# Bayesian parameter estimation: Gaussian distribution

(1.6 hours to learn)

## Summary

Using the Bayesian framework, we can infer the mean parameter of a Gaussian distribution, the scale parameter, or both. Since Gaussians are widely used in probabilistic modeling, the computations that go into this are common motifs in Bayesian machine learning more generally.

## Context

This concept has the prerequisites:

- Bayesian parameter estimation
- Gaussian distribution
- gamma distribution (The inverse gamma distribution is the conjugate prior for the scale parameter.)
- Student-t distribution (The predictive distribution, with the scale parameter integrated out, is a student-t distribution.)

## Goals

- Derive the conjugate priors for three cases:
- unknown mean, but known variance
- known mean, but unknown variance
- unknown mean and unknown variance

- Derive the posterior distribution and the posterior predictive distribution for each of these cases.

## Core resources (read/watch one of the following)

## -Free-

→ Information Theory, Inference, and Learning Algorithms

A graudate-level textbook on machine learning and information theory.

## -Paid-

→ Pattern Recognition and Machine Learning

A textbook for a graduate machine learning course, with a focus on Bayesian methods.

Location:
Sections 2.3.6-2.3.7 (except for the part about multivariate Gaussians), pgs. 97-105

## See also

-No Additional Notes-