Slide 1 Copyright © 2001, Andrew W. Moore Probabilistic and Bayesian Analytics Brigham S. Anderson...

Probabilistic and Bayesian Analytics

Brigham S. Anderson

School of Computer Science

Carnegie Mellon University

www.cs.cmu.edu/~brigham

brigham@cmu.edu

Probability

• The world is a very uncertain place

• 30 years of Artificial Intelligence and Database research danced around this fact

• And then a few AI researchers decided to use some ideas from the eighteenth century

What we’re going to do

• We will review the fundamentals of probability.

• It’s really going to be worth it

• You’ll see examples of probabilistic analytics in action: • Inference, • Anomaly detection, and • Bayes Classifiers

Discrete Random Variables

• A is a Boolean-valued random variable if A denotes an event, and there is some degree of 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 influenza

Probabilities

• We write P(A) as “the probability that A is true”

• We could at this point spend 2 hours on the philosophy of this.

• We’ll spend slightly less...

Sample Space

Definition 1.The set, S, of all possible outcomes of a particular experiment is called the sample space for the experiment

The elements of the sample space are called outcomes.

Sample Spaces

Sample space of a coin flip:

S = {H, T}

Sample Spaces

Sample space of a die roll:

S = {1, 2, 3, 4, 5, 6}

Sample Spaces

Sample space of three die rolls?

S = {111,112,113,…,

…,664,665,666}

Sample SpacesSample space of a single draw from a

deck of cards:

S={As,Ac,Ah,Ad,2s,2c,2h,…

…,Ks,Kc,Kd,Kh}

So Far…

Definition ExampleThe sample space is the set of all possible worlds.

{As,Ac,Ah,Ad,2s,2c,2h,… …,Ks,Kc,Kd,Kh}

An outcome is an element of the sample space.

Events

Definition 2.An event is any subset of S (including S itself)

Events

Event: “Jack”

Sample Space of card draw

• The Sample Space is the set of all outcomes.

• An Outcome is a possible world.

• An Event is a set of outcomes

Events

Event: “Hearts”

Events

Event: “Red and Face”

Definitions

Definition Example

The sample space is the set of all possible worlds.

An outcome is a single point in the sample space.

An event is a set of outcomes from the sample space.

{2h,2c,2s,2d}

Events

Definition 3.Two events A and B are mutually exclusive if A^B=Ø.

Definition 4.If A1, A2, … are mutually exclusive and A1 A2 … = S, then the collection A1, A2, … forms a partition of S.

hearts spades

diamonds

Probability

Definition 5.Given a sample space S, a probability function is a function that maps each event in S to a real number, and satisfies

• P(A) ≥ 0 for any event A in S• P(S) = 1• For any number of mutually exclusive events A1, A2, A3 …, we have P(A1 A2 A3 …) = P(A1) + P(A2) + P(A3) +…

** This definition of the domain of this function is

not 100% sufficient, but it’s close enough for our purposes… (I’m sparing you Borel Fields)

Definitions

Definition Example

The sample space is the set of all possible worlds.

An outcome is a single point in the sample space.

An event is a set of one or more outcomes

Card is “Red”

P(E) maps event E to a real number and satisfies the axioms of probability

P(Red) = 0.50P(Black) = 0.50

Misconception

• The relative area of the events determines their probability

• …in a Venn diagram it does, but not in general.• However, the “area equals probability” rule is guaranteed

to result in axiom-satisfying probability functions.

We often assume, for example, that the probability of “heads” is equal to

“tails” in absence of other information…

But this is totally outside the axioms!

Creating a Valid P()

• One convenient way to create an axiom-satisfying probability function:

1. Assign a probability to each outcome in S

2. Make sure they sum to one

3. Declare that P(A) equals the sum of outcomes in event A

Everyday Example

Assume you are a doctor.

This is the sample space of “patients you might see on any given day”.

Non-smoker, female, diabetic, headache, good insurance, etc…

Smoker, male, herniated disk, back pain, mildly schizophrenic, delinquent medical bills, etc…

Outcomes

Everyday Example

Number of elements in the “patient space”:

100 jillion

Are these patients equally likely to occur?

Again, generally not. Let’s assume for the moment that they are, though.

…which roughly means “area equals probability”

Everyday Example

jillion100

jillion2

Event: Patient has Flu

Size of set “F”:2 jillion(Exactly 2 jillion of the points in the sample space have flu.)

