• Table of Contents

Decision theory is a framework which attempts to answer the question of making optimal decisions in the face of uncertainty. It treats statistics as a 2-person game: the statistician vs nature/chance.

• Treat this as another way to approach problems
• Has some interesting connections with Bayesian inference

Decision Theory Intro

<aside> 📖 From a probabilistic perspective, decision theory involves assigning probabilities to outcomes to make informed decisions based on the decision maker’s preferences.

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This idea can be implemented in a variety of different ways, most commonly: minimizing expected loss.

General overview:

1. Determine loss of all outcomes
2. Find probability of all outcomes given each action
3. Choose action with the smallest expected cost

Motivating example

Example

• Consider a bacterial infection, you have the following choices:

  1. Do nothing
  2. Take penicillin
  3. Take more effective but toxic antibiotics

• First list all possible outcomes, their values, probability of occurring given an action

  Outcome	Value
  No infection
  No toxin	0 days sick
  Infection
  No toxin	5 days sick
  No infection
  toxin present	1 day sick
  Infection
  toxin present	8 days sick

  | $P(event|action)$ | None | Pen | Tox | | --- | --- | --- | --- | | No infection No toxin | 0.1 | 0.9 | 0 | | Infection No toxin | 0.9 | 0.1 | 0 | | No infection toxin present | 0 | 0 | 0.99 | | Infection toxin present | 0 | 0 | 0.01 |

• Finally compute the sum of all values times their probabilities

  1. Nothing: $0.9\times 5 = 4.5$ expected days sick
  2. Penicillin: $0.1*5=0.5$ expected days sick
  3. Toxic antibiotics: $0.99\times 1 + 0.01\times 8 = 1.07$ expected days sick

• Penicillin is the option with the lowest expected number of sick days

Formalizing the Procedure