Artificial Intelligence Uncertainty Chapter 13. Uncertainty What shall an agent do when not all is...
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Transcript of Artificial Intelligence Uncertainty Chapter 13. Uncertainty What shall an agent do when not all is...
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Artificial Intelligence
UncertaintyChapter 13
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Uncertainty
What shall an agent do when not all is crystal clear?
Different types of uncertainty effecting an agent:The state of the world?The effect of actions?
Uncertain knowledge of the world:Inputs missingLimited precision in the sensorsIncorrect model: action state due to the complexityA changing world
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Uncertainty
Let action At = leave for airport t minutes before flightWill At get me there on time?
Problems:1) partial observability (road state, other drivers' plans,
etc.)2) noisy sensors (traffic reports)3) uncertainty in action outcomes (flat tire, etc.)4) immense complexity of modeling and predicting traffic
(A1440 might reasonably be said to get me there on time but I'd have
to stay overnight in the airport ...)
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Rational Decisions
A rational decision must consider:– The relative importance of the sub-goals– Utility theory– The degree of belief that the sub-goals will be
achieved
– Probability theory
Decision theory = probability theory + utility theory : the agent is rational if and only if it chooses the action
that yields the highest expected utility, averaged over
all possible outcomes of the action”
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Using FOL for (Medical) Diagnosis
p Symptom(p, Toothache) Disease(p, Cavity)Not correct...
p Symptom(p, Toothache) Disease(p, Cavity) Disease(p, GumDisease) Disease(p, WisdomTooth)
Not complete...
p Disease(p, Cavity) Symptom(p, Toothache)Not correct...
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Handling Uncertain Knowledge
Problems using first-order logic for diagnosis:Laziness: Too much work to make complete rules. Too much work to use themTheoretical ignorance: Complete theories are rarePractical ignorance: We can’t run all tests anyway
Probability can be used to summarize the lazinessand ignorance !
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Probability
Compare the following:1) First-order logic:
“The patient has a cavity”2) Probabilistic:
“The probability that the patient has a cavity is 0.8”
1) Is either valid or not, depending on the state of the world
2) Validity depends on the agents perception history, the evidence
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Probability
Subjective or Bayesian probability:Probabilities relate propositions to one's own state of
knowledgee.g., P(A25 | no reported accidents) = 0.06
These are not claims of some probabilistic tendency in the current situation (but might be learned from past experience
of similar situations)
Probabilities of propositions change with new evidence:e.g., P(A25 | no reported accidents, 5 a.m.) = 0.15
(Analogous to logical entailment status KB ╞ )
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Probability
Probabilities are either:
Prior probability (unconditional , “obetingad”)Before any evidence is obtained
Posterior probability (conditional , “betingad”) After evidence is obtained
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Probability
Notation for unconditional probability for a proposition A: P(A)
Ex: P(Cavity)=0.2 means:“the degree of belief for “Cavity” given no extra evidence is 0.2”
Axioms for probabilities:
)()()()(.3
0)(,1)(.2
1)(0.1
BAPBPAPBAP
FalsePTrueP
AP
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Probability
The axioms of probability constrain the possible assignments of probabilities to propositions. An
agent that violates the axioms will behave irrationally
in some circumstances
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Random variable
A random variable has a domain of possible valuesEach value has a assigned probability between 0 and 1The values are :
Mutually exclusive (disjoint): (only one of them are true)Complete (there is always one that is true)
Example: The random variable Weather:P(Weather=Sunny) = 0.7P(Weather=Rain) = 0.2P(Weather=Cloudy) = 0.08P(Weather=Snow) = 0.02
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Random Variable
The random variable Weather as a whole is said to have a
probability distribution which is a vector (in the discrete
case):P(Weather) = [0.7 0.2 0.08 0.02]
(Notice the bold P which is used to denote the prob.distribution)
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Random variable - Example
Example - The random variable Season: P(Season = Spring) = 0.26 or shorter: P(Spring)=0.26
P(Season = Summer)= 0.20P(Season = Autumn) = 0.28P(Season = Winter) = 0.26
The random variable Season has a domain <Spring, Summer, Autumn, Winter>and a probability distribution:P(Season) = [0.26 0.20 0.38 0.26] The values in the domain are :
Mutually exclusive (disjoint): (only one of them are true)Complete (there is always one that is true)
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Probability ModelBegin with a set - the sample spacee.g., 6 possible rolls of a die. is a sample point/possible world/atomic event
A probability space or probability model is a sample spacewith an assignment P() for every 0 P() 1 Σ P () = 1e.g., P(1) P(2) P(3) P(4) P(5) P(6) 1/6.
