Slide 1 A Quick Overview of Probability William W. Cohen Machine Learning 10-601 Jan 2008.

A Quick Overview of Probability

William W. CohenMachine Learning 10-601

Jan 2008

Probabilistic and Bayesian Analytics

Andrew W. MooreProfessor

School of Computer ScienceCarnegie Mellon University

www.cs.cmu.edu/[email protected]

412-268-7599

Note to other teachers and users of these slides. Andrew would be delighted if you found this source material useful in giving your own lectures. Feel free to use these slides verbatim, or to modify them to fit your own needs. PowerPoint originals are available. If you make use of a significant portion of these slides in your own lecture, please include this message, or the following link to the source repository of Andrew’s tutorials: http://www.cs.cmu.edu/~awm/tutorials . Comments and corrections gratefully received.

Copyright © Andrew W. Moore

[Some material pilfered from http://www.cs.cmu.edu/~awm/tutorials]

Announcements

• Recitation this week:• Matlab tutorial• Probability q/a and tutorial

• HW1 is due on Monday

Zeno’s paradox• Lance Armstrong and the

tortoise have a race• Lance is 10x faster• Tortoise has a 1m head start

at time 0

0 1

• So, when Lance gets to 1m the tortoise is at 1.1m• So, when Lance gets to 1.1m the tortoise is at 1.11m …• So, when Lance gets to 1.11m the tortoise is at 1.111m … and Lance will never catch up -?

1+0.1+0.01+0.001+0.0001+… = ?

unresolved until calculus was invented

The Problem of Induction• David Hume (1711-

1776): pointed out1. Empirically,

induction seems to work

2. Statement (1) is an application of induction.

• This stumped people for about 200 years

1. Of the Different Species of Philosophy.

2. Of the Origin of Ideas

3. Of the Association of Ideas

4. Sceptical Doubts Concerning the Operations of the Understanding

5. Sceptical Solution of These Doubts

6. Of Probability9

7. Of the Idea of Necessary Connexion

8. Of Liberty and Necessity

9. Of the Reason of Animals

10. Of Miracles

11. Of A Particular Providence and of A Future State

12. Of the Academical Or Sceptical Philosophy

A Second Problem of Induction

• A black crow seems to support the hypothesis “all crows are black”.

• A pink highlighter supports the hypothesis “all non-black things are non-crows”

• Thus, a pink highlighter supports the hypothesis “all crows are black”.

)(CROW)(BLACK

lyequivalentor

)(BLACK)(CROW

xxx

xxx

A Third Problem of Induction

• You have much less than 200 years to figure it all out.

Probability Theory• Events

• discrete random variables, boolean random variables, compound events

• Axioms of probability• What defines a reasonable theory of uncertainty

• Compound events• Independent events• Conditional probabilities• Bayes rule and beliefs• Joint probability distribution

Discrete Random Variables• A is a Boolean-valued random variable if

• A denotes an event, • there is uncertainty as to whether A occurs.

• Examples• A = The US president in 2023 will be male• A = You wake up tomorrow with a headache• A = You have Ebola• A = the 1,000,000,000,000th digit of π is 7

• Define P(A) as “the fraction of possible worlds in which A is true”• We’re assuming all possible worlds are equally probable

Discrete Random Variables• A is a Boolean-valued random variable if

• A denotes an event, • there is uncertainty as to whether A occurs.

• Define P(A) as “the fraction of experiments in which A is true”• We’re assuming all possible outcomes are equiprobable

• Examples• You roll two 6-sided die (the experiment) and get doubles

(A=doubles, the outcome)• I pick two students in the class (the experiment) and they have

the same birthday (A=same birthday, the outcome)

a possible outcome of an “experiment”

the experiment is not deterministic

Visualizing A

Event space of all possible worlds

Its area is 1Worlds in which A is False

Worlds in which A is true

P(A) = Area ofreddish oval

The Axioms of Probability

• 0 <= P(A) <= 1• P(True) = 1• P(False) = 0• P(A or B) = P(A) + P(B) - P(A and B)

The Axioms Of Probability

(This is Andrew’s joke)

These Axioms are Not to be Trifled With

• There have been many many other approaches to understanding “uncertainty”:

