# Bayesian decision theory

(1 hours to learn)

## Summary

When we use Bayesian parameter estimation techniques, often it's because we want to make a decision. In Bayesian decision theory, we make the choice which minimizes the expected loss under the posterior. When we compute a statistic like the mode or the mean of the predictive distribution, this can be interpreted as the decision theoretic solution under a particular loss function.

## Context

This concept has the prerequisites:

- Bayesian parameter estimation (We use Bayesian parameter estimation to get the posterior on which we base our decisions.)
- expectation and variance (The goal of Bayesian decision theory is to minimize expected loss under the posterior.)
- loss function (The goal of Bayesian decision theory is to minimize the expected loss under the posterior for a particular loss function.)

## Goals

- Know how the optimal decision is defined (in terms of minimizing expected loss with respect to the posterior)

- Derive the form of the estimator for some particular loss functions:
- 0-1 loss
- quadratic loss
- absolute loss

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

## -Paid-

→ Machine Learning: a Probabilistic Perspective

A very comprehensive graudate-level machine learning textbook.

Location:
Sections 5.7-5.7.1, pages 176-180

## Supplemental resources (the following are optional, but you may find them useful)

## -Paid-

→ Pattern Recognition and Machine Learning

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

Location:
Section 1.5, pages 38-48

Additional dependencies:

- linear regression

## See also

- Influence diagrams are a graphical model formalism for decision theoretic problems.