Probability Theory

• What we need is a mathematical description of uncertainty.

• Probability distributions make intuitive sense for data, because they encode what might have been.

• How is it that they can also be used to quantify degree of belief in a model, or parameter value?

The Rules of Probability Theory

Cox Axioms. Plausibilities. If the model is true, it could take this value with this probability, or that one with this probability, etc.

Marginalization. Conditional probability

Example: Probability in a Court of Law

Plausibility of guilt.

Q: predict-test-discuss the probability of guilt in some court case example

Belief, Odds and Acceptance

Degree of belief in proposition quantified as odds you would accept.

Frequentism, where only the data have a probability distribution, puts emphasis on datasets that could have been taken but were not.

Bayesian analysis puts emphasis on understanding the data you have, and gaining understanding from that data.