# Welcome to 🐙 oktopus!¶

oktopus is all about Bayes’ Law:

$\log \underbrace{p(\theta | \mathbf{y})}_\text{posterior} = \log \underbrace{p(\mathbf{y} | \theta)}_\text{likelihood} + \log \underbrace{p(\theta)}_\text{prior} + \overbrace{h(\mathbf{y})}^\text{doesn't depend on \theta}$

In other words: posterior information is a combination of prior information and the information acquired after observing data (likelihood).

With that in mind, oktopus provides an easy interface to solve problems such as:

1. Maximum Likelihood Estimator (MLE):
$\arg \min_{\theta \in \Theta} - \log p(\mathbf{y} | \theta)$
1. Fisher Information Matrix:
$\mathbb{E}\left[\nabla_\theta\log p(\mathbf{y} | \theta)\left[\nabla_\theta\log p(\mathbf{y} | \theta) \right]^{\textrm{T}} \right]$
1. Maximum a Posteriori Probability Estimator (MAP):
$\arg \min_{\theta \in \Theta} - \log p(\theta | \mathbf{y})$