Probability, Bayesian Inference, and Decision Theory

Manual: General · Subject: Artificial Intelligence

Develop probabilistic modeling, Bayesian updating, and utility-based decision-making for uncertain environments.

Uncertainty in AI

Why probability matters

Real-world AI must operate with incomplete, noisy, and ambiguous data. Probability provides a mathematically coherent framework for uncertainty, belief revision, and prediction.

Bayesian basics

Prior

Belief before observing data

Likelihood

Probability of data given a hypothesis

Posterior

Belief after observing data

Evidence

Normalization term ensuring probabilities sum to

1

Bayes update

1

Start with a prior $p(\theta)$ .
2

Compute likelihood $p(D\mid \theta)$ .
3

Form the posterior $p(\theta\mid D) \propto p(D\mid \theta)p(\theta)$ .
4

Use the posterior for prediction or decision-making.

In Bayesian inference, what does the posterior represent?

Belief before data Belief after observing data A deterministic rule set The training accuracy

What is the role of the likelihood in Bayes' rule?

Decision theory

Decision theory chooses actions by maximizing expected utility: $a^* = \arg\max_a \mathbb{E}[U(a, X)]$ where $X$ is a random outcome. This framework separates beliefs from preferences.

Risk attitudes

Risk-neutral

Optimizes expected value
Ignores variance unless encoded in utility

Risk-averse

Penalizes uncertain outcomes
Often modeled by concave utility functions

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Practical note

Approximate Bayesian methods such as variational inference and Monte Carlo sampling are often necessary because exact inference is computationally infeasible.

Expected utility theory primarily combines which two ingredients?

Data and labels Beliefs and preferences Features and gradients Memory and attention

Neuroanatomy Explorer

Memory Deck

Match the Pairs

Concept Constellation