What

Bayesian probability

An interpretation of probability as a measure of belief and certainty rather than just frequency (like in frequentist statistics).

i.e: probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief.
based on prior knowledge of related conditions.
allows for making probabilistic statements about unknown parameters.

P (H ∣ E) = \frac{P ( E ∣ H ) \cdot P ( H )}{P ( E )} = \frac{P ( H E )}{P ( H )}

$H$ : any hypothesis whose probability may be affected by the data (i.e. $E$ ). Often there are competing hypotheses, and the task is to determine which is the most probable.
$P (H)$ : the prior probability - the prior knowledge (the estimate probability) of the hypothesis $H$ before any more evidence.
$E$ : the evidence, corresponds to new data that were not used in computing the prior probability.
$P (H ∣ E)$ : posterior probability - the updated probability of the hypothesis $H$ conditional on a new evidence $E$ (i.e. after $E$ is observed) this is what we want to know.
$P (E ∣ H)$ : likelihood - probability of the evidence $E$ given the hypothesis $H$ , this indicates the compatibility of the evidence $E$ given the hypothesis $H$ .
$P (E)$ : marginal probability - the prior probability of the evidence $E$ not conditional on anything.

This provides a mathematical formula to update the probability for a hypothesis as more evidence or information become available.

A method of statistical inference
Treats probability as equivalent with certainty
Uses the Bayes’ Theorem to update the probability (belief) for a hypothesis as more evidence or information becomes available.
- fundamentally: uses prior knowledge (in the form of a prior probability distribution) to estimate the posterior probabilities.