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Decision Making Under Today’s Class
Uncertainty • Making Decisions Under Uncertainty
AI CLASS10 (CH. 15.1-15.2.1, 16.1-16.3) • Tracking Uncertainty over Time
sensors • Decision Making under Uncertainty
• Decision Theory
? environment • Utility
agent
actuators
Material from Marie desJardin, Lise Getoor, Jean-Claude
Cynthia Matuszek – CMSC 671 1 Latombe, Daphne Koller, and Paula Matuszek
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Introduction Sources of Uncertainty
• The world is not a well-defined place. • Uncertain inputs • Uncertain outputs
• Sources of uncertainty • Missing data • All uncertain:
• Uncertain inputs: What’s the temperature? • Noisy data • Reasoning-by-default
• Uncertain (imprecise) definitions: Is Trump a good • Uncertain knowledge • Abduction & induction
president? • >1 cause à >1 effect • Incomplete deductive
inference
• Uncertain (unobserved) states: What’s the top card? • Incomplete knowledge of • Result is derived
causality correctly but wrong in
• There is uncertainty in inferences • Probabilistic effects real world
• If I have a blistery, itchy rash and was gardening all
weekend I probably have poison ivy Probabilistic reasoning only gives probabilistic results
(summarizes uncertainty from various sources)
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Reasoning Under Uncertainty
PARTI: MODELING
• People constantly make decisions anyhow. UNCERTAINTYOVERTIME
• Very successfully!
• How?
• More formally: how do we reason under uncertainty
with inexact knowledge?
• Step one: understanding what we know
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States and Observations Temporal Probabilistic Agent
• Agents don’t have a continuous view of world sensors
• People don’t either!
• We see things as a series of snapshots: ?
• Observations, associated with time slices environment
• t , t , t , … agent
1 2 3 actuators
• Each snapshot contains all variables, observed or not
• X = (unobserved) state variables at time t; observation at t is E
t t t1, t2, t3, …
• This is world state at time t
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Uncertainty and Time Uncertainty and Time
• The world changes • Basic idea:
• Examples: diabetes management, traffic monitoring • Copy state and evidence variables for each time step
• Tasks: track changes; predict changes • Model uncertainty in change over time
• Incorporate new observations as they arrive
• Basic idea: • X = unobserved/unobservable state variables at time t:
• For each time step, copy state and evidence variables t
BloodSugar , StomachContents
• Model uncertainty in change over time (the Δ) t t
• E = evidence variables at time t:
• Incorporate new observations as they arrive t
MeasuredBloodSugar , PulseRate , FoodEaten
t t t
• Assuming discrete time steps
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States (more formally) Observations (more formally)
• Change is viewed as series of snapshots • Time slice (a set of random variables indexed by t):
• Time slices/timesteps 1. the set of unobservable state variables X
t
• Each describing the state of the world at a particular time 2. the set of observable evidence variables Et
• So we also refer to these as states • An observation is a set of observed variable
• Each time slice/timestep/state is represented as a instantiations at some timestep
set of random variables indexed by t: • Observation at time t: E = e
1. the set of unobservable state variables X t t
t • (for some values e)
2. the set of observable evidence variables E t
t
• X denotes the set of variables from X to X
a:b a b
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Transition and Sensor Models Markov Assumption(s)
• So how do we model change over time? • Markov Assumption:
• Transition model This can get • Xt depends on some finite (usually fixed) number of previous Xi’s
exponentially • First-order Markov process: P(X|X ) = P(X|X )
• Models how the world changes over time large… t 0:t-1 t t-1
• Specifies a probability distribution… • kth order: depends on previous k time steps
• Over state variables at time t P(X | X )
• Given values at previous times t 0:t-1
• Sensor model • Sensor Markov assumption: P(E|X , E ) = P(E|X)
• Models how evidence (sensor data) gets its values t 0:t 0:t-1 t t
• E.g.: BloodSugart àMeasuredBloodSugart • Agent’s observations depend only on actual current state of the world
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Stationary Process Complete Joint Distribution
• Infinitely many possible values of t • Given:
• Transition model: P(X|X )
• Does each timestep need a distribution? t t-1
• Sensor model: P(E|X)
• That is, do we need a distribution of what the world looks like at t t
• Prior probability: P(X )
t , given t AND a distribution for t given t AND … 0
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• Assume stationary process: • Then we can specify a complete joint distribution
• Changes in the world state are governed by laws that do of a sequence of states:
not themselves change over time P(X ,X,...,X ,E ,...,E )= P(X ) t P(X | X )P(E |X )
• Transition model P(X|X ) and sensor model P(E|X) 0 1 t 1 t 0 ∏ i i−1 i i
t t-1 t t i=1
are time-invariant, i.e., they are the same for all t
• What’s the joint probability of instantiations?
