Reasoning with Probabilities Paolo Turrini Department of Computing, - - PowerPoint PPT Presentation

Intro to AI (2nd Part)

Reasoning with Probabilities

Paolo Turrini

Department of Computing, Imperial College London

Introduction to Artificial Intelligence 2nd Part

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

The main reference

Stuart Russell and Peter Norvig Artificial Intelligence: a modern approach Chapter 14

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Today

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Today

Bayes’ rule Conditional independence Back to the Wumpus World Bayesian Networks

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ Rule

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday

Upon your return from a holiday on an exotic island, your doctor has bad news and good news.

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday

Upon your return from a holiday on an exotic island, your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate.

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday

Upon your return from a holiday on an exotic island, your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it).

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday

Upon your return from a holiday on an exotic island, your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). How worried should you be?

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional probability and Bayes’ Rule

Definition of conditional probability: P(a|b) = P(a ∧ b) P(b) if P(b) = 0

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional probability and Bayes’ Rule

Definition of conditional probability: P(a|b) = P(a ∧ b) P(b) if P(b) = 0 Product rule gives an alternative formulation: P(a ∧ b) = P(a|b)P(b) = P(b|a)P(a)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional probability and Bayes’ Rule

Definition of conditional probability: P(a|b) = P(a ∧ b) P(b) if P(b) = 0 Product rule gives an alternative formulation: P(a ∧ b) = P(a|b)P(b) = P(b|a)P(a) but then...

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ Rule

P(a|b) = P(b|a)P(a) P(b)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ Rule

Useful for assessing diagnostic probability from causal probability: P(Cause|Effect) = P(Effect|Cause)P(Cause) P(Effect)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ Rule

Useful for assessing diagnostic probability from causal probability: P(Cause|Effect) = P(Effect|Cause)P(Cause) P(Effect) E.g., let c be cold, s be sore throat: P(c|s) = P(s|c)P(c) P(s) = 0.9 × 0.001 0.005 = 0.18

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ rule

P(c|s) = P(s|c)P(c) P(s) = 0.9 × 0.001 0.005 = 0.18

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ rule

P(c|s) = P(s|c)P(c) P(s) = 0.9 × 0.001 0.005 = 0.18 We might not know the prior probability of the evidence P(S)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ rule

P(c|s) = P(s|c)P(c) P(s) = 0.9 × 0.001 0.005 = 0.18 We might not know the prior probability of the evidence P(S) In this case...

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ rule

P(c|s) = P(s|c)P(c) P(s) = 0.9 × 0.001 0.005 = 0.18 We might not know the prior probability of the evidence P(S) In this case... we compute the posterior probability for each value of the query variable (c, ¬c)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ rule

P(c|s) = P(s|c)P(c) P(s) = 0.9 × 0.001 0.005 = 0.18 We might not know the prior probability of the evidence P(S) In this case... we compute the posterior probability for each value of the query variable (c, ¬c) and then normalise

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ rule

P(c|s) = P(s|c)P(c) P(s) = 0.9 × 0.001 0.005 = 0.18 We might not know the prior probability of the evidence P(S) In this case... we compute the posterior probability for each value of the query variable (c, ¬c) and then normalise P(C|s) = α P(s|c)P(c), P(s|¬c)P(¬c)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ Rule

P(X|Y ) = α P(Y |X)P(X)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday solved

Upon your return from a holiday on an exotic island, your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it).

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday solved

Upon your return from a holiday on an exotic island, your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it).

