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Probability and Duality: From Coin Flips to High-Dimensional Geometry

2025-09-21

This article explores several seemingly unrelated probability problems, such as the probability of a path existing in a random graph and the probability that the convex hull of four points on the unit circle contains the origin, both surprisingly equal to 1/2. The author cleverly uses duality tricks and combinatorial arguments to reveal the deep connections behind these problems. By analyzing the number of cells cut out of a high-dimensional space by linear hyperplanes and studying the properties of random matrices, the author ultimately explains these probability results and poses several unsolved mathematical problems, prompting readers to ponder the curious relationship between probability, geometry, and duality.

The Surprising Secrets Hidden in the Entropy of a Mixture

2025-07-01

This article delves into the relationship between the entropy of a mixture of probability density functions and its interpolation factor. The author reveals that entropy, as a function of probabilities, is concave, and this concavity is directly tied to the mutual information between the two distributions. By introducing a Bernoulli variable and the concept of conditional entropy, the article elegantly explains how mutual information quantifies the change in the expected surprisal of a prediction given knowledge of the mixture factor. Furthermore, it introduces a novel concept, 'proclivity', connecting it to KL divergence and cross-entropy. The article also discusses Jensen-Shannon divergence and the Neyman χ² divergence appearing in higher-order Taylor expansions. Ultimately, it concludes that the entropy function of the mixture completely describes the distribution of likelihood ratios between the two probability distributions, offering a fresh perspective on understanding the relationship between probability distributions.