Symposium on current views of subjective probability: Subjective probability as the measure of a non-measurable set

authors: Irving John Good year: 1966 See in Zotero

Literature Notes

Main point

Though many subjective probabilies aren’t actually measurable, like an unmeasurable set, we can extend the measure to produce some bounds on what the measure could/should be. Then, with these bounds we can still do a lot of things with them based on the axioms provided.

Types of probability

• Physical
• Psycological - what people have in their brains
• Subjective - mature judgement + probability theory → after some updating what people “should” or “might” have in their brains
• Logical - Subjective probability if you were to know everything about everything.

Probability and our brains

This seems to be written before or as the two disciplines were combining.

Orgs are always trying to improve their judgements (to understand the world more or better) and to find inconsistencies. In order to do this, there is some /magic/ process (mostly unobservable) going on inside our brains where we take past judgements, evidence, and suggestions (from theorems, math, etc.) and make a discernment (new opinion/ judgement) that goes into our whole body of beliefs. People’s judgements (beliefs) usually aren’t concrete enough to be explained as an equality, like $P(X|Y)=x\in \mathbb{R}.$

They are usually in preferences, or thinking that something is more likely than something else: Like $P^{\prime }(X|Y)\ge P^{\prime }(Z|Y)$

Then there is a translation process that happens between the /black box/ or magic process and the judgements that we continue to believe in before/ after the process.

• Probability and measure theory Good likens these subjective probabilities that we have as beliefs as measures. They are often unmeasurable. However, we can define $P^{\ast }(X|Y)=inf\{x\in \mathbb{R}|P(X|Y)<x\}$ $P_{\ast }(X|Y)=sup\{x\in \mathbb{R}|P(X|Y)>x\}$

These act similarly to outer and inner measures.