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday solved

Upon your return from a holiday on an exotic island, your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). E.g., let D be disease, P be that you scored positive at the test:

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday solved

Upon your return from a holiday on an exotic island, your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). E.g., let D be disease, P be that you scored positive at the test: P(d|p)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday solved

Upon your return from a holiday on an exotic island, your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). E.g., let D be disease, P be that you scored positive at the test: P(d|p) =

P(p|d)P(d) P(p|d)P(d)+P(p|¬d)P(¬d)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday solved

Upon your return from a holiday on an exotic island, your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). E.g., let D be disease, P be that you scored positive at the test: P(d|p) =

P(p|d)P(d) P(p|d)P(d)+P(p|¬d)P(¬d) = 0.99×0.0001 0.99×0.0001+0.01×0.9999

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday solved

Upon your return from a holiday on an exotic island, your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). E.g., let D be disease, P be that you scored positive at the test: P(d|p) =

P(p|d)P(d) P(p|d)P(d)+P(p|¬d)P(¬d) = 0.99×0.0001 0.99×0.0001+0.01×0.9999 = 0.0098

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Holiday solved

Upon your return from a holiday on an exotic island, your doctor has bad news and good news. The bad news is that you’ve been diagnosed a serious disease and the test is 99% accurate. The good news is that the disease is very rare (1 in 10.000 get it). E.g., let D be disease, P be that you scored positive at the test: P(d|p) =

P(p|d)P(d) P(p|d)P(d)+P(p|¬d)P(¬d) = 0.99×0.0001 0.99×0.0001+0.01×0.9999 = 0.0098

Notice:posterior probability of disease still quite small!

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Back to joint distributions: combining evidence

Paolo Turrini Intro to AI (2nd Part)

Start with the joint distribution: P(Cavity|toothache ∧ catch) =

Intro to AI (2nd Part)

Back to joint distributions: combining evidence

Paolo Turrini Intro to AI (2nd Part)

Start with the joint distribution: P(Cavity|toothache ∧ catch) = α 0.108, 0.016

Intro to AI (2nd Part)

Back to joint distributions: combining evidence

Paolo Turrini Intro to AI (2nd Part)

Start with the joint distribution: P(Cavity|toothache ∧ catch) = α 0.108, 0.016 =

Intro to AI (2nd Part)

Back to joint distributions: combining evidence

Paolo Turrini Intro to AI (2nd Part)

Start with the joint distribution: P(Cavity|toothache ∧ catch) = α 0.108, 0.016 = 0.871, 0.129

Intro to AI (2nd Part)

Back to joint distributions: combining evidence

Paolo Turrini Intro to AI (2nd Part)

Start with the joint distribution: It doesn’t scale up to a large number of variables Absolute Independence is very rare Can we use Bayes’ rule?

Intro to AI (2nd Part)

Back to joint distributions: combining evidence

Paolo Turrini Intro to AI (2nd Part)

Start with the joint distribution: P(Cavity|toothache ∧ catch) = α P(toothache ∧ catch|Cavity)P(Cavity)

Intro to AI (2nd Part)

Back to joint distributions: combining evidence

Paolo Turrini Intro to AI (2nd Part)

Start with the joint distribution: P(Cavity|toothache ∧ catch) = α P(toothache ∧ catch|Cavity)P(Cavity) Still not good: with n evidence variables 2n possible combinations for which we would need to know the conditional probabilities

Intro to AI (2nd Part)

Back to joint distributions: combining evidence

We can’t use absolute independence:

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Back to joint distributions: combining evidence

We can’t use absolute independence: Toothache and Catch are not independent:

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Back to joint distributions: combining evidence

We can’t use absolute independence: Toothache and Catch are not independent: If the probe catches in the tooth then it is likely the tooth has a cavity, which means that toothache is likely too.

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Back to joint distributions: combining evidence

We can’t use absolute independence: Toothache and Catch are not independent: If the probe catches in the tooth then it is likely the tooth has a cavity, which means that toothache is likely too. But they are independent given the presence or the absence

• f cavity!

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Back to joint distributions: combining evidence

We can’t use absolute independence: Toothache and Catch are not independent: If the probe catches in the tooth then it is likely the tooth has a cavity, which means that toothache is likely too. But they are independent given the presence or the absence

• f cavity! Toothache depends on the state of the nerves in the

tooth, catch depends on the dentist’s skills, to which toothache is irrelevant

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional Independence

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence

1 P(catch|toothache, cavity) = P(catch|cavity), the same

independence holds if I haven’t got a cavity:

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence

1 P(catch|toothache, cavity) = P(catch|cavity), the same

independence holds if I haven’t got a cavity:

