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The calculations, as in [link] , are simple but can be tedious. We have an m-procedure called bayes to perform the calculations easily. The probabilities P ( A i ) are put into a matrix PA and the conditional probabilities P ( E | A i ) are put into matrix PEA. The desired probabilities P ( A i | E ) and P ( A i | E c ) are calculated and displayed

Matlab calculations for [link]

>>PEA = [0.10 0.02 0.06];>>PA = [0.2 0.5 0.3];>>bayes Requires input PEA = [P(E|A1) P(E|A2) ... P(E|An)]and PA = [P(A1) P(A2) ... P(An)] Determines PAE = [P(A1|E) P(A2|E) ... P(An|E)]and PAEc = [P(A1|Ec) P(A2|Ec) ... P(An|Ec)] Enter matrix PEA of conditional probabilities PEAEnter matrix PA of probabilities PA P(E) = 0.048P(E|Ai) P(Ai) P(Ai|E) P(Ai|Ec) 0.1000 0.2000 0.4167 0.18910.0200 0.5000 0.2083 0.5147 0.0600 0.3000 0.3750 0.2962Various quantities are in the matrices PEA, PA, PAE, PAEc, named above

The procedure displays the results in tabular form, as shown. In addition, the various quantities are in the workspace in the matrices named, so that theymay be used in further calculations without recopying.

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The following variation of Bayes' rule is applicable in many practical situations.

(CP3*) Ratio form of Bayes' rule P ( A | C ) P ( B | C ) = P ( A C ) P ( B C ) = P ( C | A ) P ( C | B ) P ( A ) P ( B )

The left hand member is called the posterior odds , which is the odds after knowledge of the occurrence of the conditioning event. The second fraction in the right hand member is the prior odds , which is the odds before knowledge of the occurrence of the conditioning event C . The first fraction in the right hand member is known as the likelihood ratio . It is the ratio of the probabilities (or likelihoods) of C for the two different probability measures P ( | A ) and P ( | B ) .

A performance test

As a part of a routine maintenance procedure, a computer is given a performance test. The machine seems to be operating so well that the prior odds it is satisfactory are takento be ten to one. The test has probability 0.05 of a false positive and 0.01 of a false negative. A test is performed. The result is positive. What are the posterior odds the device isoperating properly?

SOLUTION

Let S be the event the computer is operating satisfactorily and let T be the event the test is favorable. The data are P ( S ) / P ( S c ) = 10 , P ( T | S c ) = 0 . 05 , and P ( T c | S ) = 0 . 01 . Then by the ratio form of Bayes' rule

P ( S | T ) P ( S c | T ) = P ( T | S ) P ( T | S c ) P ( S ) P ( S c ) = 0 . 99 0 . 05 10 = 198 so that P ( S | T ) = 198 199 = 0 . 9950
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The following property serves to establish in the chapters on "Independence of Events" and "Conditional Independence" a number of important properties for the concept of independence and of conditional independence of events.

(CP4) Some equivalent conditions If 0 < P ( A ) < 1 and 0 < P ( B ) < 1 , then

P ( A | B ) * P ( A ) iff P ( B | A ) * P ( B ) iff P ( A B ) * P ( A ) P ( B ) and
P ( A B ) * P ( A ) P ( B ) iff P ( A c B c ) * P ( A c ) P ( B c ) iff P ( A B c ) P ( A ) P ( B c )

where * is < , , = , , or > and is > , , = , , or < , respectively.

Because of the role of this property in the theory of independence and conditional independence, we examine the derivation of these results.

VERIFICATION of (CP4)

  1. P ( A B ) * P ( A ) P ( B ) iff P ( A | B ) * P ( A ) (divide by P ( B ) — may exchange A and A c )
  2. P ( A B ) * P ( A ) P ( B ) iff P ( B | A ) * P ( B ) (divide by P ( A ) — may exchange B and B c )
  3. P ( A B ) * P ( A ) P ( B ) iff [ P ( A ) - P ( A B c ) ] * P ( A ) [ 1 - P ( B c ) iff - P ( A B c ) * - P ( A ) P ( B c ) iff P ( A B c ) P ( A ) P ( B c )
  4. We may use c to get P ( A B ) * P ( A ) P ( B ) iff P ( A B c ) P ( A ) P ( B c ) iff P ( A c B c ) * P ( A c ) P ( B c )

A number of important and useful propositons may be derived from these.

  1. P ( A | B ) + P ( A c | B ) = 1 , but, in general, P ( A | B ) + P ( A | B c ) 1 .
  2. P ( A | B ) > P ( A ) iff P ( A | B c ) < P ( A ) .
  3. P ( A c | B ) > P ( A c ) iff P ( A | B ) < P ( A ) .
  4. P ( A | B ) > P ( A ) iff P ( A c | B c ) > P ( A c ) .

VERIFICATION — Exercises (see problem set)

Repeated conditioning

Suppose conditioning by the event C has occurred. Additional information is then received that event D has occurred. We have a new conditioning event C D . There are two possibilities:

  1. Reassign the conditional probabilities. P C ( A ) becomes
    P C ( A | D ) = P C ( A D ) P C ( D ) = P ( A C D ) P ( C D )
  2. Reassign the total probabilities: P ( A ) becomes
    P C D ( A ) = P ( A | C D ) = P ( A C D ) P ( C D )

Basic result : P C ( A | D ) = P ( A | C D ) = P D ( A | C ) . Thus repeated conditioning by two events may be done in any order, or may be done in one step. This result extends easilyto repeated conditioning by any finite number of events. This result is important in extending the concept of "Independence of Events" to "Conditional Independence" . These conditions are important for many problems of probable inference.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Good
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
Samson Reply

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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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