# Bayesian Probability ## The Idea in Brief Bayesian probability is a way of understanding probability as a **degree of belief**, which updates as new evidence is introduced. Rooted in **Bayes’ Theorem**, it contrasts with the frequentist view by focusing on subjective probabilities and prior knowledge. This approach has become increasingly influential in statistics, data science, and decision-making under uncertainty. --- ## Key Concepts ### 1. Bayes’ Theorem Bayes’ Theorem is the mathematical rule at the heart of Bayesian reasoning. It allows one to update the probability of a hypothesis $H$, given new evidence $E$: $ P(H \mid E) = \frac{P(E \mid H) \cdot P(H)}{P(E)} $ - **$P(H)$**: Prior probability – initial belief in the hypothesis before seeing the evidence. - **$P(E \mid H)$**: Likelihood – probability of observing the evidence if the hypothesis is true. - **$P(E)$**: Marginal probability – total probability of the evidence under all possible hypotheses. - **$P(H \mid E)$**: Posterior probability – updated belief after observing the evidence. --- ### 2. Subjective Probability Bayesian probability treats uncertainty as a **personal belief** rather than an objective frequency. This means different people might assign different probabilities to the same event, depending on their prior knowledge. --- ### 3. The Prior A **prior** is an initial assumption or estimate about a situation. It can be based on past data, expert opinion, or even be *uninformative* (flat) to reflect lack of knowledge. Choosing a prior carefully is crucial in Bayesian analysis. --- ### 4. Posterior Updating As more data becomes available, the **prior is updated** to form the **posterior**. This process is iterative: the posterior from one step becomes the prior for the next, leading to increasingly refined beliefs. --- ### 5. Comparison with Frequentism | Aspect | Bayesian Approach | Frequentist Approach | |-------------------------|----------------------------------------|------------------------------------| | Probability Meaning | Degree of belief | Long-run frequency | | Use of Prior Information| Yes | No | | Model Flexibility | High (adapts with new data) | Rigid (fixed hypothesis testing) | | Typical Methods | Bayesian inference, MCMC, Bayes factors| p-values, confidence intervals | --- ## See in Field Notes - [Decision Architecture](https://www.anishpatel.co/decision-architecture/) — Bayesian updating applied to organisations: score evidence proportionally, don't lurch in response to noise ---