2 P(catch|toothache, ¬cavity) = P(catch|¬cavity) Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence

1 P(catch|toothache, cavity) = P(catch|cavity), the same

independence holds if I haven’t got a cavity:

2 P(catch|toothache, ¬cavity) = P(catch|¬cavity)

Catch is conditionally independent of Toothache given Cavity:

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence

P(Catch|Toothache, Cavity) = P(Catch|Cavity)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence

P(Catch|Toothache, Cavity) = P(Catch|Cavity) Equivalent statements:

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence

P(Catch|Toothache, Cavity) = P(Catch|Cavity) Equivalent statements: P(Toothache|Catch, Cavity) = P(Toothache|Cavity)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence

P(Catch|Toothache, Cavity) = P(Catch|Cavity) Equivalent statements: P(Toothache|Catch, Cavity) = P(Toothache|Cavity) P(Toothache, Catch|Cavity) = P(Toothache|Cavity)P(Catch|Cavity)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence contd.

Write out full joint distribution using chain rule: P(Toothache, Catch, Cavity)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence contd.

Write out full joint distribution using chain rule: P(Toothache, Catch, Cavity) = P(Toothache|Catch, Cavity)P(Catch, Cavity)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence contd.

Write out full joint distribution using chain rule: P(Toothache, Catch, Cavity) = P(Toothache|Catch, Cavity)P(Catch, Cavity) = P(Toothache|Catch, Cavity)P(Catch|Cavity)P(Cavity)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence contd.

Write out full joint distribution using chain rule: P(Toothache, Catch, Cavity) = P(Toothache|Catch, Cavity)P(Catch, Cavity) = P(Toothache|Catch, Cavity)P(Catch|Cavity)P(Cavity) = P(Toothache|Cavity)P(Catch|Cavity)P(Cavity)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence contd.

Write out full joint distribution using chain rule: P(Toothache, Catch, Cavity) = P(Toothache|Catch, Cavity)P(Catch, Cavity) = P(Toothache|Catch, Cavity)P(Catch|Cavity)P(Cavity) = P(Toothache|Cavity)P(Catch|Cavity)P(Cavity) I.e., 2 + 2 + 1 = 5 independent numbers (equations 1 and 2 remove 2)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence contd.

In most cases, the use of conditional independence reduces the size

• f the representation of the joint distribution from exponential in n

to linear in n.

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Conditional independence contd.

In most cases, the use of conditional independence reduces the size

• f the representation of the joint distribution from exponential in n

to linear in n. Conditional independence is our most basic and robust form of knowledge about uncertain environments.

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ Rule and conditional independence

P(Cavity|toothache ∧ catch)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ Rule and conditional independence

P(Cavity|toothache ∧ catch) = α P(toothache ∧ catch|Cavity)P(Cavity)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ Rule and conditional independence

P(Cavity|toothache ∧ catch) = α P(toothache ∧ catch|Cavity)P(Cavity) = α P(toothache|Cavity)P(catch|Cavity)P(Cavity)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ Rule and conditional independence

P(Cavity|toothache ∧ catch) = α P(toothache ∧ catch|Cavity)P(Cavity) = α P(toothache|Cavity)P(catch|Cavity)P(Cavity)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ Rule and conditional independence

P(Cavity|toothache ∧ catch) = α P(toothache ∧ catch|Cavity)P(Cavity) = α P(toothache|Cavity)P(catch|Cavity)P(Cavity) This is an example of a naive Bayes model: P(Cause, Effect1, . . . , Effectn) = P(Cause)ΠiP(Effecti|Cause)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Bayes’ Rule and conditional independence

P(Cavity|toothache ∧ catch) = α P(toothache ∧ catch|Cavity)P(Cavity) = α P(toothache|Cavity)P(catch|Cavity)P(Cavity) This is an example of a naive Bayes model: P(Cause, Effect1, . . . , Effectn) = P(Cause)ΠiP(Effecti|Cause) Total number of parameters is linear in n

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

The Wumps World

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

The Wumpus World

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

The Wumpus World

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

The Wumpus World

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

The Wumpus World

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

The Wumpus World

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

The Wumpus World

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

The Wumpus World

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

The Wumpus World

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

The Wumpus World

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Wumpus World

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Wumpus World

Pij = true iff [i, j] contains a pit Bij = true iff [i, j] is breezy

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Wumpus World

Pij = true iff [i, j] contains a pit Bij = true iff [i, j] is breezy

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Specifying the probability model

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Specifying the probability model

Include only B1,1, B1,2, B2,1 in the probability model!

