Logarithmic Information & Scale

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Abstractions about entropy, information units and dimensions, logarithmic measures, geometric dispersion, and log-scaled numerical representation.

9 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Binary entropy function — The Shannon entropy of a Bernoulli variable as a concave function of its success probability.
• Geometric standard deviation — A dimensionless multiplicative spread factor obtained by exponentiating the standard deviation of logarithms.
• Information dimension — The asymptotic growth rate of the Shannon entropy of increasingly fine quantizations of a random variable or distribution.
• Logarithm — The inverse of exponentiation with a fixed admissible base, assigning to a positive input the exponent needed to produce it.
• Logarithmic mean — Average two positive numbers by their difference divided by the difference of their logarithms, using the continuous value x when the arguments coincide.
• Log–log plot — Plot positive x and y values on logarithmic axes so multiplicative ratios become equal distances and a power law y=ax^k becomes a straight line with slope k and intercept log a.
• Min-entropy — The negative logarithm of the largest outcome probability, measuring worst-case single-guess unpredictability as the order-infinity Rényi entropy.
• Nat (unit) — The unit of information associated with natural logarithms, equal to the information in an event of probability 1/e and to 1/ln 2 bits.
• Units of Information — Choose a logarithmic reference base and unit convention for information quantities so the same amount can be expressed coherently in bits or shannons, nats, hartleys, and their standardized multiples without confusing information with storage capacity.