October 2020

Normative Models

Uncertainty

The world is riddled with uncertainty.

Noise

Ambiguity

Partial Information

Probability Theory

Probability theory is nothing but common sense reduced to calculation.

Pierre-Simon Laplace (1819)

Bayes’ Theorem

Given the state $x$, the observation $y$ and the hypothesis space $\mathcal{H}$

$P(x|y, \mathcal{H}) = \frac{P(y|x, \mathcal{H}) \cdot P(x| \mathcal{H})}{P(y| \mathcal{H})}$

• Likelihood: Probability of the observations given the explanation.
• Prior: Probability of the explanation based on prior experiences.
• Posterior: Probability of the explanation given the observations.

Model Selection

What happens when we have multiple explanations $\mathcal{H}_1, \mathcal{H}_2$ for the data? $\mathcal{H}^* = arg \max_{\mathcal{H}_i} P(\mathcal{H}_i|y)$

• Find the most probable explanation by maximisation
• Automatically embodies Occams’ Razor

Expected Utility Maximisation

$a^* = arg \max_a \int p(x | y) \cdot U(a, x) dy$

• Utility function $U(x, a)$ encodes our own preferences
• Choose the action that maximises utility!

Ideal-Observer Analysis

Derive a theoretically optimal model on how to perform a given task.

Perceptual Decision Making

In simple perceptual decision tasks, humans perform close to optimal.

Process Models

Artificial Neural Networks

Study mathematical models of the brain.

Reinforcement Learning

Let artificial agents solve real tasks.

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