• Fuzzy Logic, three-valued logic, Dempster-Shafer, non-monotonic reasoning, …

• 25 years ago people in AI argued about these; now they mostly don’t• Any scheme for combining uncertain information, uncertain

“beliefs”, etc,… really should obey these axioms• If you gamble based on “uncertain beliefs”, then [you can

be exploited by an opponent] [your uncertainty formalism violates the axioms] - di Finetti 1931 (the “Dutch book argument”)

Interpreting the axioms• 0 <= P(A) <= 1• P(True) = 1• P(False) = 0• P(A or B) = P(A) + P(B) - P(A and B)

The area of A can’t get any smaller than 0

And a zero area would mean no world could ever have A true


The area of A can’t get any bigger than 1

And an area of 1 would mean all worlds will have A true


A

B


A

B

P(A or B)

BP(A and B)

Simple addition and subtraction

Theorems from the Axioms• 0 <= P(A) <= 1, P(True) = 1, P(False) = 0• P(A or B) = P(A) + P(B) - P(A and B)

P(not A) = P(~A) = 1-P(A)

P(A or ~A) = 1 P(A and ~A) = 0

P(A or ~A) = P(A) + P(~A) - P(A and ~A)

1 = P(A) + P(~A) - 0

Elementary Probability in Pictures

• P(~A) + P(A) = 1

A ~A

Side Note

• I am inflicting these proofs on you for two reasons:

1. These kind of manipulations will need to be second nature to you if you use probabilistic analytics in depth

2. Suffering is good for you (This is also Andrew’s joke)

Another important theorem• 0 <= P(A) <= 1, P(True) = 1, P(False) = 0• P(A or B) = P(A) + P(B) - P(A and B)

P(A) = P(A ^ B) + P(A ^ ~B)

A = A and (B or ~B) = (A and B) or (A and ~B)

P(A) = P(A and B) + P(A and ~B) – P((A and B) and (A and ~B))

P(A) = P(A and B) + P(A and ~B) – P(A and A and B and ~B)


• P(A) = P(A ^ B) + P(A ^ ~B)

B

~B

A ^ ~B

A ^ B

The Monty Hall Problem• You’re in a game show.

Behind one door is a prize. Behind the others, goats.

• You pick one of three doors, say #1

• The host, Monty Hall, opens one door, revealing…a goat!

3

You now can either

• stick with your guess

• always change doors

• flip a coin and pick a new door randomly according to the coin

The Monty Hall Problem• Case 1: you don’t

swap.• W = you win.

• Pre-goat: P(W)=1/3• Post-goat: P(W)=1/3

• Case 2: you swap• W1=you picked the

cash initially.• W2=you win.

• Pre-goat: P(W1)=1/3.

• Post-goat:• W2 = ~W1• Pr(W2) = 1-

P(W1)=2/3. Moral: ?

The Extreme Monty Hall/Survivor Problem

• You’re in a game show. There are 10,000 doors. Only one of them has a prize.

• You pick a door.• Over the remaining 13 weeks, the host eliminates

9,998 of the remaining doors.• For the season finale:

• Do you switch, or not?…

Some practical problems

• You’re the DM in a D&D game.• Joe brings his own d20 and throws 4 critical hits in

a row to start off• DM=dungeon master• D20 = 20-sided die• “Critical hit” = 19 or 20

• Is Joe cheating?• What is P(A), A=four critical hits?

• A is a compound event: A = C1 and C2 and C3 and C4

Independent Events

• Definition: two events A and B are independent if Pr(A and B)=Pr(A)*Pr(B).

• Intuition: outcome of A has no effect on the outcome of B (and vice versa).• We need to assume the different rolls are

independent to solve the problem.• You almost always need to assume

independence of something to solve any learning problem.


• You’re the DM in a D&D game.• Joe brings his own d20 and throws 4 critical hits in

a row to start off• DM=dungeon master• D20 = 20-sided die• “Critical hit” = 19 or 20

• What are the odds of that happening with a fair die?

• Ci=critical hit on trial i, i=1,2,3,4 • P(C1 and C2 … and C4) = P(C1)*…*P(C4) =

(1/10)^4Followup: D=pick an ace or king out of deck three times in a row: D=D1 ^ D2 ^ D3


The specs for the loaded d20 say that it has 20 outcomes, X where

• P(X=20) = 0.25

• P(X=19) = 0.25

• for i=1,…,18, P(X=i)= Z * 1/18

• What is Z?