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Specifying the probability model

Include only B1,1, B1,2, B2,1 in the probability model! The full joint distribution is P(P1,1, . . . , P4,4, B1,1, B1,2, B2,1)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Specifying the probability model

Include only B1,1, B1,2, B2,1 in the probability model! The full joint distribution is P(P1,1, . . . , P4,4, B1,1, B1,2, B2,1) Apply product rule: P(B1,1, B1,2, B2,1 | P1,1, . . . , P4,4)P(P1,1, . . . , P4,4) (Do it this way to get P(Effect|Cause).)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Specifying the probability model

Include only B1,1, B1,2, B2,1 in the probability model! The full joint distribution is P(P1,1, . . . , P4,4, B1,1, B1,2, B2,1) Apply product rule: P(B1,1, B1,2, B2,1 | P1,1, . . . , P4,4)P(P1,1, . . . , P4,4) (Do it this way to get P(Effect|Cause).) First term: 1 if pits are adjacent to breezes, 0 otherwise

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Specifying the probability model

Include only B1,1, B1,2, B2,1 in the probability model! The full joint distribution is P(P1,1, . . . , P4,4, B1,1, B1,2, B2,1) Apply product rule: P(B1,1, B1,2, B2,1 | P1,1, . . . , P4,4)P(P1,1, . . . , P4,4) (Do it this way to get P(Effect|Cause).) First term: 1 if pits are adjacent to breezes, 0 otherwise Second term: pits are placed randomly, probability 0.2 per square:

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Specifying the probability model

Include only B1,1, B1,2, B2,1 in the probability model! The full joint distribution is P(P1,1, . . . , P4,4, B1,1, B1,2, B2,1) Apply product rule: P(B1,1, B1,2, B2,1 | P1,1, . . . , P4,4)P(P1,1, . . . , P4,4) (Do it this way to get P(Effect|Cause).) First term: 1 if pits are adjacent to breezes, 0 otherwise Second term: pits are placed randomly, probability 0.2 per square: P(P1,1, . . . , P4,4) = Π

4,4 i,j = 1,1P(Pi,j) = 0.2n × 0.816−n

for n pits.

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Observations and query

We know the following facts: b = ¬b1,1 ∧ b1,2 ∧ b2,1

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Observations and query

We know the following facts: b = ¬b1,1 ∧ b1,2 ∧ b2,1 explored = ¬p1,1 ∧ ¬p1,2 ∧ ¬p2,1

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Observations and query

We know the following facts: b = ¬b1,1 ∧ b1,2 ∧ b2,1 explored = ¬p1,1 ∧ ¬p1,2 ∧ ¬p2,1 Query is P(P1,3|explored, b)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Observations and query

We know the following facts: b = ¬b1,1 ∧ b1,2 ∧ b2,1 explored = ¬p1,1 ∧ ¬p1,2 ∧ ¬p2,1 Query is P(P1,3|explored, b) Define Unexplored = Pijs other than P1,3 and Explored

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Observations and query

We know the following facts: b = ¬b1,1 ∧ b1,2 ∧ b2,1 explored = ¬p1,1 ∧ ¬p1,2 ∧ ¬p2,1 Query is P(P1,3|explored, b) Define Unexplored = Pijs other than P1,3 and Explored

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Complexity

For inference by enumeration, we have P(P1,3|explored, b) = αΣunexploredP(P1,3, unexplored, explored, b)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Complexity

For inference by enumeration, we have P(P1,3|explored, b) = αΣunexploredP(P1,3, unexplored, explored, b) There are 12 unknown squares

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Complexity

For inference by enumeration, we have P(P1,3|explored, b) = αΣunexploredP(P1,3, unexplored, explored, b) There are 12 unknown squares The summation contains 212 = 4096 terms

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Complexity

For inference by enumeration, we have P(P1,3|explored, b) = αΣunexploredP(P1,3, unexplored, explored, b) There are 12 unknown squares The summation contains 212 = 4096 terms In general the summation grows exponentiatlly with the number of squares!

