Lecture 25 of 41
Universal Planning and Reaction;
Probability Review
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Probability Review - Artificial Intelligence - Lecture Slides, Slides of Artificial Intelligence
Some concept of Artificial Intelligence are Agents and Problem Solving, Autonomy, Programs, Classical and Modern Planning, First-Order Logic, Resolution Theorem Proving, Search Strategies, Structure Learning. Main points of this lecture are: Probability Review, Universal Planning, Reaction, Simple Reflex Agents, Environment, Agent, Effectors, Condition-Action, Goals, Action
Typology: Slides
2012/2013
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Lecture 25 of 41
Universal Planning and Reaction;
Probability Review
Review:
Simple Reflex Agents
Agent Sensors
Effectors
Condition-Action
Rules
What action I
should do now
Environment
Review [3]:
Goal-Based Agents
Agent Sensors
Effectors
Goals
What action I
should do now
State^ Environment
How world evolves
What my actions do
What world is
like now
What it will be
like if I do
action A
Review [4]:
Utility-Based Agents
Agent Sensors
Effectors
Utility What action I
should do now
State^ Environment
How world evolves
What my actions do
What world is
like now
What it will be
like if I do A
How happy will
I be
Bayes’s Theorem [1]
Bayes’s Theorem [2]
- Theorem
- P ( h ) Prior Probability of Assertion (Hypothesis) h
- Measures initial beliefs (BK) before any information is obtained (hence prior)
- P ( D ) Prior Probability of Data (Observations) D
- Measures probability of obtaining sample D (i.e., expresses D )
- P ( h | D ) Probability of h Given D
- | denotes conditioning - hence P(h | D) is a conditional ( aka posterior) probability
- P ( D | h ) Probability of D Given h
- Measures probability of observing D given that h is correct (“ generative ” model)
- P ( h D ) Joint Probability of h and D
- Measures probability of observing D and of h being correct
P D
P h D
P D
P D|hP h P h|D
Choosing Hypotheses
arg max f x
x Ω
- Bayes’s Theorem
- MAP Hypothesis
- Generally want most probable hypothesis given the training data
- Define: the value of x in the sample space with the highest f ( x )
- Maximum a posteriori hypothesis, hMAP
- ML Hypothesis
- Assume that p ( hi ) = p ( hj ) for all pairs i , j (uniform priors, i.e., PH ~ Uniform)
- Can further simplify and choose the maximum likelihood hypothesis, hML
argmaxP D |h P h
P D
PD|hP h argmax
h argmaxP h|D
h H
hH
hH MAP
P D
P h D
P D
P D|hP h P h|D
i
h H
hML argmaxPD|h i
Terminology
- Introduction to Reasoning under Uncertainty
- Probability foundations
- Definitions: subjectivist, frequentist, logicist
- (3) Kolmogorov axioms
- Bayes’s Theorem
- Prior probability of an event
- Joint probability of an event
- Conditional (posterior) probability of an event
- Maximum A Posteriori (MAP) and Maximum Likelihood (ML) Hypotheses
- MAP hypothesis: highest conditional probability given observations (data)
- ML: highest likelihood of generating the observed data
- ML estimation (MLE): estimating parameters to find ML hypothesis
- Bayesian Inference: Computing Conditional Probabilities (CPs) in A Model
- Bayesian Learning: Searching Model (Hypothesis) Space using CPs