Size of “patient space”:100 jillion

= 0.02PpatientSpace(F) =

Everyday Example

jillion100

jillion2

= 0.02PpatientSpace(F) =

From now on, the subscript on P() willbe omitted…

These Axioms are Not to be Trifled With

• There have been attempts to do different methodologies for uncertainty

• Fuzzy Logic• Three-valued logic• Dempster-Shafer• Non-monotonic reasoning

• But the axioms of probability are the only system with this property:

If you gamble using them you can’t be unfairly exploited by an opponent using some other system [di Finetti 1931]

Theorems from the AxiomsAxioms• P(A) ≥ 0 for any event A in S• P(S) = 1• For any number of mutually exclusive events A1, A2, A3 …, we have P(A1 A2 A3 …) = P(A1) + P(A2) + P(A3) +…

Theorem.If P is a probability function and A is an event in S, thenP(~A) = 1 – P(A)

Proof:(1) Since A and ~A partition S, P(A ~A) = P(S) = 1

(2) Since A and ~A are disjoint, P(A ~A) = P(A) + P(~A)

Combining (1) and (2) gives the result

Multivalued 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 {A1,A2, ... Ak}, and

• The events {A1,A2,…,Ak} partition S, so

jiAAP ji if 0),(

1)...( 21 kAAAP

Elementary Probability in Pictures

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

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

),()(1

jjABPBP

Useful!

Conditional Probability

Assume once more that you are a doctor.

Again, this is the sample space of “patients you might see on any given day”.

Event: Flu

P(F) = 0.02

Event: Headache

P(H) = 0.10H

P(F) = 0.02P(H) = 0.10

…we still need to specify the interaction between flu and headache…

Define

P(H|F) = Fraction of F’s outcomes which are also in H

P(F) = 0.02P(H) = 0.10P(H|F) = 0.50

0.01 0.01

H = “headache”F = “flu”

P(H|F) = Fraction of flu worlds in which patient has a headache

= #worlds with flu and headache ------------------------------------ #worlds with flu

= Size of “H and F” region ------------------------------ Size of “F” region

= P(H, F) ---------- P(F)

0.01 0.01

Definition.If A and B are events in S, and P(B) > 0, then the conditional probability of A given B, written P(A|B), is

),()|(

BAPBAP

The Chain RuleA simple rearrangement of the above equation yields

)()|(),( BPBAPBAP Main BayesNet concept!

Probabilistic Inference

H = “Have a headache”F = “Coming down with Flu”

P(H) = 0.10P(F) = 0.02P(H|F) = 0.50

One day you wake up with a headache. You think: “Drat! 50% of flus are associated with headaches so I must have a 50-50 chance of coming down with flu”

Is this reasoning good?

Probabilistic Inference

),()|(

HFPHFP

10.01.0

)02.0()50.0(

H = “Have a headache”F = “Coming down with Flu”

P(H) = 0.10P(F) = 0.02P(H|F) = 0.50

What we just did…

P(A,B) P(A|B) P(B)

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

P(A) P(A)

This is Bayes Rule

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

More General Forms of Bayes Rule

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

)()|()|(

APABPAPABP

APABPBAP

),(),|(),|(

CAPCABPCBAP

More General Forms of Bayes Rule

APABPBAP

)()|()|(

Independence

Definition.Two events, A and B, are statistically independent if

)()(),( BPAPBAP

Which is equivalent to

)()|( APBAP

Important forBayes Nets

Representing P(A,B,C)

• How can we represent the function P(A)?• P(A,B)?• P(A,B,C)?

Recipe for making a joint distribution of M variables:

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).

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 Prob

0 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

Example: P(A, B, C)

C0.050.25

0.10 0.050.05

0.100.30

The Joint Probability Table

Using the Joint

One you have the JPT you can ask for the probability of any logical expression

PEP matching rows

)row()(

…what is P(Poor,Male)?

Using the Joint

P(Poor, Male) = 0.4654 E

PEP matching rows

)row()(

…what is P(Poor)?

Using the Joint

P(Poor) = 0.7604 E

PEP matching rows

)row()(

…what is P(Poor|Male)?

Inference with the

matching rows

and matching rows

2121 )row(

),()|(

EEPEEP

Inference with the

matching rows

and matching rows

2121 )row(

),()|(

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?

• There’s a thriving set of industries growing based around Bayesian Inference. Highlights are: Medicine, Pharma, Help Desk Support, Engine Fault Diagnosis

Where do Joint Distributions come from?

• Idea One: Expert Humans• Idea Two: Simpler probabilistic facts and some algebraExample: Suppose you knew

P(A) = 0.5P(B|A) = 0.2P(B|~A) = 0.1

P(C|A,B) = 0.1P(C|A,~B) = 0.8P(C|~A,B) = 0.3P(C|~A,~B) = 0.1

Then you can automatically compute the JPT using the chain rule

P(A,B,C) = P(A) P(B|A) P(C|A,B)

Bayes Nets are a systematic way to do this.

