Exam Questions - Block 1: Subjective Probability + Bayesian Statistics

[SebastianVeuskens]

Subjective probability and Bayesian statistics

[DesR99]

Subjective Probability

Subjective probability measures the level of plausibility assigned by a person to an uncertain event and the level of plausibility can change from person to person. The probability is not a physical property of the event and is instead based on the beliefs held by the individual. This is what makes it subjective i.e. it is based on personal opinions.

Mathematically

An individual assigns a probability to event E denoted as P(E) = p which is the price the individual considers fair to place a bet where they will win 0 if E does not occur and 1 if E occurs. The assessment is said to be coherent if no combination of events can guarantee a sure win. It is necessary that 0 ≤ P(E) ≤ 1. Where: P(E) = 0 means E is impossible and P(E) = 1 means E is certain.

Bayesian Statistics

Bayesian Statistics are based on the relationships between probability and inductive reasoning. There are a number of assumptions required for Bayesian statistics which include:

1. Series in the experiment are exchangeable.
2. The statistical inference is based on the final distribution and firstly an initial distribution must be assigned.
3. All of the individual's knowledge can be used to form the initial distribution.
4. Parameter estimation needs the description and a synthesis of the final distribution.

[SebastianVeuskens]

Great answer!

I want to emphasize the main assumption of the Bayesian paradigm:

Prior Probabilities can be assigned, starting from an initial probability distribution.

Bayes theorem allows us to update probabilites that represent our believes, after obtaining new data. It thereby ensures the coherence of the obtained solutions.

A bayesian result is only a mathematical consequence and thus cannot be formally wrong.

Here the cited source.