Uncertainty
The world is riddled with uncertainty.

Noise

Ambiguity

Partial Information

Why do we see a convex face when the mask turns?
In which direction do the bars move? Why do we see this movement in particular and not one of the other possibilities?
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.
Posterior Update

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
I give you two different models \(m_1\) and \(m_2\) .
\(m_1\) has many parameters and is thus very flexible, whereas \(m_2\) is a simple model.
Both can explain the data equally well.
Which one should I choose?
Occams’ Razor: Choose the simplest explanation compatible with the observations.
Complex explanations must spread their probability mass more thinly
How does Bayesian inference help us to study decision making?
We have seen how probability theory can be used to represent and updates our beliefs about the state of the world.

How can we use our inferences of the world to act in it?
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!
Formalize the decision problem as a game against nature.
In this game, nature picks a state \(x\) unknown to us, and then generates an observation \(y\) which we get to see.
We then have to make a decision, that is, we have to choose an action \(a\) .
Finally we have a utility function \(U(x, a)\) , which measures how compatible our action \(a\) and nature’s hidden state \(x\) is with our own preferences.
Essence of what we mean by rational behaviour.
Ideal-Observer Analysis
Derive a theoretically optimal model on how to perform a given task.

Optimality is to be understood in a statistical sense referring to a physical limit of information processing.
Does not refer to a complete absence of errors
Are humans optimal?
Perceptual Decision Making
In simple perceptual decision tasks, humans perform close to optimal.