# Summary of basic rules for probability theory

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Provides a summary of the rules of inductive reasoning, as advocated by E.T. Jaynes. Includes probability rules, and decision 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

$A+A=A$ ;

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

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:

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

Rule 2: $p\left(A\mid B\right)+p\left({A}_{c}\mid B\right)=1$

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

Rule 4: If $\left\{{A}_{1},\dots {A}_{N}\right\}$ 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:

$p\left({A}_{i}\mid B\right)=1/n,i=1\dots n$ (principle of insufficient reason)

From rule 1 we obtain Bayes’ theorem:

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

From Rule 3, if $\left\{{A}_{1},\dots {A}_{N}\right\}$ are mutually exclusive,

$p\left({A}_{1}+\dots {A}_{N}\mid B\right)=\sum _{i=1}^{n}p\left({A}_{i}\mid B\right)$

If in addition, the ${A}_{i}$ are exhaustive, we obtain the chain rule:

$p\left(B\mid C\right)=\sum _{i=1}^{n}p\left({\text{BA}}_{i}\mid C\right)=\sum _{i=1}^{n}p\left(B\mid {A}_{i}C\right)p\left({A}_{i}\mid C\right)$

## Prior probabilities

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

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 $L\left({D}_{i},{\theta }_{i}\right)$ 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 ${⟨L⟩}_{i}=\sum _{j}L\left({D}_{i},{\theta }_{j}\right)p\left({\theta }_{j}\mid \text{EX}\right)$

$\text{choose}{D}_{i}\text{such that is minimized}$

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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Sherica
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Sherica
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Tamia
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
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Asali
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Samantha
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Asali
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Cied
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Porter
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Stotaw
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Azam
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Azam
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Prasenjit
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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