Where do Joint Distributions come from?

• Idea Three: Learn them from data!

Prepare to witness an impressive learning algorithm….

Learning a JPT

Build a Joint Probability table for your attributes in which the probabilities are unspecified

Then fill in each row with

records ofnumber total

row matching records)row(ˆ P

A B C Prob

0 0 0 ?

0 0 1 ?

0 1 0 ?

0 1 1 ?

1 0 0 ?

1 0 1 ?

1 1 0 ?

1 1 1 ?

A B C Prob

0 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

Fraction of all records in whichA and B are True but C is False

Example of Learning a JPT

• This JPT was obtained by learning from three attributes in the UCI “Adult” Census Database [Kohavi 1995]

Where are we?

• We have recalled the fundamentals of probability

• We have become content with what JPTs are and how to use them

• And we even know how to learn JPTs from data.

Density Estimation

• Our Joint Probability Table (JPT) learner is our first example of something called Density Estimation

• A Density Estimator learns a mapping from a set of attributes to a Probability

DensityEstimator

ProbabilityInput

Attributes

• Given a record x, a density estimator M can tell you how likely the record is:

• Given a dataset with R records, a density estimator can tell you how likely the dataset is:(Under the assumption that all records were independently generated

from the probability function)

Evaluating a density estimator

kkR |MP|MP|MP

121 )(ˆ),,(ˆ)dataset(ˆ xxxx

)(ˆ |MP x

A small dataset: Miles Per Gallon

From the UCI repository (thanks to Ross Quinlan)

192 Training Set Records

mpg modelyear maker

good 75to78 asiabad 70to74 americabad 75to78 europebad 70to74 americabad 70to74 americabad 70to74 asiabad 70to74 asiabad 75to78 america: : :: : :: : :bad 70to74 americagood 79to83 americabad 75to78 americagood 79to83 americabad 75to78 americagood 79to83 americagood 79to83 americabad 70to74 americagood 75to78 europebad 75to78 europe

mpg modelyear maker

10 3.4

)(ˆ),,(ˆ)dataset(ˆ

kkR |MP|MP|MP xxxx

Log Probabilities

Since probabilities of datasets get so small we usually use log probabilities

kk |MP|MP|MP

)(ˆlog)(ˆlog)dataset(ˆlog xx

mpg modelyear maker

466.19

)(ˆlog)(ˆlog)dataset(ˆlog11

kk |MP|MP|MP xx

Summary: The Good News

The JPT allows us to learn P(X) from data.

• Can do inference: P(E1|E2)Automatic Doctor, Recommender, etc

• Can do anomaly detection spot suspicious / incorrect records

(e.g., credit card fraud)

• Can do Bayesian classificationPredict the class of a record

(e.g, predict cancerous/not-cancerous)

Summary: The Bad News

• Density estimation with JPTs is trivial, mindless and dangerous

Using a test set

An independent test set with 196 cars has a much worse log likelihood than it had on the training set

(actually it’s a billion quintillion quintillion quintillion quintillion times less likely)

….Density estimators can overfit. And the JPT estimator is the overfittiest of them all!

Overfitting Density Estimators

If this ever happens, it means there are certain combinations that we learn are “impossible”

Using a test set

The only reason that our test set didn’t score -infinity is that Andrew’s code is hard-wired to always predict a probability of at least one in 1020

We need Density Estimators that are less prone to overfitting

Is there a better way?

The problem with the JPT is that it just mirrors the training data.

In fact, it is just another way of storing the data: we could reconstruct the original dataset perfectly from it!

We need to represent the probability function with fewer parameters…

Aside:Bayes Nets

Bayes Nets• What are they?

• Bayesian nets are a framework for representing and analyzing models involving uncertainty

• What are they used for?• Intelligent decision aids, data fusion, 3-E feature recognition,

intelligent diagnostic aids, automated free text understanding, data mining

• How are they different from other knowledge representation and probabilistic analysis tools?• Uncertainty is handled in a mathematically rigorous yet efficient

and simple way

Bayes Net Concepts

1.Chain RuleP(A,B) = P(A) P(B|A)

2.Conditional IndependenceP(A|B,C) = P(A|B)

A Simple Bayes Net

• Let’s assume that we already have P(Mpg,Horse)

How would you rewrite this using the Chain rule?

