S 2.1 Probability and information: the fundamental notation

• The generic task: we need to attach a probability to one `proposition' given some other `proposition'.

• The word `proposition' expresses the role of probability as an extension of the rules of logic into the grey area lying between the extremes of `truth' and `falsehood'.

• We shall denote the proposition that is `given' by I, standing for information .

• We shall denote the proposition whose probability is at issue by A, standing for `assertion'.

• There are many other words which may (in specific circumstances) be more appropriate than these two.

  • Other words for `assertion': `result', `event', `inference', `prediction'...

  • Other words for `information': `premise', `context', `data', `experiment', `hypothesis'...

  Examples:
  

  • I: `It rained yesterday.'

    A: `It will rain today.'

  • I: `I have selected one card from this pack'

    A: `The card will be the ace of spades.'

  • I: `I bought a lottery ticket.'

    A: `I will win the lottery this week.'

  • I: `This is an atom of Argon'

    A: `It will be found in its lowest energy state'

• Clearly, the probability attached to A will depend upon I.

  Example:
  Go back to the above examples; click on the `more information' button Consider the implications of the additional information for the probability to be attached to A.

• It is essential that the notation shows this dependence.

• We denote the probability of the `assertion' A in the light of the `information' I by P(A|I). Thus

  Key Point 2.1
  

  P(A|I) is the probability of A given I.

• The truthfulness of I is irrelevant to the value of P(A|I) but essential to its use.

• The `|I' is sometimes omitted for notational simplicity, but at the risk of confusions or paradoxes (see eg EQ1 ).

• The loose definitions of probabilities provided in S1.2 can now be written more formally:

  Key Point 2.2
  

  Classical (frequentist) definition:

  $$P(A\vert I) = lim_{N(I)\rightarrow \infty} \frac{ N(A)} {N(I)}$$

  where N(I) is the number of times the circumstances described in I are realised and N(A) is the number (of these) for which proposition A is found to be true. Thus P(A|I) is the fraction of instances of I that give A, in the `long-run'.

  Key Point 2.3
  

  Degree-of-belief definition:
  

  P(A|I) is the degree of rational belief that, in the circumstances specified in I, the proposition A is true.

Questions for you to do at this point: EQ1