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Complexity

For inference by enumeration, we have P(P1,3|explored, b) = αΣunexploredP(P1,3, unexplored, explored, b) There are 12 unknown squares The summation contains 212 = 4096 terms In general the summation grows exponentiatlly with the number of squares! And now?

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Using conditional independence

Basic insight: observations are conditionally independent of other hidden squares given neighbouring hidden squares

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Using conditional independence

Basic insight: observations are conditionally independent of other hidden squares given neighbouring hidden squares

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Using conditional independence

Basic insight: observations are conditionally independent of other hidden squares given neighbouring hidden squares Define Unexplored = Fringe ∪ Other

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Using conditional independence

Basic insight: observations are conditionally independent of other hidden squares given neighbouring hidden squares Define Unexplored = Fringe ∪ Other P(b|P1,3, Explored, Unexplored) = P(b|P1,3, Explored, Fringe)

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Using conditional independence

Basic insight: observations are conditionally independent of other hidden squares given neighbouring hidden squares Define Unexplored = Fringe ∪ Other P(b|P1,3, Explored, Unexplored) = P(b|P1,3, Explored, Fringe) Manipulate query into a form where we can use this!

Paolo Turrini Intro to AI (2nd Part)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

P(P1,3|explored, b)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

P(P1,3|explored, b) = α

unexplored P(P1,3, unexplored, explored, b)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

Inference by enumeration P(P1,3|explored, b) = α

unexplored P(P1,3, unexplored, explored, b)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α

unexplored P(P1,3, unexplored, explored, b)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α

unexplored P(P1,3, unexplored, explored, b)

= α

unexplored P(b|explored, P1,3, unexplored)×

×P(P1,3, explored, unexplored)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

Product rule

α

unexplored P(P1,3, unexplored, explored, b)

= α

unexplored P(b|explored, P1,3, unexplored)×

×P(P1,3, explored, unexplored)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α

unexplored P(b|P1,3, unexplored, explored)P(P1,3, unexplored, explored)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α

unexplored P(b|P1,3, unexplored, explored)P(P1,3, unexplored, explored)

= α

fringe

• ther P(b|explored, P1,3, fringe, other)×

×P(P1,3, explored, fringe, other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

Distinguishing the unknown

α

unexplored P(b|P1,3, unexplored, explored)P(P1,3, unexplored, explored)

= α

fringe

• ther P(b|explored, P1,3, fringe, other)×

×P(P1,3, explored, fringe, other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α

fringe

• ther P(b|explored, P1,3, fringe, other)×

×P(P1,3, explored, fringe, other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α

fringe

• ther P(b|explored, P1,3, fringe, other)×

×P(P1,3, explored, fringe, other) = α

fringe

• ther P(b|explored, P1,3, fringe)×

×P(P1,3, explored, fringe, other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

Conditional Independence

α

fringe

• ther P(b|explored, P1,3, fringe, other)×

×P(P1,3, explored, fringe, other) = α

fringe

• ther P(b|explored, P1,3, fringe)×

×P(P1,3, explored, fringe, other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α

fringe

• ther P(b|explored, P1,3, fringe)×

×P(P1,3, explored, fringe, other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α

fringe

• ther P(b|explored, P1,3, fringe)×

×P(P1,3, explored, fringe, other) = α

fringeP(b|explored, P1,3, fringe)×

×

• therP(P1,3, explored, fringe, other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