0.480.12bad

0.040.36good

highlowP(good, low) = 0.36P(good,high) = 0.04P( bad, low) = 0.12P( bad,high) = 0.48

P(Mpg, Horse) =

Review: Chain Rule

0.480.12bad

0.040.36good

highlow

P(Mpg, Horse)

P(good, low) = 0.36P(good,high) = 0.04P( bad, low) = 0.12P( bad,high) = 0.48

P(Mpg, Horse) P(good) = 0.4P( bad) = 0.6

P( low|good) = 0.89P( low| bad) = 0.21P(high|good) = 0.11P(high| bad) = 0.79

P(Mpg)

P(Horse|Mpg)

Review: Chain Rule

0.480.12bad

0.040.36good

highlow

P(Mpg, Horse)

P(Mpg, Horse) P(good) = 0.4P( bad) = 0.6

P(Mpg)

P(Horse|Mpg)

= P(good) * P(low|good) = 0.4 * 0.89

= P(good) * P(high|good) = 0.4 * 0.11

= P(bad) * P(low|bad) = 0.6 * 0.21

= P(bad) * P(high|bad) = 0.6 * 0.79

How to Make a Bayes Net

P(Mpg, Horse) = P(Mpg) * P(Horse | Mpg)

P(good) = 0.4P( bad) = 0.6

P(Mpg)

P(Horse|Mpg)

P(good) = 0.4P( bad) = 0.6

P(Mpg)

P(Horse|Mpg)

• Each node is a probability function

• Each arc denotes conditional dependence

So, what have we accomplished thus far?

Nothing; we’ve just “Bayes Net-ified” the

P(Mpg, Horse) JPT using the Chain rule.

…the real excitement starts when we wield conditional independence

P(Mpg)

P(Horse|Mpg)

Before we continue, we need a worthier opponent than puny P(Mpg, Horse)…

We’ll use P(Mpg, Horse, Accel):

P(good, low,slow) = 0.37P(good, low,fast) = 0.01P(good,high,slow) = 0.02P(good,high,fast) = 0.00P( bad, low,slow) = 0.10P( bad, low,fast) = 0.12P( bad,high,slow) = 0.16P( bad,high,fast) = 0.22

P(Mpg,Horse,Accel)

* Note: I made these up…

Step 1: Rewrite joint using the Chain rule.

P(Mpg, Horse, Accel) = P(Mpg) P(Horse | Mpg) P(Accel | Mpg, Horse)

Note:Obviously, we could have written this 3!=6 different ways…

P(M, H, A) = P(M) * P(H|M) * P(A|M,H) = P(M) * P(A|M) * P(H|M,A) = P(H) * P(M|H) * P(A|H,M) = P(H) * P(A|H) * P(M|H,A) = … = …

Step 1: Rewrite joint using the Chain rule.

P(Mpg, Horse, Accel) = P(Mpg) P(Horse | Mpg) P(Accel | Mpg, Horse)

P(Mpg)

P(Horse|Mpg)

P(Accel|Mpg,Horse)

P(good) = 0.4P( bad) = 0.6

P(Mpg)

P(Horse|Mpg)

P(Accel|Mpg,Horse)

* Note: I made these up too…

P(good) = 0.4P( bad) = 0.6

P(Mpg)

P(Horse|Mpg)

P(Accel|Mpg,Horse)

A Miracle Occurs!

You are told by God (or another domain expert)that Accel is independent of Mpg given Horse!

i.e., P(Accel | Mpg, Horse) = P(Accel | Horse)

P(good) = 0.4P( bad) = 0.6

P(Mpg)

P(Horse|Mpg)

P(slow| low) = 0.22P(slow|high) = 0.64P(fast| low) = 0.78P(fast|high) = 0.36

P(Accel|Horse)

P(good) = 0.4P( bad) = 0.6

P(Mpg)

P(Horse|Mpg)

P(Accel|Horse)

Thank you, domain expert!

Now I only need to learn 5 parameters

instead of 7 from my data!

My parameter estimateswill be more accurate as

a result!

Independence“The Acceleration does not depend on the Mpg once I know the Horsepower.”

This can be specified very simply:

P(Accel Mpg, Horse) = P(Accel | Horse)

This is a powerful statement!

It required extra domain knowledge. A different kind of knowledge than numerical probabilities. It needed an understanding of causation.

Bayes Nets Formalized

A Bayes net (also called a belief network) is an augmented directed acyclic graph, represented by the pair V , E where:

• V is a set of vertices.• E is a set of directed edges joining vertices. No loops

of any length are allowed.

Each vertex in V contains the following information:• A Conditional Probability Table (CPT) indicating how

this variable’s probabilities depend on all possible combinations of parental values.