Multivalued Discrete Random Variables

• Suppose A can take on more than 2 values• A is a random variable with arity k if it can take

on exactly one value out of {v1,v2, .. vk}• Thus…

jivAvAP ji if 0)(

1)( 21 kvAvAvAP

More about Multivalued Random Variables

• Using the axioms of probability…0 <= P(A) <= 1, P(True) = 1, P(False) = 0P(A or B) = P(A) + P(B) - P(A and B)

• And assuming that A obeys…

• It’s easy to prove that

jivAvAP ji if 0)(1)( 21 kvAvAvAP

)()(1

21

i

jji vAPvAvAvAP

More about Multivalued Random Variables

• Using the axioms of probabilityand assuming that A obeys…

• It’s easy to prove that

jivAvAP ji if 0)(1)( 21 kvAvAvAP

)()(1

21

i

jji vAPvAvAvAP

• And thus we can prove

1)(1

k

jjvAP


1)(1

k

jjvAP

A=1

A=2

A=3

A=4

A=5

Some practical problemsThe specs for the loaded d20 say that it has 20 outcomes, X

• P(X=20) = P(X=19) = 0.25

• for i=1,…,18, P(X=i)= Z * 1/18 and what is Z?

5.018

11825.025.0)(1

1)(

18

1

1

ZvAP

vAP

jj

k

jj

Some practical problems• You (probably) have 8

neighbors and 5 close neighbors.

• What is Pr(A), A=one or more of your neighbors has the same sign as you?• What’s the

experiment?• What is Pr(B), B=you

and your close neighbors all have different signs?• What about

neighbors?

n c n

c * c

n c n

Moral: ?

Some practical problemsI bought a loaded d20 on EBay…but it didn’t come with any specs. How can I find out how it behaves?

Frequency

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Face Shown

P(X=20) = P(X=19) = 0.25

for i=1,…,18, P(X=i)= 0.5 * 1/18


• I have 3 standard d20 dice, 1 loaded die.

• Experiment: (1) pick a d20 uniformly at random then (2) roll it. Let A=d20 picked is fair and B=roll 19 or 20 with that die. What is P(B)?

P(B) = P(B and A) + P(B and ~A)= 0.1*0.75 + 0.5*0.25 = 0.2

using Andrew’s “important theorem” P(A) = P(A ^ B) + P(A ^ ~B)


• P(A) = P(A ^ B) + P(A ^ ~B)

B

~B

A ^ ~B

A ^ B

Followup:

What if I change the ratio of fair to loaded die in the experiment?


• I have lots of standard d20 die, lots of loaded die, all identical.

• Experiment is the same: (1) pick a d20 uniformly at random then (2) roll it. Can I mix the dice together so that P(B)=0.137 ?P(B) = P(B and A) + P(B and ~A)= 0.1*λ + 0.5*(1- λ) = 0.137

λ = (0.5 - 0.137)/0.4 = 0.9075“mixture model”

Another picture for this problem

A (fair die) ~A (loaded)

A and B ~A and B

It’s more convenient to say• “if you’ve picked a fair die then …” i.e. Pr(critical hit|fair die)=0.1• “if you’ve picked the loaded die then….” Pr(critical hit|loaded die)=0.5

Conditional probability:Pr(B|A) = P(B^A)/P(A)

P(B|A) P(B|~A)

Definition of Conditional Probability

P(A ^ B) P(A|B) = ----------- P(B)

Corollary: The Chain Rule

P(A ^ B) = P(A|B) P(B)



• Experiment: (1) pick a d20 uniformly at random then (2) roll it. Let A=d20 picked is fair and B=roll 19 or 20 with that die. What is P(B)?

P(B) = P(B|A) P(A) + P(B|~A) P(~A) = 0.1*0.75 + 0.5*0.25 = 0.2

“marginalizing out” A


A and B ~A and B P(B|A) P(B|~A)

P(A) P(~A)P(B) = P(B|A)P(A) + P(B|~A)P(~A)



• Experiment: (1) pick a d20 uniformly at random then (2) roll it. Let A=d20 picked is fair and B=roll 19 or 20 with that die.

