# Gaussian processes
###### tags: `Machine Learning II`

## Elements and distributions in probabilistic learning
When doing probabilistic machine learning, our **ouput is** not a value, but a **probability density function**.

Elements
* **Observations/Data** $D$ :

>[!TLDR]
>

>[!missing]
>What is the Hypothesis space $H$.

Distributions
* **Prior**
* **Likelihood**
* **Posterior**

There is also the **posterior predictive distribution**:

$$p(y^*| \mathbf{x}^*, D) = \sum_h p(y^* = h(\mathbf{x}^*)|\mathbf{x}^*,h)\cdot p(h|D)$$

>[!tip]
>Use priors that are easy to use mathematically

## Bayesian linear regression
This form of regression assumes that the data was generated from a data distribution with parameters $\mathbf{w}$, but was contaminated with noise $\sigma_n$.

>[!info]
>The product of grouping several gaussians is a multivariate gaussian.
>Recall that the marginals and conditionals of a Gaussian are also Gaussian themselves.