Pushing the sums inwards

α

fringe

• ther P(b|explored, P1,3, fringe)×

×P(P1,3, explored, fringe, other) = α

fringeP(b|explored, P1,3, fringe)×

×

• therP(P1,3, explored, fringe, other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α

fringeP(b|explored, P1,3, fringe)×

• therP(P1,3, explored, fringe, other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α

fringeP(b|explored, P1,3, fringe)×

• therP(P1,3, explored, fringe, other)

= α

fringe P(b|explored, P1,3, fringe)×

• ther P(P1,3)P(explored)P(fringe)P(other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

Independence

α

fringeP(b|explored, P1,3, fringe)×

• therP(P1,3, explored, fringe, other)

= α

fringe P(b|explored, P1,3, fringe)×

• ther P(P1,3)P(explored)P(fringe)P(other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α

fringe P(b|explored, P1,3, fringe)×

×

• ther P(P1,3)P(explored)P(fringe)P(other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α

fringe P(b|explored, P1,3, fringe)×

×

• ther P(P1,3)P(explored)P(fringe)P(other)

= α P(explored)P(P1,3)× ×

fringe P(b|explored, P1,3, fringe)P(fringe)

• ther P(other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

Reordering and pushing sums inwards

α

fringe P(b|explored, P1,3, fringe)×

×

• ther P(P1,3)P(explored)P(fringe)P(other)

= α P(explored)P(P1,3)× ×

fringe P(b|explored, P1,3, fringe)P(fringe)

• ther P(other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α P(explored)P(P1,3)× ×

fringe P(b|explored, P1,3, fringe)P(fringe)

• ther P(other)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

α P(explored)P(P1,3)× ×

fringe P(b|explored, P1,3, fringe)P(fringe)

• ther P(other)

= α′ P(P1,3)

fringe P(b|explored, P1,3, fringe)P(fringe)

Intro to AI (2nd Part)

Using conditional independence

Paolo Turrini Intro to AI (2nd Part)

Simplifying

α P(explored)P(P1,3)× ×

fringe P(b|explored, P1,3, fringe)P(fringe)

• ther P(other)

= α′ P(P1,3)

fringe P(b|explored, P1,3, fringe)P(fringe)

Intro to AI (2nd Part)

Using conditional independence contd.

Paolo Turrini Intro to AI (2nd Part)

P(b|explored, P1,3, fringe)

Intro to AI (2nd Part)

Using conditional independence contd.

Paolo Turrini Intro to AI (2nd Part)

P(b|explored, P1,3, fringe) = 1 when the frontier is consistent with the observations = 0 otherwise

Intro to AI (2nd Part)

Using conditional independence contd.

Paolo Turrini Intro to AI (2nd Part)

P(b|explored, P1,3, fringe) = 1 when the frontier is consistent with the observations = 0 otherwise We can sum over the possible configurations for the frontier variables that are consistent with the known facts.

Intro to AI (2nd Part)

Using conditional independence contd.

Paolo Turrini Intro to AI (2nd Part)

P(P1,3|explored, b)=

Intro to AI (2nd Part)

Using conditional independence contd.

Paolo Turrini Intro to AI (2nd Part)

P(P1,3|explored, b)= α′ 0.2(0.04 + 0.16 + 0.16), 0.8(0.04 + 0.16)

Intro to AI (2nd Part)

Using conditional independence contd.

Paolo Turrini Intro to AI (2nd Part)

P(P1,3|explored, b)= α′ 0.2(0.04 + 0.16 + 0.16), 0.8(0.04 + 0.16) ≈ 0.31, 0.69

Intro to AI (2nd Part)

Using conditional independence contd.

Paolo Turrini Intro to AI (2nd Part)

P(P1,3|explored, b)= α′ 0.2(0.04 + 0.16 + 0.16), 0.8(0.04 + 0.16) ≈ 0.31, 0.69 P(P2,2|explored, b) ≈ 0.86, 0.14

Intro to AI (2nd Part)

Using conditional independence contd.

Paolo Turrini Intro to AI (2nd Part)

P(P1,3|explored, b)= α′ 0.2(0.04 + 0.16 + 0.16), 0.8(0.04 + 0.16) ≈ 0.31, 0.69 P(P2,2|explored, b) ≈ 0.86, 0.14