Download Introduction to Probability II

Introduction to Probability II
Eleisa Heron
Neuropsychiatric Genetics Research Group
Trinity College Dublin
22/10/08
Review – Probability I
– Origins of probability – 17th C. gambling, frequentist and
Bayesian interpretations
– Definitions – trial, outcome, event, sample space
– Probability Rules – [0,1], impossible and certain events
– Mutually/Non-Mutually Exclusive Events – OR means add rule (ME)
Venn diagrams
– Conditional Probability – conditional probability fallacy
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Bayes’ Theorem
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Conditional probability for event
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Using this we can say
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Giving Bayes’ theorem
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given that event
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has occurred
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Bayes’ Theorem and Bayesian Inference
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Sometimes, Bayes’ theorem is written in the following way
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In Bayesian inference, given data and parameters
want to estimate the parameters
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of a model, we
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Bayes’ Theorem Example
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Court Setting
Suppose a juror wants to combine DNA evidence that is presented to him/her
together with their own prior beliefs in order to decide if defendant is guilty or not
Evidence: DNA that matches the defendant’s DNA was found at the crime scene
•
•
•
is the event the defendant is guilty
is the event the defendant is not guilty
i is the event that the defendant’s DNA matches the DNA found at the crime
scene
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Interested in
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Probability of being guilty before any evidence is presented
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Probability of the event
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Bayes’ Theorem Example
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From Bayes’ theorem we have
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Suppose a male committed the crime in a town with a male population of
20,000
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Juror uses this for his prior
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This gives
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Heads and tails – 0’s and 1’s
10110011001111111110
11010110000111111010
00100111000010000001
00101111001001000011
10110100110000110010
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1010010100101001011
0
0101011011010101101
0
0010010101001011010
1
1010101100101001011
0
0100110101001011001
0
Which sequence have I made up and which is a true simulation?
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Boys and Girls
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Five children in a family,
assume probability of a boy (B) = probability of a girl (G) = 0.5
GGGGG
BBGBG
BBBBB
Which has higher probability?
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Independent Events
• Two events
and
are independent if
• Toss a coin and get a head
Toss the coin again
Probability of getting a head again is independent of the first toss and
the fact that we got a head
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Independent Events
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For independent events
and
then
AND means MULTIPLY for independent events
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Toss two coins
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Independent Events
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Can mutually exclusive events be independent?
Two events A and B
Mutually exclusive
Not mutually exclusive
Independent
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Dependent
Assuming A and B are non-trivial events
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Marginal Probability
•
.
the marginal probability of an event
probability of the event
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The probability of the event regardless of what other events occurred
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Disease screening example
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is the unconditional
Positive
Negative
Disease
P(Dis, Pos)
P(Dis, Neg)
P(Disease)
Well
P(Well, Pos)
P(Well, Neg)
P(Well)
P(Positive)
P(Negative)
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Odds
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Odds of an event are defined as the ratio of the probability that the
event will occur to the probability that the event will not occur
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To convert from odds to probability
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Probability more intuitive, but odds used in gambling and in analysis of
binary outcome variables (logistic regression) for example
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Odds Cont’d
•
Odds lie between
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and
, since probability lies between
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and
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Odds Ratio
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Odds ratio (OR) is a measure of effect size
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OR defined as the ratio of the odds of an event occurring in one group
to the odds of the event occurring in another group
is the probability of the event in the first group
is the probability of the event in the second group
OR =1: the condition or event is equally likely in both groups
OR >1: the condition or event is more likely in the first group
OR <1: the condition or event is less likely in the first group
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Again, just like odds, OR’s lie between
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Pink and Blue Cards
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Pick a card
– What is the probability card is blue on both sides?
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Pick a card, card is blue on one side
– What is the probability card is blue on the other side?
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Pink and Blue Cards Cont’d
Card Picked
Pink Pink
1/3
1/3
First
Second
1/2
Pink Pink
1/2
Pink Pink
1/2
Pink Blue
1/2
Blue Pink
1/2
Blue Blue
1/2
Blue Blue
Pink Blue
1/3
Blue Blue
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Pink and Blue Cards Cont’d
•
is the event that the first card is blue and
second card is blue
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is the event that the
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Goats and Car – Monty Hall Problem
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Game Show
– 3 doors, a car behind one and a goat behind each of the others
– Contestant picks one of the 3 doors
– Game show host (who knows which door conceals the car) opens
one of the remaining two doors to reveal a goat
– Contestant offered the chance to swap the door they originally
chose for the remaining door
Should the contestant swap?
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Goats and Car – Monty Hall Problem Cont’d
Car location
Host Opens
1/3
Door 2
1/3
Door 3
Don’t Swap Swap
1/2
Door 2
1/6
Car
Goat
1/2
Door 3
1/6
Car
Goat
Door 3
1/3
Goat
Car
Door 2
1/3
Goat
Car
Door 1
1/3
Prob. Total
1
1
Assume Door 1 is chosen
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Goats and Car – Monty Hall Problem Cont’d
Assume contestant chooses door 1 and the presenter opens door 3
is the event that the presenter opens door 3
is the event that the car is behind door 1, similarly
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and
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Goats and Car – Monty Hall Problem Cont’d
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References
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Most introductory statistics books have a probability section at the start
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Statistics for Technology by C. Chatfield
Basic introductory probability
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Introduction to Probability by Charles M. Grinstead, J. Laurie Snell (1997)
More in depth
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Weighing the Odds: A Course in Probability and Statistics by D. Williams (2001)
More advanced, looking at both frequentist and Bayesian approaches
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