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Example Inference Tasks
Rt-1 P(Rt| Rt-1) Weather has a 30% chance • Filtering or monitoring: P(X|e ,…,e ):
t 0.7 of changing and a 70% t 1 t
f 0.3 chance of staying the same. • Compute the current belief state, given all evidence to date
• Prediction: P(X |e ,…,e ):
Raint-1 Raint Raint+1 t+k 1 t
• Compute the probability of a future state
• Smoothing: P(X |e ,…, ):
k 1 et
Umbrellat-1 Umbrellat Umbrellat+1 • Compute the probability of a past state (hindsight)
Rt P(Ut| Rt) • Most likely explanation: arg max P(x ,…,x |e ,…,e )
t 0.9 x1,..xt 1 t 1 t
f 0.2 • Given a sequence of observations, find the sequence of states that is
most likely to have generated those observations
Fully worked out HMM for rain: www2.isye.gatech.edu/~yxie77/isye6416_17/Lecture6.pdf 18
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Examples Filtering
• Filtering: What is the probability that it is raining today, • Maintain a current state estimate and update it
given all of the umbrella observations up through today? • Instead of looking at all observed values in history
• Prediction: What is the probability that it will rain the day • Also called state estimation
after tomorrow, given all of the umbrella observations up
through today? • Given result of filtering up to time t, agent must
• Smoothing: What is the probability that it rained yesterday, compute result at t+1 from new evidence e :
given all of the umbrella observations through today? t+1
• Most likely explanation: If the umbrella appeared the first P(Xt+1 | e1:t+1) = f(et+1, P(Xt | e1:t))
three days but not on the fourth, what is the most likely … for some function f.
weather sequence to produce these umbrella sightings?
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Recursive Estimation Recursive Estimation
• P(X | e ) as a function of e and P(X | e ):
1. Project current state forward (t à t+1) t+1 1:t+1 t+1 t 1:t
P(X |e )=P(X |e ,e ) dividing up evidence
2. Update state using new evidence e t+1 1:t+1 t+1 1:t t+1
t+1 =αP(e |X ,e )P(X |e )
t+1 t+1 1:t t+1 1:t Bayes rule
=αP(e |X )P(X |e ) sensor Markov assumption
P(X | e ) as function of e and P(X | e ): t+1 t+1 t+1 1:t
t+1 1:t+1 t+1 t 1:t • P(e | X ) updates with new evidence (from sensor)
P(X+1 | e ) = P(X | e ,e ) t+1 1:t+1
t 1:t+1 t+1 1:t t+1 • One-step prediction by conditioning on current state X:
=αP(e |X )∑P(X |x)P(x |e )
t+1 t+1 t+1 t t 1:t
xt
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Recursive Estimation Group Exercise: Filtering
P(X | e ) = α P(e | X ) P(X | X ) P(X |e ) We got here, but I don’t know that
• One-step prediction by conditioning on current state X: t +1 1:t+1 t +1 t +1 ∑ t+1 t t 1:t they really understood it. Spent
=αP(e |X )∑P(X |x)P(x |e ) Xt Rt-1 P(Rt|Rt-1) time on the class exercise and told
t+1 t+1 t+1 t t 1:t T 0.7
xt transition current F 0.3 them to do it outside. Definitely
model state Raint-1 Raint Raint+1 one for HW3/final exam.
• …which is what we wanted! €
• So, think of P(X | e ) as a “message” f
t 1:t 1:t+1 Didn’t even start decision making.
• Carried forward along the time steps Umbrellat-1 Umbrellat Umbrellat+1
• Modified at every transition, updated at every new observation Rt P(Ut|Rt)
• This leads to a recursive definition: What is the probability of rain on T 0.9
f = aFORWARD(f , e ) Day 2, given a uniform prior of rain F 0.2
1:t+1 1:t t+1 on Day 0, U1 = true, and U2 = true?
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