An event A is any subset of
P(A) = Σ{ A} P()
E.g., P(die roll < 4) = 1/6 + 1/6 + 1/6 = 1/2
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The Joint Probability Distribution
Assume that an agent describing the world using therandom variables X1, X2, ..Xn.
The joint probability distribution (or ”joint”) assigns values for all combinations of values on X1, X2, ..Xn.
Notation: P(X1, X2, ..Xn) (i.e. P bold)
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The Joint Probability Distribution
Joint probability distribution for a set of r.v.s gives the probability of
every atomic event on those r.v.s (i.e., every sample point)
P(Weather,Cavity) = a 4 x 2 matrix of values:
Every question about a domain can be answered by the joint distribution because every event is a sum of sample points
Weather = sunny
rain cloudy
snow
Cavity = true
0.144
0.02
0.016 0.02
Cavity = false
0.576
0.08
0.064 0.08
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WeatherWeather Season Season SunSun RainRain CloudCloud SnowSnow
SpringSpring 0.070.07 0.030.03 0.100.10 0.060.06 0.260.26
P(Spring)P(Spring) SummerSummer 0.130.13 0.010.01 0.050.05 0.010.01 ++0.200.20
P(Summer)P(Summer) AutumnAutumn 0.050.05 0.050.05 0.150.15 0.030.03 ++0.280.28
P(Autumn)P(Autumn) WinterWinter 0.050.05 0.010.01 0.100.10 0.100.10 ++0.260.26
P(Winter)P(Winter)0.300.30 0.100.10 0.400.40 0.200.20P(Sun)+P(Rain)+P(Cloud)+P(Snow) P(Sun)+P(Rain)+P(Cloud)+P(Snow) ==
1.001.00
ExExampleample:: P(Weather=Sun P(Weather=Sun Season=Summer) = 0.13 Season=Summer) = 0.13
Example of ”Joint” P(Season, Weather)
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Conditional ProbabilityThe Posterior prob. (conditional prob.) after obtaining evidence:Notation: P( A|B ) means: ”The probability of A given that all we know is B”. Example:P( Sunny | Summer ) = 0.65
Is defined as: if P(B) 0
Can be rewritten as the product rule:
P(A B) = P(A|B)P(B) = P(B|A)P(A)
”For A and B to be true, B has to be true, and A has to be true given B”
)()(
)|(BPBAP
BAP
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Conditional Probability
For the entire random variables:
should be interpreted as a set of equations for allpossible values on the random variables A and B.Example:
)|()(),( BABBA PPP
)|()(),( SeasonWeatherSeasonSeasonWeather PPP
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Conditional Probability
A general version holds for whole distributions, e.g.,P(Weather,Cavity) = P(Weather|Cavity) P(Cavity)
(View as a 4 x 2 set of equations)
Chain rule is derived by successive application of product
rule:P(X1,...,Xn) = P(X1,...,Xn-1) P(Xn | X1,...,Xn-1)
= P(X1,...,Xn-2)P(Xn-1 | X1,...,Xn-2) P(Xn | X1,...,Xn-1)
= ... = n
i=1 P(Xi | X1,...,Xi-1)
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Bayes’ Rule
The left side of the product rule is symmetric w.r.t B and A:
Equating the two right-hand sides yields Bayes’ rule:
)()()|(
)|(AP
BPBAPABP
)|()()(
)|()()(
BAPBPBAP
ABPAPBAP
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Bayes' Rule
Useful for assessing diagnostic probability from causal probability:
P(Cause|Effect) = P(Effect|
Cause)P(Cause)
P(Effect)
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Example of Medical Diagnosis using Bayes’ rule
Known facts:Meningitis causes stiff neck 50% of the time.The probability of a patient having meningitis (M) is
1/50.000.The probability of a patient having stiff neck (S) is 1/20.Question:What is the probability of meningitis given stiff neck ?Solution:P(S|M)=0.5P(M) = 1/50.000 Note: posterior probability of
P(S) = 1/20 meningitis still very small!!0002.0
20/1
50000/15.0
P(S)
P(M)M)|P(SS)|P(M
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Bayes' Rule
In distribution form
P(Y|X) = = P(X|Y) P(Y)P(X|Y) P(Y)
P(X)
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Combining Evidence
