topic dimension, revealing which subjects a party communicates substantively about and which they systematically avoid. A party who is truthful about most topics but evasive about one will show high Fisher information across most dimensions but conspicuously low information in the sensitive area. This selective obfuscation signature identifies not just that a party is withholding but precisely what they are withholding about — a diagnostic with obvious relevance to legal discovery and cross-examination strategy.