COMP 2208 Dr. Long Tran-Thanh University of Southampton Bayes’ Theorem, Bayesian Reasoning, and Bayesian Networks

Classification Environment Perception Behaviour Categorize inputs Update belief model Update decision making policy Decision making Perception Behaviour

Reasoning Environment Perception Behaviour Categorize inputs Update belief model Update decision making policy Decision making Perception Behaviour

Reasoning Logic /Rule based build up basic rules (axioms) using some form of logic Other rules (reasoning) can be derived from the above Functional or declarative programming (LISP, ML, Prolog, etc…) Stochastic reasoning Frequentist (non-Bayesian) Bayesian Some bridging efforts: E.g., Markov logic (see, e.g., Pedro Domingos)

The right way to do reasoning? Debate 1: logic based vs. stochastic E.g., Noam Chomsky vs. peter Norvig Debate 2: frequentist vs. Bayesian Many vs. many Today we talk about Bayesian (because it’s simple to understand and elegant)

The Bayesian way Bayes’ Theorem Bayesian belief update Inference in Bayesian networks

Some probability theory Space of all possible world models = area equal to 1

Some probability theory Probability of event A = fraction of worlds in which A happens A A What does it mean that P(A) = 0.2? What does it mean that P(A) = 0? or = 1?

Some probability theory Probability of A not happening = complement of P(A) A A

Some probability theory A A B B

Basic axioms in probability theory Domain of the probability value: Constants: Connection of AND and OR:

Conditional probability A A B B Only consider worlds in which A happens -> new space of worlds B|A Consider worlds in which B happens, but only within the new space P(B|A) = fraction of worlds with B within the new space

Conditional probability A A B B B|A

Conditional probability We have: Chain rule: Law of total probability:

Bayes’ Theorem (Bayes’ rule) Use chain rule twice for P(A and B): The right hand sides must be the same! Bayes’ rule:

The beauty of Bayes’ Theorem A = evidence (observation); B = hypothesis Prior: captures our prior knowledge/belief Likelihood: how likely to observe the evidence, if the hypothesis was true P(evidence) = probability of observing the evidence in general (aggregated over all possible hypotheses) We update our belief after observing some evidences

Example: the Monty Hall problem The game: At the beginning, all doors are closed The prize is behind 1 door (with equal probability) You choose 1 door (say Door 1) The host opens a door (say Door 2) which has a goat behind it Let’s make a deal: would you swap your choice to Door 3?

Solution of the Monty Hall problem Your choice: Door 1; Offer: choose Door 3 instead What are the chances of each option for winning (getting the prize)? A = hypothesis: prize behind Door 1 B = host chooses Door 2 to open (and we see a goat) Bayes’ rule: = 1/3 = 1/2 = ½ (why?) Chance of winning the prize for staying with Door 1 = 1/3 Chance of winning the prize for switching to Door 3 = 2/3

Calculating the denominator Bayes’ rule: A = prize behind Door 1; B = host chooses Door 2 (between Doors 2 and 3) Use law of total probability: X = door with prize X= Door 1: X= Door 2: X= Door 3:

Example 2: the HIV test problem HIV lab tests are quite accurate: 99% sensitivity: if a patient is HIV+, then probability that the test has positive results is % specificity: if a patient is HIV-, then probability that the test has negative results is also 0.99 HIV is rare in patients in our population: about 1 out of 1000 (even among those who get tested) Situation: A patient does a HIV test and gets a positive result Question: what are the chances that the patient is indeed HIV+?

Solution for the HIV test problem A = test was positive; B = patient is HIV+ We want to calculate P(B|A) Prior: P(B) = (1 out of 1000 is HIV+) Likelihood: P(A|B) = 0.99 (99% sensitivity) What about P(A) ?

Solution for the HIV test problem Calculation of P(A) Use law of total probability Term 1: Term 2:

Solution for HIV test Calculating P(B|A) P(B) = 0.001; P(A|B) = 0.99; P(A) = This means that even if the test is positive, it’s only 9% that the patient is HIV+

Some discussions Only 15% of doctors gets this right Most doctors think that if a HIV test is positive, there’s a high chance that the patient is HIV+ Why? They typically focus on the accuracy (sensitivity) of the test They're neglecting the background or base rate of HIV prevalence (prior)

Bonus question Russian roulette with 2 bullets You put 2 bullets into a revolver, such that they are next to each other Your opponent spins and pull the trigger … and survives. Now it’s your turn! Question: should you spin the revolver as well, or you shouldn’t spin it? Question 2: what if there’s only 1 bullet in the revolver?

Belief update London – Rome flight

Belief update Near London Near Rome Near Paris Near Monaco You look out the window, and you see… land…and sea…and high mountains But you know that you’ve been flying for a while unlikely maybe unlikely maybe unlikely maybe unlikely probably unlikely You don’t know over which area you are flying …

Bayesian belief update Prior: probability that the model is true (before the observation) Likelihood: how likely to have the observed event, if the model was true Denominator: marginal likelihood (or model evidence) Left hand side = posterior: probability that the model is true, after we have seen the observations Belief: probability distribution over all the possible models Captures our knowledge + uncertainty about the true world model How to update our belief after each observation?

