Log10PmaxPmin

Log10PmaxPmin is the base-10 logarithm of the ratio between the largest and smallest probabilities in a discrete probability distribution. If p_i denote category probabilities with p_i ≥ 0 and sum p_i = 1, then Pmax = max_i p_i and Pmin = min_i p_i. The quantity is defined as log10(Pmax/Pmin) = log10(Pmax) − log10(Pmin). It is dimensionless and equals zero when the distribution is uniform (Pmax = Pmin). If any category has zero probability, Pmin = 0 and the ratio is unbounded and the logarithm is undefined; in practice, smoothing or pseudo-counts are used to avoid infinities.

Interpretation: Log10PmaxPmin measures the dynamic range or concentration of probabilities. Larger values indicate greater disparity among category probabilities, while smaller values indicate a more even distribution. It is sensitive to the most and least probable categories but ignores intermediate probabilities.

Calculation considerations: The statistic depends on the chosen support of the distribution; changing the number of categories changes Pmax and Pmin. It is simple to compute but can be unstable when Pmin is estimated with limited data. Example: for [0.5, 0.3, 0.2], Pmax=0.5, Pmin=0.2, ratio=2.5, log10 ≈ 0.398. For a uniform [1/3, 1/3, 1/3], the value is 0.

Contexts and related measures: Log10PmaxPmin is used as a compact descriptor of distributional concentration in fields such as machine learning evaluation, ecology, and reliability analysis. It complements entropy, perplexity, and the Gini coefficient by focusing on the extremes rather than overall uncertainty.