Introduction

This module was adapted from E.T. Jaynes’ manuscript entitled: “Probability Theory with Applications to Science and Engineering – A Series of Informal Lectures”, 1974. The entire manuscript is available at http://bayes.wustl.edu/etj/science.pdf.html.

A second and significantly expanded edition of this manuscript is available on Amazon. The first 3 chapters of the second edition are available here http://bayes.wustl.edu/etj/prob/book.pdf.

Deductive Logic (Boolean Algebra)

Denote propositions by A, B, etc., their denials by AcAc size 12{A rSub { size 8{c} } } {}, BcBc size 12{B rSub { size 8{c} } } {} etc. Define the logical product and logical sum by

AB≡AB≡ size 12{ ital "AB" equiv } {} “Both A and B are true”

A+B≡A+B≡ size 12{A+B equiv } {} “At least one of the propositions, A, B are true”

Deductive reasoning then consists of applying relations such as

A+A=AA+A=A size 12{A+A=A} {};

A (B+C)=(AB)+(AC)A (B+C)=(AB)+(AC) size 12{ ital "A " \( B+C \) = \( ital "AB" \) + \( ital "AC" \) } {};

if D = AcBc D = AcBc size 12{ ital "D "= ital " A" rSub { size 8{c} } B rSub { size 8{ ital "c "} } } {}then Dc= A+BDc= A+B size 12{D rSub { size 8{c} } = ital " A"+B} {}.

Inductive Logic (Probability Theory)

Inductive logic is the extension of deductive logic, describing the reasoning of an idealized “robot”, who represents degrees of plausibility of a logical proposition by real numbers:

p(A∣B)p(A∣B) size 12{p \( A \lline B \) } {}= probability of A, given B.

We use the original term “robot” advocated by Jaynes, it is intended to mean the use of inductive logic that follows a set of consistent rules that can be agreed upon. In this formulation of probability theory, conditional probabilities are fundamental. The elementary requirements of common sense and consistency determine these basic rules of reasoning (see Jaynes for the derivation).

In these rules, one can think of the proposition CC size 12{C} {} being the prior information that is available to assign probabilities to logical propositions, but these rules are true without this interpretation.

Rule 1: p(AB∣C)= p(A∣BC)p(B∣C)=p(B∣AC)p(A∣C)p(AB∣C)= p(A∣BC)p(B∣C)=p(B∣AC)p(A∣C) size 12{p \( ital "AB" \lline C \) = ital " p" \( A \lline ital "BC" \) p \( B \lline C \) =p \( B \lline ital "AC" \) p \( A \lline C \) } {}

Rule 2: p(A∣B)+p(Ac∣B)=1p(A∣B)+p(Ac∣B)=1 size 12{p \( A \lline B \) +p \( A rSub { size 8{c} } \lline B \) = 1} {}

Rule 3: p(A+B∣C)=p(A∣C)+p(B∣C)−p(AB∣C)p(A+B∣C)=p(A∣C)+p(B∣C)−p(AB∣C) size 12{p \( A+B \lline C \) =p \( A \lline C \) +p \( B \lline C \) - p \( ital "AB" \lline C \) } {}

Rule 4: If {A1,…AN}{A1,…AN} size 12{ lbrace A rSub { size 8{1} } , dotslow A rSub { size 8{N} } rbrace } {}are mutually exclusive and exhaustive, and information BB size 12{B} {}is indifferent to tem; i.e. if BB size 12{B} {} gives no preference to one over any other then:

p(Ai∣B)=1/n,i=1…np(Ai∣B)=1/n,i=1…n size 12{p \( A rSub { size 8{i} } \lline B \) =1/n,i=1 dotslow n} {} (principle of insufficient reason)