• Suppose B happens (e.g., I roll a 20). What is the chance the die I rolled is fair? i.e. what is P(A|B) ?

P(B|A) * P(A)

P(B)P(A|B) =

P(A|B) * P(B)

P(A)P(B|A) =

Bayes, Thomas (1763) An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53:370-418

…by no means merely a curious speculation in the doctrine of chances, but necessary to be solved in order to a sure foundation for all our reasonings concerning past facts, and what is likely to be hereafter…. necessary to be considered by any that would give a clear account of the strength of analogical or inductive reasoning…

Bayes’ rule

priorposterior

More General Forms of Bayes Rule

)(~)|~()()|(

)()|()|(

APABPAPABP

APABPBAP

)(

)()|()|(

XBP

XAPXABPXBAP

More General Forms of Bayes Rule

An

kkk

iii

vAPvABP

vAPvABPBvAP

1

)()|(

)()|()|(

Useful Easy-to-prove facts

1)|()|( BAPBAP

1)|(1

An

kk BvAP

More about Bayes rule• An Intuitive Explanation of Bayesian Reasoning:

Bayes' Theorem for the curious and bewildered; an excruciatingly gentle introduction - Eliezer Yudkowsky

• Problem: Suppose that a barrel contains many small plastic eggs. Some eggs are painted red and some are painted blue. 40% of the eggs in the bin contain pearls, and 60% contain nothing. 30% of eggs containing pearls are painted blue, and 10% of eggs containing nothing are painted blue. What is the probability that a blue egg contains a pearl?

Some practical problems• Joe throws 4 critical hits in a row, is Joe cheating?• A = Joe using cheater’s die• C = roll 19 or 20; P(C|A)=0.5, P(C|~A)=0.1• B = C1 and C2 and C3 and C4• Pr(B|A) = 0.0625 P(B|~A)=0.0001

)(~)|~()()|(

)()|()|(

APABPAPABP

APABPBAP

))(1(*0001.0)(*0625.0

)(*0625.0)|(

APAP

APBAP

What’s the experiment and outcome here?

• Outcome A: Joe is cheating• Experiment:

• Joe picked a die uniformly at random from a bag containing 10,000 fair die and one bad one.

• Joe is a D&D player picked uniformly at random from set of 1,000,000 people and n of them cheat with probability p>0.

• I have no idea, but I don’t like his looks. Call it P(A)=0.1

Remember: Don’t Mess with The Axioms

• A subjective belief can be treated, mathematically, like a probability• Use those axioms!

• There have been many many other approaches to understanding “uncertainty”:

• Fuzzy Logic, three-valued logic, Dempster-Shafer, non-monotonic reasoning, …

• 25 years ago people in AI argued about these; now they mostly don’t• Any scheme for combining uncertain information, uncertain

“beliefs”, etc,… really should obey these axioms• If you gamble based on “uncertain beliefs”, then [you can be

exploited by an opponent] [your uncertainty formalism violates the axioms] - di Finetti 1931 (the “Dutch book argument”)


• Joe throws 4 critical hits in a row, is Joe cheating?

• A = Joe using cheater’s die• C = roll 19 or 20; P(C|A)=0.5, P(C|~A)=0.1• B = C1 and C2 and C3 and C4• Pr(B|A) = 0.0625 P(B|~A)=0.0001

)(/)()|(

)(/)()|(

)|(

)|(

BPAPABP

BPAPABP

BAP

BAP

)(

)()|()|(

BP

APABPBAP

)(

)(

)|(

)|(

AP

AP

ABP

ABP

)(

)(

0001.0

0625.0

AP

AP

)(

)(250,6

AP

AP

Moral: with enough evidence the prior P(A) doesn’t really matter.

Some practical problemsI bought a loaded d20 on EBay…but it didn’t come with any specs. How can I find out how it behaves?

Frequency

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Face Shown

One solutionI bought a loaded d20 on EBay…but it didn’t come with any specs. How can I find out how it behaves?