Bayes Nets Summary

• Bayes nets are a factorization of the full JPT which uses the chain rule and conditional independence.

• They can do everything a JPT can do (like quick, cheap lookups of probabilities)

The good news

We can do inference.

We can compute any conditional probability:

P( Some variable Some other variable values )

matching entriesjoint

and matching entriesjoint

2121 )entryjoint (

)entryjoint (

EEPEEP

The good news

We can do inference.

We can compute any conditional probability:

P( Some variable Some other variable values )

matching entriesjoint

and matching entriesjoint

2121 )entryjoint (

)entryjoint (

EEPEEP

Suppose you have m binary-valued variables in your Bayes Net and expression E2 mentions k variables.

How much work is the above computation?

The sad, bad news

Doing inference “JPT-style” by enumerating all matching entries in the joint are expensive:

Exponential in the number of variables.

But perhaps there are faster ways of querying Bayes nets?• In fact, if I ever ask you to manually do a Bayes Net inference, you’ll find

there are often many tricks to save you time.• So we’ve just got to program our computer to do those tricks too, right?

Sadder and worse news:General querying of Bayes nets is NP-complete.

Case Study I

Pathfinder system. (Heckerman 1991, Probabilistic Similarity Networks, MIT Press, Cambridge MA).

• Diagnostic system for lymph-node diseases.

• 60 diseases and 100 symptoms and test-results.

• 14,000 probabilities

• Expert consulted to make net.

• 8 hours to determine variables.

• 35 hours for net topology.

• 40 hours for probability table values.

• Apparently, the experts found it quite easy to invent the causal links and probabilities.

• Pathfinder is now outperforming the world experts in diagnosis. Being extended to several dozen other medical domains.

Bayes Net Info

GUI Packages:• Genie -- Free • Netica -- $$• Hugin -- $$

Non-GUI Packages:• All of the above have APIs• BNT for MATLAB• AUTON code (learning extremely large networks of tens

of thousands of nodes)

Bayes Nets andMachine Learning

Machine Learning Tasks

ClassifierData point x

AnomalyDetector

Data point x P(x)

P(C | x)

Inference

Engine

Evidence e1P(e2 | e1) Missing Variables e2

What is an Anomaly?

• An irregularity that cannot be explained by simple domain models and knowledge

• Anomaly detection only needs to learn from examples of “normal” system behavior.

• Classification, on the other hand, would need examples labeled “normal” and “not-normal”

Anomaly Detection in Practice

• Monitoring computer networks for attacks.

• Monitoring population-wide health data for outbreaks or attacks.

• Looking for suspicious activity in bank transactions

• Detecting unusual eBay selling/buying behavior.

JPTAnomaly Detector

• Suppose we have the following model:

P(Mpg, Horse) =

• We’re trying to detect anomalous cars.

• If the next example we see is <good,high>, how anomalous is it?

JPTAnomaly Detector

),(),(

highgoodPhighgoodlikelihood

P(Mpg, Horse) = How likely is

<good,high>?

Could not be easier! Just look up the entry in the JPT!

Smaller numbers are more anomalous in that themodel is more surprised to see them.

Bayes NetAnomaly Detector

),(),(

goodhighPgoodP

How likely is

<good,high>?

P(good) = 0.4P( bad) = 0.6

P(Mpg)

P(Horse|Mpg)

Bayes NetAnomaly Detector

),(),(

goodhighPgoodP

How likely is

<good,high>?

P(good) = 0.4P( bad) = 0.6

P(Mpg)

P(Horse|Mpg)

Again, trivial!We need to do one tiny lookup

for each variable in the network!

Machine Learning Tasks

AnomalyDetector

Data point x P(x)

P(C | x)

Inference

Engine

Evidence e1P(E2 | e1) Missing Variables E2

Bayes Classifiers

• A formidable and sworn enemy of decision trees

Bayes Classifiers in 1 Slide

Bayes classifiers just do inference.

That’s it.

The “algorithm”1. Learn P(class,X)

2. For a given input x, infer P(class|x)

3. Choose the class with the highest probability

JPTBayes Classifier

• Suppose we have the following model:

P(Mpg, Horse) =

• We’re trying to classify cars as Mpg = “good” or “bad”

• If the next example we see is Horse = “low”, how do we classify it?

JPTBayes Classifier

),()|(

lowgoodPlowgoodP

P(Mpg, Horse) =

),(),(

lowbadPlowgoodP

lowgoodP

739.012.036.0

How do we classify

<Horse=low>?