From rule 1 we obtain Bayes’ theorem:

p ( A ∣ BC ) = p ( A ∣ C ) p ( B ∣ AC ) p ( B ∣ C ) p ( A ∣ BC ) = p ( A ∣ C ) p ( B ∣ AC ) p ( B ∣ C ) size 12{p \( A \lline ital "BC" \) =p \( A \lline C \) { {p \( B \lline ital "AC" \) } over {p \( B \lline C \) } } } {}

From Rule 3, if {A1,…AN}{A1,…AN} size 12{ lbrace A rSub { size 8{1} } , dotslow A rSub { size 8{N} } rbrace } {}are mutually exclusive,

p ( A 1 + … A N ∣ B ) = ∑ i = 1 n p ( A i ∣ B ) p ( A 1 + … A N ∣ B ) = ∑ i = 1 n p ( A i ∣ B ) size 12{p \( A rSub { size 8{1} } + dotslow A rSub { size 8{N} } \lline B \) = Sum cSub { size 8{i=1} } cSup { size 8{n} } {p \( A rSub { size 8{i} } \lline B \) } } {}

If in addition, the AiAi size 12{A rSub { size 8{i} } } {}are exhaustive, we obtain the chain rule:

p ( B ∣ C ) = ∑ i = 1 n p ( BA i ∣ C ) = ∑ i = 1 n p ( B ∣ A i C ) p ( A i ∣ C ) p ( B ∣ C ) = ∑ i = 1 n p ( BA i ∣ C ) = ∑ i = 1 n p ( B ∣ A i C ) p ( A i ∣ C ) size 12{p \( B \lline C \) = Sum cSub { size 8{i=1} } cSup { size 8{n} } {p \( ital "BA" rSub { size 8{i} } \lline C \) } = Sum cSub { size 8{i=1} } cSup { size 8{n} } {p \( B \lline A rSub { size 8{i} } C \) } p \( A rSub { size 8{i} } \lline C \) } {}

Prior Probabilities

The initial information available to the robot at the beginning of any problem is denoted by XX size 12{X} {}. p(A∣X)p(A∣X) size 12{p \( A \lline X \) } {}is then the prior probability of AA size 12{A} {}. Applying Bayes’ theorem to take account of new evidence EE size 12{E} {}yields the posterior probability p(A∣EX)p(A∣EX) size 12{p \( A \lline ital "EX" \) } {}. In a posterior probability we sometimes leave off the XX size 12{X} {} for brevity: p(A∣E)≡p(A∣EX).p(A∣E)≡p(A∣EX). size 12{p \( A \lline E \) equiv p \( A \lline ital "EX" \) "." } {}

Prior probabilities are determined by Rule 4 when applicable; or more generally by the principle of maximum entropy.

Decision Theory

Enumerate the possible decisions D1,…DkD1,…Dk size 12{D rSub { size 8{1} } , dotslow D rSub { size 8{k} } } {}and introduce the loss function L(Di,θi)L(Di,θi) size 12{L \( D rSub { size 8{i} } ,θ rSub { size 8{i} } \) } {}representing the “loss” incurred by making decision DiDi size 12{D rSub { size 8{i} } } {}if θjθj size 12{θ rSub { size 8{j} } } {}is the true state of nature. After accumulating new evidence E, make that decision DiDi size 12{D rSub { size 8{i} } } {}which minimizes the expected loss over the posterior distribution of θjθj size 12{θ rSub { size 8{j} } } {} :

Choose the decision DiDi size 12{D rSub { size 8{i} } } {}which minimizes 〈L〉i=∑jL(Di,θj)p(θj∣EX)〈L〉i=∑jL(Di,θj)p(θj∣EX) size 12{ langle L rangle rSub { size 8{i} } = Sum cSub { size 8{j} } {L \( D rSub { size 8{i} } ,θ rSub { size 8{j} } \) p \( θ rSub { size 8{j} } \lline ital "EX" \) } } {}

choose D i such that is minimized choose D i such that is minimized size 12{"choose "D rSub { size 8{i} } " such that is minimized"} {}