Frequency

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Face Shown

P(1)=0

P(2)=0

P(3)=0

P(4)=0.1

…

P(19)=0.25

P(20)=0.2

MLE = maximumlikelihood estimate

A better solutionI bought a loaded d20 on EBay…but it didn’t come with any specs. How can I find out how it behaves?

Frequency

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Face Shown

P(A=i|B) = P(B|A=i)*P(A=i) / P(B)

B = observed counts

P(A=i) = prior probability (subjective)

… and we’ll need some tricks to make this simple to compute

MAP = maximuma posteriori estimate


• I have 1 standard d6 die, 2 loaded d6 die.

• Loaded high: P(X=6)=0.50 Loaded low: P(X=1)=0.50

• Experiment: pick one d6 uniformly at random (A) and roll it. What is more likely – rolling a seven or rolling doubles?

Three combinations: HL, HF, FLP(D) = P(D ^ A=HL) + P(D ^ A=HF) + P(D ^ A=FL) = P(D | A=HL)*P(A=HL) + P(D|A=HF)*P(A=HF) + P(A|A=FL)*P(A=FL)


• I have 1 standard d6 die, 2 loaded d6 die.

• Loaded high: P(X=6)=0.50 Loaded low: P(X=1)=0.50

• Experiment: pick one d6 uniformly at random (A) and roll it. What is more likely – rolling a seven or rolling doubles?

1 2 3 4 5 6

1 D 7

2 D 7

3 D 7

4 7 D

5 7 D

6 7 D

Three combinations: HL, HF, FL

Roll 1

Roll

2

A brute-force solutionA Roll 1 Roll 2 P

FL 1 1 1/3 * 1/6 * ½

FL 1 2 1/3 * 1/6 * 1/10

FL 1 … …

… 1 6

FL 2 1

FL 2 …

… … …

FL 6 6

HL 1 1

HL 1 2

… … …

HF 1 1

…

Comment

doubles

seven

doubles

doubles

A joint probability table shows P(X1=x1 and … and Xk=xk) for every possible combination of values x1,x2,…., xk

With this you can compute any P(A) where A is any boolean combination of the primitive events (Xi=Xk), e.g.

• P(doubles)

• P(seven or eleven)

• P(total is higher than 5)

• ….

The Joint Distribution

Recipe for making a joint distribution of M variables:

Example: Boolean variables A, B, C



1. Make a truth table listing all combinations of values of your variables (if there are M Boolean variables then the table will have 2M rows).


A B C0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1




2. For each combination of values, say how probable it is.


A B C Prob0 0 0 0.30

0 0 1 0.05

0 1 0 0.10

0 1 1 0.05

1 0 0 0.05

1 0 1 0.10

1 1 0 0.25

1 1 1 0.10




2. For each combination of values, say how probable it is.

3. If you subscribe to the axioms of probability, those numbers must sum to 1.


A B C Prob0 0 0 0.30

0 0 1 0.05

0 1 0 0.10

0 1 1 0.05

1 0 0 0.05

1 0 1 0.10

1 1 0 0.25

1 1 1 0.10

A

B

C0.050.25

0.10 0.050.05

0.10

0.100.30

Using the Joint

One you have the JD you can ask for the probability of any logical expression involving your attribute

E

PEP matching rows

)row()(

Using the Joint

P(Poor Male) = 0.4654 E

PEP matching rows

)row()(

Using the Joint

P(Poor) = 0.7604 E

PEP matching rows

)row()(

Inference with the

Joint

2

2 1

matching rows

and matching rows

2

2121 )row(

)row(

)(

)()|(

E

EE

P

P

EP

EEPEEP

Inference with the

Joint

2

2 1

matching rows

and matching rows

2

2121 )row(

)row(

)(

)()|(

E

EE

P

P

EP

EEPEEP

P(Male | Poor) = 0.4654 / 0.7604 = 0.612

Inference is a big deal

• I’ve got this evidence. What’s the chance that this conclusion is true?• I’ve got a sore neck: how likely am I to have

meningitis?• I see my lights are out and it’s 9pm. What’s the

chance my spouse is already asleep?

Slide 1 A Quick Overview of Probability William W. Cohen Machine Learning 10-601 Jan 2008.

Documents

Transcript of Slide 1 A Quick Overview of Probability William W. Cohen Machine Learning 10-601 Jan 2008.