The P(good | low) = 0.75,so we classify the example

as “good”

Bayes Net Classifier

P(good) = 0.4P( bad) = 0.6

P(Mpg)

P(Horse|Mpg)

P(Accel|Horse)

• We’re trying to classify cars as Mpg = “good” or “bad”

• If the next example we see is <Horse=low,Accel=fast> how do we classify it?

112Suppose we get a

<Horse=low, Accel=fast> example?

),,(),|(

fastlowP

fastlowgoodPfastlowgoodP

P(good) = 0.4P( bad) = 0.6

P(Mpg)

P(Horse|Mpg)

P(Accel|Horse)

)|()|()(

fastlowP

lowfastPgoodlowPgoodP

0178.0

)05.0)(89.0)(4.0(

fastlowPfastlowP

),,(),,(

0178.0

fastlowbadPfastlowgoodP

75.0 Note: this is not exactly 0.75 because I rounded some of the CPT numbers earlier…

Bayes NetBayes Classifier

P(good) = 0.4P( bad) = 0.6

P(Mpg)

P(Horse|Mpg)

P(Accel|Horse)

The P(good | low, fast) = 0.75,so we classify the example

as “good”.

…but that seems somehow familiar…

Wasn’t that the same answer asP(Mpg=good | Horse=low)?

Bayes NetBayes Classifier

Bayes Classifiers

• OK, so classification can be posed as inference

• In fact, virtually all machine learning tasks are a form of inference

• Anomaly detection: P(x)• Classification: P(Class | x)• Regression: P(Y | x)• Model Learning: P(Model | dataset)• Feature Selection: P(Model | dataset)

The Naïve Bayes Classifier

ASSUMPTION: all the attributes are conditionally independent

given the class variable

At least 256 parameters!You better have the data

to support them…

A mere 25 parameters!(the CPTs are tiny because the attribute

nodes only have one parent.)

The Naïve Bayes Advantage

What is the Probability Functionof the Naïve Bayes?

P(Mpg,Cylinders,Weight,Maker,…) =

P(Mpg) P(Cylinders|Mpg) P(Weight|Mpg) P(Maker|Mpg) …

What is the Probability Functionof the Naïve Bayes?

i classxPclassPclassP )|()(),( x

This is another great feature of Bayes Nets; you can graphically

see your model assumptions

Bayes Classifier Results: “MPG”:

392 records

The Classifier

learned by “Naive BC”

Bayes Classifier Results: “MPG”:

40 records

More Facts About Bayes Classifiers

• Many other density estimators can be slotted in

• Density estimation can be performed with real-valued inputs

• Bayes Classifiers can be built with real-valued inputs

• Rather Technical Complaint: Bayes Classifiers don’t try to be maximally discriminative---they merely try to honestly model what’s going on

• Zero probabilities are painful for Joint and Naïve. A hack (justifiable with the magic words “Dirichlet Prior”) can help.

• Naïve Bayes is wonderfully cheap. And survives 10,000 attributes cheerfully!

Summary

• Axioms of Probability

• Bayes nets are created by• chain rule• conditional independence

• Bayes Nets can do• Inference• Anomaly Detection• Classification

Using Bayes Rule to Gamble

The “Win” envelope has a dollar and four beads in it

The “Lose” envelope has three beads and no money

Trivial question: someone draws an envelope at random and offers to sell it to you. How much should you pay?

Using Bayes Rule to Gamble

The “Win” envelope has a dollar and four beads in it

The “Lose” envelope has three beads and no moneyInteresting question: before deciding, you are allowed to see

one bead drawn from the envelope.Suppose it’s black: How much should you pay? Suppose it’s red: How much should you pay?

Calculation…$1.00

Probability Model Uses

AnomalyDetector

Data point x P(x)

P(C | x)

Inference

Engine

How do we evaluate a particular density estimator?

Probability Models

Full Prob.Table Naïve Prob.

• No assumptions• Overfitting-prone• Scales horribly

• Strong assumptions• Overfitting-resistant• Scales incredibly well

Bayes Nets

• Carefully chosen assumptions• Overfitting and scaling

properties depend on assumptions

What you should know

• Probability• Fundamentals of Probability and Bayes Rule• What’s a Joint Distribution• How to do inference (i.e. P(E1|E2)) once you have a JD

• Density Estimation• What is DE and what is it good for• How to learn a Joint DE• How to learn a naïve DE

How to build a Bayes Classifier• Assume you want to predict output Y which has arity nY and values v1, v2, …

• Assume there are m input attributes called X1, X2, … Xm

• Break dataset into nY smaller datasets called DS1, DS2, … DSny.

• Define DSi = Records in which Y=vi

• For each DSi , learn Density Estimator Mi to model the input distribution among the Y=vi records.