Bayesian belief update Prior probabilityPrior distribution = prior belief Posterior probabilityPosterior distribution = posterior belief

Bayesian belief update example Search for a crashed airplane using Bayesian updating Imagine you're designing a search-and- rescue UAV. Its job is to autonomously look for aircraft wreckage It is easier to detect wreckage in some terrain types than others

Bayesian belief update example Difficulty model: what’s the probability to find the wreckage in the area

Bayesian belief update example Prior belief: based on the last known location

Bayesian belief update example Search: always go to the point with highest probability At the beginning:

Bayesian belief update example Search: always go to the point with highest probability After 10 steps:

Bayesian belief update example Search: always go to the point with highest probability After 50 steps:

Bayesian belief update example Search: always go to the point with highest probability After 250 steps:

Bayesian belief update example Search: always go to the point with highest probability After 500 steps:

Bayesian belief update example Search: always go to the point with highest probability After 1000 steps:

Bayesian belief update example Search: always go to the point with highest probability After 2000 steps:

Complex knowledge representation So far we deal with simple correlations between probabilities A A B B We use probabilities to capture uncertainty in our knowledge What if we have much more complicated network of correlations? Inference: derive extra information/conclusions from observed data How to do inference in complex networks? How to use Bayes’ rule there? Inference in simple systems: use Bayes’ rule

Inference with joint distribution The simplest way to do inference is to look at the joint distribution of the probability variables Joint distribution captures all the interconnections + dependencies An example from employment survey M: the person is a male? L: does the person work long hours? R: is the person rich? But before Bayesian nets …

Inference with joint distribution Truth table: if all the variables are binaries

Inference with joint distribution P(the person is rich) = ? = 0.26

Inference with joint distribution P(L | M) = ? P (L|M) = P(L and M)/P(M) = ( )/( ) = 0.35

Inference with joint distribution We can do any inference from joint distribution Issues: doesn’t scale well in practice (brute force solution) E.g., with 30 variables, we need 2^30 probabilities … (1 billion) In theory: doesn’t show the relationships We might want to exploit the structure of relationships to simplify the calculations E.g., if R (rich) is independent from M (male) and L (working long hours), then we can drop M and L when we do inference about R

Independence Definition: Two random variables are independent if their joint probability is the product of their probabilities Similarly: Another property: If P(A), P(B) > 0

Bayesian networks Use graphical representation to capture the dependencies between the random variables Studied for the exam Lecturer is in good mood High exam result P(M) = 0.3 P(cM) = 0.7 P(S) = 0.8 P(cS) = 0.2 P(H|M, S) = 0.9 P(H|cM,S) = 0.4 P(H|M, cS) = 0.5 P(H|cM,cS) = 0.05

Bayesian networks Inference: given high exam result, what is the probability that the lecturer was in a good mood? (P(M|H) = ?) Studied for the exam Lecturer is in good mood High exam result P(M) = 0.3 P(cM) = 0.7 P(S) = 0.8 P(cS) = 0.2 P(H|M, S) = 0.9 P(H|cM,S) = 0.4 P(H|M, cS) = 0.5 P(H|cM,cS) = 0.05

Bayesian networks P(M) = 0.3 P(cM) = 0.7 P(S) = 0.8 P(cS) = 0.2 P(H|M, S) = 0.9 P(H|cM,S) = 0.4 P(H|M, cS) = 0.5 P(H|cM,cS) = 0.05 P(M| H) = ? P(M) = 0.3 P(H|M) = P(H|M, S)P(S) + P(H|M,cS)P(cS) = 0.9* *0.2 = 0.73 P(H) = ?

Bayesian networks P(H) = P(H|M,S)P(M and S) + P(H|M,cS)P(M and cS) + P(H|cM,S)P(cM and S) + P(H|cM,cS)P(cM and cS) M and S are independent = P(H|M,S)P(M)P(S) + P(H|M,cS)P(M)P(cS) + P(H|cM,S)P(cM)P(S) + P(H|cM,cS)P(cM)P(cS) P(H)= 0.9*0.3* *0.3* *0.7* *0.7*0.2 = = 0.477

Bayesian networks P(M) = 0.3 P(cM) = 0.7 P(S) = 0.8 P(cS) = 0.2 P(H|M, S) = 0.9 P(H|cM,S) = 0.4 P(H|M, cS) = 0.5 P(H|cM,cS) = 0.05 P(M| H) = ? P(M) = 0.3 P(H|M) = 0.73 P(H) = P(M| H) = 0.3*0.73/0.477 = 0.459

Building Bayesian networks Bayesian nets are sometimes built manually, consulting domain experts for structure and probabilities. More often, the structure is supplied by domain experts (i.e., they specify what affects what) but the probabilities are learned from data. Sometimes both structure and probabilities are learned from data. Difficult problem: puts the AI program in a similar position to a scientist trying out different hypotheses. Need a method to reward the proposed net structure for matching the data, but to penalize excessive complexity (Occam’s razor). Building from data: With domain expert:

Properties of Bayesian networks Bayesian networks must be directed acyclic graphs. The major efficiency of the Bayesian network is that we have economized on memory. They are also easier for human beings to interpret than the raw joint distribution.