Copyright © 2006, Brigham S. Anderson Machine Learning and the Axioms of Probability Brigham S. Anderson School.

2 Copyright © 2006, Brigham S. Anderson 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

3 Copyright © 2006, Brigham S. Anderson 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

4 Copyright © 2006, Brigham S. Anderson 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

6 Copyright © 2006, Brigham S. Anderson 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.

11 Copyright © 2006, Brigham S. Anderson So Far … DefinitionExample The 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. 2c

15 Copyright © 2006, Brigham S. Anderson Events Event: “Red and Face” 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

16 Copyright © 2006, Brigham S. Anderson Definitions DefinitionExample The sample space is the set of all possible worlds. {As,Ac,Ah,Ad,2s,2c,2h,… …,Ks,Kc,Kd,Kh} An outcome is a single point in the sample space. 2c An event is a set of outcomes from the sample space. {2h,2c,2s,2d}

17 Copyright © 2006, Brigham S. Anderson Events Definition 3. Two events A and B are mutually exclusive if A^B=Ø. Definition 4. If A 1, A 2, … are mutually exclusive and A 1 A 2 … = S, then the collection A 1, A 2, … forms a partition of S. clubs hearts spades diamonds

18 Copyright © 2006, Brigham S. Anderson 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)

19 Copyright © 2006, Brigham S. Anderson Definitions DefinitionExample The sample space is the set of all possible worlds. {As,Ac,Ah,Ad,2s,2c,2h,… …,Ks,Kc,Kd,Kh} An outcome is a single point in the sample space. 4c 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) = 0.50 P(Black) = 0.50

20 Copyright © 2006, Brigham S. Anderson A ~A 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!

21 Copyright © 2006, Brigham S. Anderson 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

22 Copyright © 2006, Brigham S. Anderson 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

23 Copyright © 2006, Brigham S. Anderson 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”

24 Copyright © 2006, Brigham S. Anderson Everyday Example F 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 = 0.02 P patientSpace (F) =

26 Copyright © 2006, Brigham S. Anderson 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]

27 Copyright © 2006, Brigham S. Anderson Theorems from the Axioms Axioms 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, then P(~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

28 Copyright © 2006, Brigham S. Anderson 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 {A 1,A 2,... A k }, and The events {A 1,A 2,…,A k } partition S, so

36 Copyright © 2006, Brigham S. Anderson Conditional Probability P(F) = 0.02 P(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 H F

38 Copyright © 2006, Brigham S. Anderson Conditional Probability 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) F 0.01 0.89 0.09 H

39 Copyright © 2006, Brigham S. Anderson Conditional Probability 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 The Chain Rule A simple rearrangement of the above equation yields Main Bayes Net concept!

40 Copyright © 2006, Brigham S. Anderson Probabilistic Inference H = “Have a headache” F = “Coming down with Flu” P(H) = 0.10 P(F) = 0.02 P(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? H F

42 Copyright © 2006, Brigham S. Anderson 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

47 Copyright © 2006, Brigham S. Anderson 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 2 M 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. ABCProb 0000.30 0010.05 0100.10 0110.05 100 1010.10 1100.25 1110.10 Example: P(A, B, C) A B C 0.05 0.25 0.100.05 0.10 0.30 The Joint Probability Table

53 Copyright © 2006, Brigham S. Anderson 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

54 Copyright © 2006, Brigham S. Anderson Where do Joint Distributions come from? Idea One: Expert Humans Idea Two: Simpler probabilistic facts and some algebra Example: Suppose you knew P(A) = 0.5 P(B|A) = 0.2 P(B|~A) = 0.1 P(C|A,B) = 0.1 P(C|A,~B) = 0.8 P(C|~A,B) = 0.3 P(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.

56 Copyright © 2006, Brigham S. Anderson Learning a JPT Build a Joint Probability table for your attributes in which the probabilities are unspecified Then fill in each row with ABCProb 000? 001? 010? 011? 100? 101? 110? 111? ABC 0000.30 0010.05 0100.10 0110.05 100 1010.10 1100.25 1110.10 Fraction of all records in which A and B are True but C is False

58 Copyright © 2006, Brigham S. Anderson 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.

59 Copyright © 2006, Brigham S. Anderson 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 Density Estimator Probability Input Attributes

60 Copyright © 2006, Brigham S. Anderson 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

66 Copyright © 2006, Brigham S. Anderson Summary: The Good News The JPT allows us to learn P(X) from data. Can do inference: P(E 1 |E 2 ) Automatic Doctor, Recommender, etc Can do anomaly detection spot suspicious / incorrect records (e.g., credit card fraud) Can do Bayesian classification Predict the class of a record (e.g, predict cancerous/not-cancerous)

68 Copyright © 2006, Brigham S. Anderson 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!

70 Copyright © 2006, Brigham S. Anderson 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 10 20 We need Density Estimators that are less prone to overfitting

