Unlocking Bayes' Theorem: From Basics to Real-World Applications

Bayes' Theorem

We first look at the magical Bayes' theorem. It may seem ordinary, but this is all you need to master Bayesian statistics. Understanding how the theorem is derived helps grasp its meaning.

From the multiplication rule in probability theory we obtain an expression that can be rearranged into Bayes' theorem.

The formula shows that the probability of a hypothesis and the probability of data are not necessarily equal—a point often missed even by those familiar with statistics. For example, the probability of being human given two legs differs from the probability of having two legs given being human.

If we interpret H as a hypothesis and D as data, Bayes' theorem tells us how to compute the probability of the hypothesis given the data. The hypothesis is incorporated via probability distributions—essentially the model parameters—so it is more accurate to speak of a model rather than a vague hypothesis.

Bayes' theorem is built from four key components:

Prior : the distribution reflecting our knowledge about parameters before observing data. When we know nothing, a uniform distribution can be used.

Likelihood : the probability of the observed data under given parameters, indicating how well the parameters explain the data.

Posterior : the result of Bayesian analysis, representing our updated knowledge after combining prior and likelihood. The posterior is proportional to prior × likelihood, not a single value but a distribution.

Evidence (or marginal likelihood): the normalizing factor obtained by integrating the product of prior and likelihood over all possible parameter values.

The posterior from one analysis can serve as the prior for a subsequent analysis, making Bayesian methods especially suitable for sequential data processing, such as real‑time weather or satellite data for early disaster warning.

Reference: Osvaldo Martin, "Python Bayesian Analysis".