Bayesian Updates

A Bayesian update consists of updating the probability distribution that represents our knowledge of a random variable after an event, using Bayes theorem, $\mathbb{P}[X|E]=\frac{\mathbb{P}[E|X]\mathbb{P}[X]}{\mathbb{P}[E]}$, where:

• $\mathbb{P}[X]$ is the original probability we assigned to the random variable
• $\mathbb{P}[X|E]$ is the updated probability
• $\mathbb{P}[E|X]$ is the probability of the observed event IF we know the value of the random variable $X$.
• $\mathbb{P}[E]$ is the probability of the event.

Since the resulting value is normalised by dividing by $\mathbb{P}[E]$, we can often simplify calculations by working with the likelihood $\mathbb{P}[X|E]\propto \mathbb{P}[E|X]\mathbb{P}[X]$.

• Build table of Bayesian updates for multiple probability distributions 🔽
• Build distribution relations based on this twit :🔽

It might also be useful to treat this as the construction of a Probability Monoid