71 Copyright © 2006, Brigham S. Anderson 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…

73 Copyright © 2006, Brigham S. Anderson 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

75 Copyright © 2006, Brigham S. Anderson A Simple Bayes Net Let’s assume that we already have P(Mpg,Horse) How would you rewrite this using the Chain rule? 0.480.12 bad 0.040.36 good highlow P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 P(Mpg, Horse) =

76 Copyright © 2006, Brigham S. Anderson Review: Chain Rule 0.480.12 bad 0.040.36 good highlow P(Mpg, Horse) P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 P(Mpg, Horse) P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.89 P( low| bad) = 0.21 P(high|good) = 0.11 P(high| bad) = 0.79 P(Mpg) P(Horse|Mpg) *

77 Copyright © 2006, Brigham S. Anderson Review: Chain Rule 0.480.12 bad 0.040.36 good highlow P(Mpg, Horse) P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 P(Mpg, Horse) P(good) = 0.4 P( bad) = 0.6 P( low|good) = 0.89 P( low| bad) = 0.21 P(high|good) = 0.11 P(high| bad) = 0.79 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

80 Copyright © 2006, Brigham S. Anderson How to Make a Bayes Net Mpg Horse P(good) = 0.4 P( bad) = 0.6 P(Mpg) P( low|good) = 0.90 P( low| bad) = 0.21 P(high|good) = 0.10 P(high| bad) = 0.79 P(Horse|Mpg) Each node is a probability function Each arc denotes conditional dependence

81 Copyright © 2006, Brigham S. Anderson How to Make a Bayes Net 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 Mpg Horse P(Mpg) P(Horse|Mpg)

82 Copyright © 2006, Brigham S. Anderson How to Make a Bayes Net Before we continue, we need a worthier opponent than puny P(Mpg, Horse)… We’ll use P(Mpg, Horse, Accel): P(good, low,slow) = 0.37 P(good, low,fast) = 0.01 P(good,high,slow) = 0.02 P(good,high,fast) = 0.00 P( bad, low,slow) = 0.10 P( bad, low,fast) = 0.12 P( bad,high,slow) = 0.16 P( bad,high,fast) = 0.22 P(Mpg,Horse,Accel) * Note: I made these up…

83 Copyright © 2006, Brigham S. Anderson How to Make a Bayes Net 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) = …

90 Copyright © 2006, Brigham S. Anderson 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.

91 Copyright © 2006, Brigham S. Anderson 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.

92 Copyright © 2006, Brigham S. Anderson 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)

94 Copyright © 2006, Brigham S. Anderson The good news We can do inference. We can compute any conditional probability: P( Some variable  Some other variable values ) Suppose you have m binary-valued variables in your Bayes Net and expression E 2 mentions k variables. How much work is the above computation?

95 Copyright © 2006, Brigham S. Anderson 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.

96 Copyright © 2006, Brigham S. Anderson 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.

97 Copyright © 2006, Brigham S. Anderson 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)

100 Copyright © 2006, Brigham S. Anderson 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”

101 Copyright © 2006, Brigham S. Anderson 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.

102 Copyright © 2006, Brigham S. Anderson JPT Anomaly Detector Suppose we have the following model: P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 P(Mpg, Horse) = We’re trying to detect anomalous cars. If the next example we see is, how anomalous is it?

103 Copyright © 2006, Brigham S. Anderson JPT Anomaly Detector P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 P(Mpg, Horse) = How likely is ? Could not be easier! Just look up the entry in the JPT! Smaller numbers are more anomalous in that the model is more surprised to see them.

105 Copyright © 2006, Brigham S. Anderson Bayes Net Anomaly Detector How likely is ? Mpg Horse P(good) = 0.4 P( bad) = 0.6 P(Mpg) P( low|good) = 0.90 P( low| bad) = 0.21 P(high|good) = 0.10 P(high| bad) = 0.79 P(Horse|Mpg) Again, trivial! We need to do one tiny lookup for each variable in the network!

108 Copyright © 2006, Brigham S. Anderson 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

109 Copyright © 2006, Brigham S. Anderson JPT Bayes Classifier Suppose we have the following model: P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 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?

110 Copyright © 2006, Brigham S. Anderson JPT Bayes Classifier P(good, low) = 0.36 P(good,high) = 0.04 P( bad, low) = 0.12 P( bad,high) = 0.48 P(Mpg, Horse) = How do we classify ? The P(good | low) = 0.75, so we classify the example as “good”

114 Copyright © 2006, Brigham S. Anderson 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)

116 Copyright © 2006, Brigham S. Anderson 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

121 Copyright © 2006, Brigham S. Anderson 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!

Copyright © 2006, Brigham S. Anderson Machine Learning and the Axioms of Probability Brigham S. Anderson School.

Similar presentations

Presentation on theme: "Copyright © 2006, Brigham S. Anderson Machine Learning and the Axioms of Probability Brigham S. Anderson School."— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Copyright © 2006, Brigham S. Anderson Machine Learning and the Axioms of Probability Brigham S. Anderson School.

Similar presentations

Presentation on theme: "Copyright © 2006, Brigham S. Anderson Machine Learning and the Axioms of Probability Brigham S. Anderson School."— Presentation transcript:

Similar presentations

About project

Feedback