Task: Compute P(Cavity|Toothache Catch )
1. Rewrite using the definition and use the joint. With N evidence variables, the “joint” will be an N dimensional table. It is often impossible to compute probabilities for all entries in the table.
2. Rewrite using Bayes’ rule. This also requires a lot of cond.prob. to be estimated. Other methods are to prefer.
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Bayes' Rule and Conditional Independence
P(Cavity|toothache catch) = P(toothache catch|Cavity) P(Cavity) = P(toothache|Cavity) P(catch|Cavity) P(Cavity)
This is an example of a naive Bayes model:
P(Cause,Effect1,...,Effectn) = P(Cause) i P(Effecti|Cause)
Total number of parameters is linear in n
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Summary
Probability can be used to reason about uncertainty
Uncertainty arises because of both laziness and ignorance and it is inescapable
in complex, dynamic, or inaccessible worlds.
Probabilities summarize the agent's beliefs.
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Summary
Basic probability statements include prior probabilities and conditional probabilities over simple and complex propositions
The full joint probability distribution specifies the probability of
each complete as assignment of values to random variables. It is
usually too large to create or use in its explicit form
The axioms of probability constrain the possible assignments of probabilities to propositions. An agent that violates the axioms
will behave irrationally in some circumstances
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Summary
Bayes' rule allows unknown probabilities to be computed from known conditional probabilities, usually in the causal direction.
With many pieces of evidence it will in general run into the same scaling problems as does the full joint distribution
Conditional independence brought about by direct causal relationships in the domain might allow the full joint distribution to
be factored into smaller, conditional distributions
The naive Bayes model assumes the conditional independence of all effect variables, given a single cause variable, and grows linearly with the number of effects
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Next!
Bayesian Network!Chapter 14
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Artificial Intelligence
Bayesian NetworkChapter 14
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Introduction
Most application requires a way of handling uncertainty to be able to make adequate decisions
Bayesian decision theory is a fundamental statistical approach to the problem of complex decision making
The expert system developed in the middle of the 1970s used probabilistic techniques
Promising results - but not sufficient because of the exponential number of probabilities required in the full joint distribution
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Bayesian Network - Syntax
A Bayesian network (or belief network) holds certain properties:
1. A set of random variables U={A1,…,An}, where each variable has a finite set of mutually exclusive states: Aj={a1,…, am}
2. A set of directed links or arrow connects pairs of nodes {A1 An}
3. Each node has a conditional probability table (CPT) that quantifies the effect that the parents have on the node
4. The graph is a directed acyclic graph (DAG)
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Example
Topology of network encodes conditional independence assertions:
Weather is independent of the other variables
Toothache and Catch are conditionally independent given Cavity
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Example
I'm at work, neighbor John calls to say my alarm is ringing, but
neighbor Mary doesn't call. Sometimes it is set off by minorearthquakes. Is there a burglar?
