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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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
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
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele 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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