• Mi estimates P(X1, X2, … Xm | Y=vi )

• Idea: When a new set of input values (X1 = u1, X2 = u2, …. Xm = um) come along to be evaluated predict the value of Y that makes P(X1, X2, … Xm | Y=vi ) most likely

)|(argmax 11predict vYuXuXPY mm

Is this a good idea?

• Idea: When a new set of input values (X1 = u1, X2 = u2, …. Xm = um) come along to be evaluated predict the value of Y that makes P(X1, X2, … Xm | Y=vi ) most likely

This is a Maximum Likelihood classifier.

It can get silly if some Ys are very unlikely

• Idea: When a new set of input values (X1 = u1, X2 = u2, …. Xm = um) come along to be evaluated predict the value of Y that makes P(Y=vi | X1, X2, … Xm) most likely

)|(argmax 11predict

uXuXvYPY

Much Better Idea

Terminology

• MLE (Maximum Likelihood Estimator):

• MAP (Maximum A-Posteriori Estimator):

)|(argmax 11predict

uXuXvYPY

Getting what we need

)|(argmax 11predict

uXuXvYPY

Getting a posterior probability

vYPvYuXuXP

uXuXvYP

Bayes Classifiers in a nutshell

)()|(argmax

)|(argmax

11predict

vYPvYuXuXP

uXuXvYPY

1. Learn the distribution over inputs for each value Y.

2. This gives P(X1, X2, … Xm | Y=vi ).

3. Estimate P(Y=vi ). as fraction of records with Y=vi .

4. For a new prediction:

Bayes Classifiers in a nutshell

)()|(argmax

)|(argmax

11predict

vYPvYuXuXP

uXuXvYPY

1. Learn the distribution over inputs for each value Y.

2. This gives P(X1, X2, … Xm | Y=vi ).

3. Estimate P(Y=vi ). as fraction of records with Y=vi .

4. For a new prediction:

We can use our favorite Density Estimator here.

Right now we have two options:

•Joint Density Estimator•Naïve Density Estimator

Joint Density Bayes Classifier

)()|(argmax 11predict vYPvYuXuXPY mm

In the case of the joint Bayes Classifier this degenerates to a very simple rule:

Ypredict = the most common value of Y among records in which X1 = u1, X2 = u2, …. Xm = um.

Note that if no records have the exact set of inputs X1 = u1, X2 = u2, …. Xm = um, then P(X1, X2, … Xm | Y=vi ) = 0 for all values of Y.

In that case we just have to guess Y’s value

Naïve Bayes Classifier

In the case of the naive Bayes Classifier this can be simplified:

vvYuXPvYPY

predict )|()(argmax

Naïve Bayes Classifier

In the case of the naive Bayes Classifier this can be simplified:

vvYuXPvYPY

predict )|()(argmax

Technical Hint:If you have 10,000 input attributes that product will underflow in floating point math. You should use logs:

vvYuXPvYPY

predict )|(log)(logargmax

What you should know

• Bayes Classifiers• How to build one• How to predict with a BC• Contrast between naïve and joint BCs

Where are we now?

• We have a methodology for building Bayes nets.

• We don’t require exponential storage to hold our probability table. Only exponential in the maximum number of parents of any node.

• We can compute probabilities of any given assignment of truth values to the variables. And we can do it in time linear with the number of nodes.

• So we can also compute answers to any questions.

E.G. What could we do to compute P(R T,~S)?

P(s)=0.3P(M)=0.6

P(RM)=0.3P(R~M)=0.6

P(TL)=0.3P(T~L)=0.8

P(LM^S)=0.05P(LM^~S)=0.1P(L~M^S)=0.1P(L~M^~S)=0.2

Where are we now?

P(s)=0.3P(M)=0.6

P(RM)=0.3P(R~M)=0.6

P(TL)=0.3P(T~L)=0.8

Step 1: Compute P(R ^ T ^ ~S)

Step 2: Compute P(~R ^ T ^ ~S)

Step 3: Return

P(R ^ T ^ ~S)

-------------------------------------

P(R ^ T ^ ~S)+ P(~R ^ T ^ ~S)

Where are we now?

P(s)=0.3P(M)=0.6

P(RM)=0.3P(R~M)=0.6

P(TL)=0.3P(T~L)=0.8

Step 3: Return

P(R ^ T ^ ~S)

-------------------------------------

P(R ^ T ^ ~S)+ P(~R ^ T ^ ~S)

Sum of all the rows in the Joint that match R ^ T ^ ~S

Sum of all the rows in the Joint that match ~R ^ T ^ ~S

Where are we now?

