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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 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

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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signal and information processing for sonar. OpenStax CNX. Dec 04, 2007 Download for free at http://cnx.org/content/col10422/1.5
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