Bayes’ Theorem — A Probability Rule for Updating Beliefs with New Evidence

The most important mathematical formula in AI — it updates prior probabilities when new evidence appears, allowing more accurate probability estimation.

10/28/2025

Chain Rule of Probability — A Rule for Decomposing Joint Probability into a Product of Conditional Probabilities

Chain Rule of Probability is a rule that expresses a joint probability as a product of conditional probabilities. As the number of random variables grows, it becomes increasingly difficult to handle the full joint distribution all at once. The chain rule turns that complexity into an ordered computational structure, which makes probabilistic modeling and inference much more tractable.

04/08/2026

Conditional Dependence — How Variables Remain Related After Accounting for Conditions

Conditional Dependence describes a probabilistic relationship where two variables \(X\) and \(Y\) remain connected even after a condition \(Z\) is taken into account. While Conditional Independence means that the relationship between variables disappears once additional information is provided, Conditional Dependence means that useful information about one variable still remains in another variable after considering the given condition.

08/02/2026

Conditional Independence — How Variable Relationships Disappear After Considering Conditions

Conditional Independence describes a probabilistic structure where the dependency between two variables \(X\) and \(Y\) disappears once another variable \(Z\) is taken into account. In other words, even if \(X\) and \(Y\) appear to move together, knowing \(Z\) means that neither variable provides additional information for explaining the other. The important point is that \(X\) and \(Y\) are not always independent; they can be treated as independent only under the condition that \(Z\) is known. This concept provides a foundation for decomposing complex joint probabilities into simpler conditional probabilities and plays a key role in building efficient probabilistic models.

08/02/2026

Conditional Probability — Recomputing event likelihood once a condition is given

Conditional probability is a way to recompute the probability of an event after you are told that another event has already occurred. It is a core idea in probabilistic reasoning because it updates decisions after observing data.

10/29/2025

Continuous Uniform Distribution — Why Probability Is Proportional to Interval Length

Continuous Uniform Distribution is a continuous probability distribution where every value within the interval [a, b] has the same probability density. It is one of the simplest continuous distributions, built on a symmetry assumption over a finite interval. Because no value inside the range is considered more likely than another, it is often used as a baseline model when representing a state of limited information. In this distribution, probability is not assigned to individual points but is determined by the area over a given interval.

08/02/2026

Discrete Uniform Distribution — Why Equal Probability Matters When No Prior Information Exists

Discrete Uniform Distribution is a probability model for situations where a finite set of possible outcomes exists and each outcome is assigned the same probability mass. The model relies on a symmetry assumption over the sample space: when there is no evidence that one outcome is more likely than another, all possibilities are treated equally. This makes it a natural baseline assumption when no prior preference or additional information is available.

08/02/2026

Emission Probability — How Hidden States Generate Observable Data

Emission Probability represents the likelihood that a particular hidden state produces a specific observation. In a Hidden Markov Model (HMM), the underlying state cannot be observed directly. Instead, we only see the resulting observations. Emission Probability measures how likely each observation is under a given hidden state, making it possible to infer the most probable state from the data we actually observe. In other words, it serves as the key probabilistic link between invisible internal states and visible outputs.

08/08/2026

Expected Value — A Probability-Weighted Average

Expected Value is an average benchmark that summarizes many possible outcomes into a single number. Unlike an ordinary average, it does not treat every outcome equally. Instead, outcomes that happen more often carry more weight, while rare outcomes contribute less. That is why expected value is one of the most fundamental tools for thinking clearly about uncertainty.

03/16/2026

Law of Large Numbers — Why Sample Averages Converge to Expected Values

Law of Large Numbers describes the expectation consistency property that appears when repeated experiments satisfy the i.i.d. assumption, meaning each observation is independent and follows the same probability distribution. Although individual outcomes can vary randomly, averaging a large number of observations reduces the impact of random fluctuations. As a result, the sample mean becomes increasingly stable and approaches the distribution’s expected value. This makes the law more than an empirical pattern; it is a fundamental limit theorem explaining why averages converge toward the center of a probability distribution.

08/02/2026