Variables: Burglar, Earthquake, Alarm, JohnCalls, MaryCallsNetwork topology reflects “causal” knowledge:
A burglar can set the alarm offAn earthquake can set the alarm offThe alarm can cause Mary to callThe alarm can cause John to call
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Example
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Bayesian Network - Semantics
Every Bayesian network provides a complete description of the
domain and has a joint probability distribution:
In order to construct a Bayesian network with the correct structure for the domain, we need to choose parents for each node such that this property holds
The parents of node Xi should contain all those nodes in
X1,…,Xi-1 that directly influence Xi
.))(|(),...,(1
1
n
iiin XParentsXPXXP
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Example - Semantics
"Global" semantics defines the full joint distribution as
the product of the local conditional distributions:
e.g., P(j m a b e) = P(j|a) P(m|a) P(a|b,e) P(b) P(e)
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Bayesian Network - Complexity
Since a Bayesian network is a complete and nonredundant representation of the domain it is often more compact than the full joint
General property of locally structure systems
In a locally structured, each subcomponent interacts directly with only a bounded number of other components
It is reasonable to suppose that in most domains each random variable is directly influenced by at most k others for some constant k
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Bayesian network - Complexity
Consider Boolean variable for simplicity:
then the amount of information need to specify the CPT for a node will be at most 2k numbers, so the complete network can be specified by n2k numbers. The full joint contains 2n
For example, suppose we have 20 nodes (n=20) and each
node has 5 parents (k=5). Then the Bayesian network requires 640 numbers, but the full joint will require over a million
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Bayesian network - Complexity
I.e., grows linearly with n, vs. O(2n) for the full joint distribution
For burglary net, 1 + 1 + 4 + 2 + 2 = 10 numbers (vs. 25-1 = 31)
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Example
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Conditional Independence
It is important to be aware of that the links between the variables represent a direct causal relationship
New information can be “transmitted” trough nodes in the
same and opposite directions of the links
This effects the properties of conditional dependency and
independency in Bayesian belief networks
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Local Semantics
Local semantics: each node is conditionally independentof its nondescendants given its parents
Theorem: Local semantics global semantics
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Constructing Bayesian Networks
Need a method such that a series of locally testable assertions ofconditional independence guarantees the required global semantics
Choose the set of relevant variables that describe the domainChoose an ordering of variables X1,...,Xn
For i = 1 to nadd Xi to the network
select parents from X1,...,Xi-1 such thatP(Xi|Parents(Xi)) = P(Xi|X1,..., Xi-1)
This choice of parents guarantees the global semantics:
P(X1,...,Xn) = ni = 1 P(Xi | X1,..., Xi-1) (chain rule)
= ni = 1 P(Xi|Parents(Xi))(by construction)
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Learning with BN - Problems
Steps are often intermingled in practice
Judgments of conditional independence and/or cause and effect
can influence problem formulation
Assessments in probability may lead to changes in the network
structure
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Learning with BN
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Learning with BN
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Learning with BN
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Learning with BN
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Inference
The basic task for any probabilistic inference system is to compute the posterior probability distribution for a set of query variables, given exact values for some evidence variables
Bayesian network are flexible enough so that any node can
serve as either a query or an evidence variable
By adding the evidence we get the posterior probability distribution our query variable will give
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Inference
The task of interest is to compute: P(X|E)
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Summary
Bayesian networks, a well-developed representation for uncertain knowledge
A Bayesian network is a directed acyclic graph whose nodes correspond to random variables; each node has a conditional distribution for the node, given its parents
Bayesian networks provide a concise way to represent conditional independence relationships in the domain
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Summary
A Bayesian network specifies a full joint distribution; Each joint entry is defined as the product of the corresponding entries in the local conditional distributions
A Bayesian network is often exponentially smaller than the full joint distribution
Learning in Bayesian networks involves: Parametric estimation, parametric optimization, learning structures, parametric optimization combined with learning
structures
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Next Time!
Repetition!