Summary of basic rules for probability theory

“Probability theory is nothing but common sense reduced to calculation” (Laplace).

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 (External Link) .

A second and significantly expanded edition of this manuscript is available on Amazon. The first 3 chapters of the second edition are available here (External Link) .

Deductive logic (boolean algebra)

Denote propositions by A, B, etc., their denials by $A_c$ , $B_c$ etc. Define the logical product and logical sum by

$\begin{array}{}\text{AB}\equiv \\ \end{array}$ “Both A and B are true”

$\begin{array}{}A+B\equiv \\ \end{array}$ “At least one of the propositions, A, B are true”

Deductive reasoning then consists of applying relations such as

$A+A=A$ ;

$\text{A}(B+C)=(\text{AB})+(\text{AC})$ ;

if $\text{D}={\text{A}}_c{B}_{\text{c}}$ then $D_c=\text{A}+B$ .

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:

$p(A\mid B)$ = 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 $C$ being the prior information that is available to assign probabilities to logical propositions, but these rules are true without this interpretation.

Rule 1: $p(\text{AB}\mid C)=\text{p}(A\mid \text{BC})p(B\mid C)=p(B\mid \text{AC})p(A\mid C)$

Rule 2: $p(A\mid B)+p(A_c\mid B)=1$

Rule 3: $p(A+B\mid C)=p(A\mid C)+p(B\mid C)-p(\text{AB}\mid C)$

Rule 4: If $\{A_1,\dots A_N\}$ are mutually exclusive and exhaustive, and information $B$ is indifferent to tem; i.e. if $B$ gives no preference to one over any other then:

$p(A_i\mid B)=1/n,i=1\dots n$ (principle of insufficient reason)

From rule 1 we obtain Bayes’ theorem:

$p(A\mid \text{BC})=p(A\mid C)\frac{p(B\mid \text{AC})}{p(B\mid C)}$

From Rule 3, if $\{A_1,\dots A_N\}$ are mutually exclusive,

$p(A_1+\dots A_N\mid B)=\sum _{i=1}^{n}p(A_i\mid B)$

If in addition, the $A_i$ are exhaustive, we obtain the chain rule:

$p(B\mid C)=\sum _{i=1}^{n}p({\text{BA}}_i\mid C)=\sum _{i=1}^{n}p(B\mid A_{i}C)p(A_i\mid C)$

Prior probabilities

The initial information available to the robot at the beginning of any problem is denoted by $X$ . $p(A\mid X)$ is then the prior probability of $A$ . Applying Bayes’ theorem to take account of new evidence $E$ yields the posterior probability $p(A\mid \text{EX})$ . In a posterior probability we sometimes leave off the $X$ for brevity: $p(A\mid E)\equiv p(A\mid \text{EX})\text{.}$

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

Decision theory

Enumerate the possible decisions $D_1,\dots D_k$ and introduce the loss function $L(D_i,{\theta }_i)$ representing the “loss” incurred by making decision $D_i$ if ${\theta }_j$ is the true state of nature. After accumulating new evidence E, make that decision $D_i$ which minimizes the expected loss over the posterior distribution of ${\theta }_j$ :

Choose the decision $D_i$ which minimizes ${⟨L⟩}_i=\sum _{j}L(D_i,{\theta }_j)p({\theta }_j\mid \text{EX})$

$\text{choose}{D}_i\text{such that is minimized}$