P(s)=0.3P(M)=0.6

P(RM)=0.3P(R~M)=0.6

P(TL)=0.3P(T~L)=0.8

Step 3: Return

P(R ^ T ^ ~S)

-------------------------------------

P(R ^ T ^ ~S)+ P(~R ^ T ^ ~S)

Sum of all the rows in the Joint that match R ^ T ^ ~S

Sum of all the rows in the Joint that match ~R ^ T ^ ~S

Each of these obtained by the “computing a joint probability entry” method of the earlier slides

4 joint computes

IndependenceWe’ve stated:

P(M) = 0.6P(S) = 0.3P(S M) = P(S)

M S Prob

And since we now have the joint pdf, we can make any queries we like.

From these statements, we can derive the full joint pdf.

Classic Machine Learning Tasks

AnomalyDetector

Data point x P(x)

P(C | x)

Inference

Engine

Anomaly Detection on 1 page

A note about independence

• Assume A and B are Boolean Random Variables. Then

“A and B are independent”

if and only if

P(A|B) = P(A)

• “A and B are independent” is often notated as

Independence Theorems

• Assume P(A|B) = P(A)• Then P(A^B) =

= P(A) P(B)

• Assume P(A|B) = P(A)• Then P(B|A) =

= P(B)

Independence Theorems

• Assume P(A|B) = P(A)• Then P(~A|B) =

= P(~A)

• Assume P(A|B) = P(A)• Then P(A|~B) =

= P(A)

Multivalued Independence

For multivalued Random Variables A and B,

BAif and only if

)()|(:, uAPvBuAPvu from which you can then prove things like…

)()()(:, vBPuAPvBuAPvu )()|(:, vBPvAvBPvu

Back to Naïve Density Estimation

• Let x[i] denote the i’th field of record x:• Naïve DE assumes x[i] is independent of {x[1],x[2],..x[i-1], x[i+1],…x[M]}• Example:

• Suppose that each record is generated by randomly shaking a green dice and a red dice

• Dataset 1: A = red value, B = green value

• Dataset 2: A = red value, B = sum of values

• Dataset 3: A = sum of values, B = difference of values

• Which of these datasets violates the naïve assumption?

Using the Naïve Distribution

• Once you have a Naïve Distribution you can easily compute any row of the joint distribution.

• Suppose A, B, C and D are independently distributed. What is P(A^~B^C^~D)?

Using the Naïve Distribution

• Once you have a Naïve Distribution you can easily compute any row of the joint distribution.

• Suppose A, B, C and D are independently distributed. What is P(A^~B^C^~D)?

= P(A|~B^C^~D) P(~B^C^~D)

= P(A) P(~B^C^~D)

= P(A) P(~B|C^~D) P(C^~D)

= P(A) P(~B) P(C^~D)

= P(A) P(~B) P(C|~D) P(~D)

= P(A) P(~B) P(C) P(~D)

Naïve Distribution General Case

• Suppose x[1], x[2], … x[M] are independently distributed.

kkM ukxPuMxuxuxP

121 )][()][,]2[,]1[(

• So if we have a Naïve Distribution we can construct any row of the implied Joint Distribution on demand.

• So we can do any inference • But how do we learn a Naïve Density

Estimator?

Learning a Naïve Density Estimator

records ofnumber total

][in which records#)][(ˆ uixuixP

Another trivial learning algorithm!

Contrast

Joint DE Naïve DE

Can model anything Can model only very boring distributions

No problem to model “C is a noisy copy of A”

Outside Naïve’s scope

Given 100 records and more than 6 Boolean attributes will screw up badly

Given 100 records and 10,000 multivalued attributes will be fine

A tiny part of the DE

learned by “Joint”

Empirical Results: “MPG”The “MPG” dataset consists of 392 records and 8 attributes

The DE learned by

“Naive”

A tiny part of the DE

learned by “Joint”

Empirical Results: “MPG”The “MPG” dataset consists of 392 records and 8 attributes

The DE learned by

“Naive”

The DE learned by

“Joint”

Empirical Results: “Weight vs. MPG”Suppose we train only from the “Weight” and “MPG” attributes

The DE learned by

“Naive”

The DE learned by

“Joint”

Empirical Results: “Weight vs. MPG”Suppose we train only from the “Weight” and “MPG” attributes

The DE learned by

“Naive”

The DE learned by

“Joint”

“Weight vs. MPG”: The best that Naïve can do

The DE learned by

“Naive”

The Axioms Of Probability

Slide 1 Copyright © 2001, Andrew W. Moore Probabilistic and Bayesian Analytics Brigham S. Anderson...

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