3.
Bayes' Theorem
• also called, Bayes' law or Bayes' rule
• based on probability theory,
mathematics and Statistics
• named after Thomas Bayes (/ˈbeɪz/
1701–1761)
• Bayes suggested the theorem to
update beliefs
• Significantly edited and updated by
Richard Price
• Bayes' theorem “is to the theory of
probability what Pythagoras's
theorem is to geometry” -Harold
Jeffreys
4.
probability theory
rooted in,
• the science of
Statistics
• the science of
Mathematics
• the probability of an
event occurring
• the analysis of
random phenomena
5.
statistical theory
gives,
–ways of comparing
statistical
procedures
–a best possible
procedure for
statistical problems
–guidance on the
choice between
alternative
procedures
Passenger Identities Number
China/Taiwan 154
Malaysia 38
India 5
Indonesia 7
Australia 6
France 4
USA 3
New Zealand 2
Canada 2
Russia 1
Italy 1
Netherlands 1
Austria 1
6.
statistical theory
covers
– Statistical decision
methods and problems
– Statistical inferences or
conclusions
– actions and deductions
To satisfy the basic principles
stated for these different
approaches.
STATISTICS(FACTS)
Aircraft: Malaysian Air Lines
Aircraft Age: 11years 10
months
Aircraft Type: Boeing 777 ER
Flight No: MH370(shared with
China Southern Airlines)
Passengers travelled: 227
Crew Travelled: 12
Nationalities: 14
Take Off Time: 12.41 am
From: Kuala Lumpur, Malaysia
To: Beijing, China
7.
Statistics
• is the study of the
collection, organization,
analysis, interpretation and
presentation of information
or data
• all aspects of data including
the planning of data
collection in terms of the
design of surveys and
experiments are used
8.
mathematical roots
• Two mathematical results of Bayes Theorem
are,
–THE LAW OF LARGE NUMBERS, and
–THE CENTRAL LIMIT THEOREM
9.
law of large numbers (LLN)
• Is a theorem
• describes the result of
performing the same
experiment a large
number of times
• thus the average of results
obtained should be close
to the expected value, and
• The result will tend to
become closer to reality
10.
central limit theorem (CLT)
• the arithmetic mean of a
sufficiently large number of
iterates of independent
random variables, each with a
well-defined expected value
and well-defined variance,
will be approximately
normally distributed
• This will probably lead to a
conclusion of the reality
11.
normal distribution
• suppose
– sample contains a large number of
observations,
– each observation being randomly
generated,
– in a way that does not depend on the
values of the other observations
– that the arithmetic average of the
observed values is computed,
– and if this procedure is performed
many times,
– then central limit theorem says the
computed values of the average will be
a “normal distribution“
– The normal distribution is commonly
known as a "bell curve"
12.
Bayes' theorem
is,
• a result that is of importance
in the mathematical
manipulation of conditional
probabilities
• It is a result that derives from
the more basic axioms of
probability
• It is a result that can lead to
realistic conclusions
13.
Bayesian (or epistemological)
interpretation
• probability measures a degree of belief
• Bayes' theorem then links the degree of
belief
• in a proposition before and after
accounting for evidence.
• example, take a proposal that a biased
coin is twice as likely to land heads than tails.
– Degree of belief in this might initially be
50%.
– The coin is then flipped a number of
times to collect evidence.
– Belief may rise to 70% if the evidence
supports the proposition
14.
axioms of probability
probability is,
• a game of chance
• measures the real, physical
tendency of something to occur
• Measures how strongly one
believes it will occur
• draws on both these elements?
• interprets the probability values
of probability theory to answer
the above questions
15.
real life probability
Event Or Occurrence Non-Random event Random Event
Weather Seasonal Change, Local
Climate
Precipitation, Temperature on
Specific Days
Car Accidents Safe or Unsafe Driving
Practices
Specific Cars or Conditions
Met on the Road
Customers at Mall Hours Open, Time of the Day Specific Pattern of Customer
Arrival
State Lottery Decision About Games
Available, Prices and Awards
Numbers Drawn or Winning
Patterns on Tickets
Plane Crash Ability of Flight Captains,
Security Measures
Plane Engine Failure,
Terrorists Intrusion
Student Grades Amount of Time of study,
Revision and Class
Preparation
Appearance of Specific
questions on Tests
17.
why MH370 flight missing?
• Its loss is a mystery
• Many theories, probabilities
and interpretations still
remain
• Inconclusive, inadequate
and intriguing
• Probably Bayes Theoretical
approach would help
• Scientists tend to believe in
this theory as a probable
lead to discovery of the
missing plane
18.
the theory that would not die
• “It‟s a very short,
simple equation that
says you can start out
with hypothesis about
something - and it
doesn‟t matter how
good the hypothesis
is,”
• - Sharon Bertsch McGrayne
19.
how
because of,
• this character of the
formula
• forcing researchers to
change their hypothesis
with each new information
• that the probability
becomes more accurate
20.
the proof
• Bayes’ Theorem “allows the
organisation of available data with
associated uncertainties and
computation of the PDF
• Scientists applied Bayes’ Theorem in
the Air France incident of 2009
• French scientist Pierre Simon Laplace,
helped locate German U-boats during
World War II and spot Soviet
submarines during the Cold War
• More recently, it is used in Google's
"driverless cars" project and in stock
market predictions
• PDF or the probability distribution
function, identified the target
location given these data"
21.
Why Not?
• Advocates of this theory said it
was also used in the search for the
black box of the ill-fated Air France
flight 447, which crashed in the
vast Atlantic Ocean in June 2009
• What took two years for other
experts in the search for the black
box, took only five days for
consultants who applied the Bayes’
Theorem, to finally find the device
12,000 feet under water.
• In the current search for flight
MH370, it is “highly unlikely” that
Bayes’ Theorem is being applied.
Air France Accident summary
Date 1 June 2009
Summary Entered high altitude stall,
impacted ocean
Site Atlantic Ocean
near waypoint TASIL
[1]
3°03′57″N 30°33′42″WCoordina
tes: 3°03′57″N 30°33′42″W
Passengers 216
Crew 12
Fatalities 228 (all)
Survivors 0
Aircraft type Airbus A330-203
Operator Air France
Registration F-GZCP
Flight origin Rio de Janeiro–Galeão Airport
Destination Paris-Charles de Gaulle Airport
22.
Inconclusive Theories about MH 370
• Sudden shut down of all
flight systems
• Sabotage by suspects
• Hijackers on board
• ‘Undesirable clients’ in the
cockpit
• Deep dive into ocean bed
• Disgruntled or insane flight
officer
• Ill equipped management
and security systems
• Etc., etc., etc…..
23.
learn from experience
• Bayes' Theorem,
–is all about learning
from experience
–one would need
"reasonably accurate
past experiences" for
the theorem to work
–in other words, to
calculate accurately to
locate the plane
24.
find the plane
• „go find the
plane, with
science in your
bag, enough of
spiritual and
emotional beliefs
and traditional
methods‟-masi
25.
thanks
• Malaysian government reports/statements
• Wikipedia.com
• Malaysiakini Radio/TV and the Media
• Photo Gallery
• Gayesian experts like Sharon Bertsch McGrayne
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