Transcript CS 4100 Artificial Intelligence Prof. C. Hafner Class Notes March 20, 2012

CS 4100 Artificial Intelligence
Prof. C. Hafner
Class Notes March 20, 2012
Outline
• Midterm planning problem: solution
http://www.ccs.neu.edu/course/cs4100sp12/classnotes/midterm-planning.doc
• Discuss term projects
• Continue uncertain reasoning in AI
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Probability distribution (review)
Conditional Probability and the Chain Rule (cont.)
Bayes’ Rule
Independence, “Expert” systems and the combinatorics of
joint probabilities
– Bayes networks
– Assignment 6
Term Projects – The Process
1. Form teams of 3 or 4 people – 10-12 teams
2. Before next class (Mar 20) each team send an email
a. Name and a main contact person (email)
b. All team members’ names and email addresses
c. You can reserve a topic asap (first request)
3. Brief written project proposal due Fri March 23 10pm (email)
4. Each team will
a.
b.
c.
submit a written project report (due April 17, last day of class)
a running computer application (due April 17, last day of class)
make a presentation of 15 minutes on their project (April 12 & 17)
5. Attendance is required and will be taken on April 12 & 17
Term Projects – The Content
1. Select a domain
2. Model the domain
a.
b.
c.
“Logical/state model” : define an ontology w/ example world state
Implementation in Protégé – demo with some queries
“Dynamics model” (of how the world changes)
Using Situation Calculus formalism or STRIPS-type operators
3. Define and solve example planning problems: initial state  goal
state
a.
b.
Specify planning axioms or STRIPS-type operators
Show (on paper) a proof or derivation of a trivial plan and then a
more challenging one using resolution or the POP algorithm
Ontology Design Example: Protege
• Simplest example: Dog project
• Cooking ontology
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Overall Design
Implement class taxonomy
Slots representing data types
Slots containing relationships
Cooking ontology (for meal or party planning)
• FoodItem – taxonomy can include Dairy, Meat, Starch, Veg,
Fruit, Sweets. A higher level can be Protein, Carbs. Should
include nuts due to possible allergies
• A Dish – taxonomy can be Appetizer, Main Course, Salad,
Dessert. A Dish has Ingredients which are instances of
FoodItem
• A Recipe
– Has Servings (a number)
– Has steps
• Each step includes a FoodItem, Amount, and Prep
• An Amount is a number and units
• Prep is a string
• Relationships
Bayes' Rule
• Product rule P(ab) = P(a | b) P(b) = P(b | a) P(a)
 Bayes' rule: P(a | b) = P(b | a) P(a) / P(b)
• or in distribution form
P(Y|X) = P(X|Y) P(Y) / P(X) = αP(X|Y) P(Y)
• Useful for assessing diagnostic probability from causal
probability:
P(Cause|Effect) = P(Effect|Cause) P(Cause) / P(Effect)
P(Disease|Symptom) = P(Symptom|Diease) P(Symptom) / (Disease)
– E.g., let M be meningitis, S be stiff neck:
P(m|s) = P(s|m) P(m) / P(s) = 0.8 × 0.0001 / 0.1 = 0.0008
– Note: posterior probability of meningitis still very small!
Bayes' Rule and conditional independence
P(Cavity | toothache  catch)
= αP(toothache  catch | Cavity) P(Cavity)
= αP(toothache | Cavity) P(catch | Cavity) P(Cavity)
• We say: “toothache and catch are independent, given cavity”.
This is an example of a naïve Bayes model. We will study this
later as our simplest machine learning application
P(Cause,Effect1, … ,Effectn) = P(Cause) πiP(Effecti|Cause)
• Total number of parameters is linear in n (number of
symptoms). This is our first Bayesian inference net.
Conditional independence
• P(Toothache, Cavity, Catch) has 23 – 1 = 7 independent entries
• If I have a cavity, the probability that the probe catches in it doesn't
depend on whether I have a toothache:
(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:
P(Catch | Toothache,Cavity) = P(Catch | Cavity)
• Equivalent statements (from original definitions of independence):
P(Toothache | Catch, Cavity) = P(Toothache | Cavity)
P(Toothache, Catch | Cavity) = P(Toothache | Cavity) P(Catch | Cavity)
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
• In most cases, the use of conditional independence reduces the
size of 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.
Remember this examples
Example of conditional independence
Test your understanding of the Chain Rule
This is our second Bayesian inference net
How to construct a Bayes Net
Test your understanding: design a Bayes net with plausible
numbers
Calculating using Bayes’ Nets