<< Chapter < Page Chapter >> Page >
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 size 12{A rSub { size 8{c} } } {} , B c size 12{B rSub { size 8{c} } } {} etc. Define the logical product and logical sum by

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

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

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

if D = A c B c size 12{ ital "D "= ital " A" rSub { size 8{c} } B rSub { size 8{ ital "c "} } } {} then D c = 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 ) 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 C 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 ) 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 ( A c 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 ) size 12{p \( A+B \lline C \) =p \( A \lline C \) +p \( B \lline C \) - p \( ital "AB" \lline C \) } {}

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

p ( A i 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 ) 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 { A 1 , A N } 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 ) 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 A i 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 ) 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 X size 12{X} {} . p ( A X ) size 12{p \( A \lline X \) } {} is then the prior probability of A size 12{A} {} . Applying Bayes’ theorem to take account of new evidence E size 12{E} {} yields the posterior probability p ( A EX ) size 12{p \( A \lline ital "EX" \) } {} . In a posterior probability we sometimes leave off the X size 12{X} {} for brevity: 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 D 1 , D k size 12{D rSub { size 8{1} } , dotslow D rSub { size 8{k} } } {} and introduce the loss function L ( D i , θ i ) size 12{L \( D rSub { size 8{i} } ,θ rSub { size 8{i} } \) } {} representing the “loss” incurred by making decision D i size 12{D rSub { size 8{i} } } {} if θ j size 12{θ rSub { size 8{j} } } {} is the true state of nature. After accumulating new evidence E, make that decision D i size 12{D rSub { size 8{i} } } {} which minimizes the expected loss over the posterior distribution of θ j size 12{θ rSub { size 8{j} } } {} :

Choose the decision D i size 12{D rSub { size 8{i} } } {} which minimizes L i = j L ( D i , θ j ) p ( θ j 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 size 12{"choose "D rSub { size 8{i} } " such that is minimized"} {}

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signal and information processing for sonar. OpenStax CNX. Dec 04, 2007 Download for free at http://cnx.org/content/col10422/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signal and information processing for sonar' conversation and receive update notifications?

Ask