probabilistic reasoning over time n.
Download
Skip this Video
Download Presentation
Probabilistic Reasoning over Time

Loading in 2 Seconds...

play fullscreen
1 / 11

Probabilistic Reasoning over Time - PowerPoint PPT Presentation


  • 98 Views
  • Uploaded on

Probabilistic Reasoning over Time. CMSC 671 – Fall 2010 Class #19 – Monday, November 8 Russell and Norvig, Chapter 15.1-15.2. material from Lise Getoor, Jean-Claude Latombe, and Daphne Koller. sensors. environment. ?. agent. actuators. Temporal Probabilistic Agent. t 1 , t 2 , t 3 , ….

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Probabilistic Reasoning over Time' - hashim-reed


Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
probabilistic reasoning over time

Probabilistic Reasoning over Time

CMSC 671 – Fall 2010

Class #19 – Monday, November 8

Russell and Norvig, Chapter 15.1-15.2

material from Lise Getoor, Jean-Claude Latombe, and Daphne Koller

temporal probabilistic agent

sensors

environment

?

agent

actuators

Temporal Probabilistic Agent

t1, t2, t3, …

time and uncertainty
Time and Uncertainty
  • The world changes, we need to track and predict it
  • Examples: diabetes management, traffic monitoring
  • Basic idea:
    • Copy state and evidence variables for each time step
    • Model uncertainty in change over time, incorporating new observations as they arrive
    • Reminiscent of situation calculus, but for stochastic domains
  • Xt – set of unobservable state variables at time t
    • e.g., BloodSugart, StomachContentst
  • Et – set of evidence variables at time t
    • e.g., MeasuredBloodSugart, PulseRatet, FoodEatent
  • Assumes discrete time steps
states and observations
States and Observations
  • Process of change is viewed as series of snapshots, each describing the state of the world at a particular time
  • Each time slice is represented by a set of random variables indexed by t:
      • the set of unobservable state variables Xt
      • the set of observable evidence variables Et
  • The observation at time t is Et = et for some set of values et
  • The notation Xa:b denotes the set of variables from Xa to Xb
stationary process markov assumption
Stationary Process/Markov Assumption
  • Markov Assumption:
    • Xt depends on some previous Xis
  • First-order Markov process: P(Xt|X0:t-1) = P(Xt|Xt-1)
  • kth order: depends on previous k time steps
  • Sensor Markov assumption:P(Et|X0:t, E0:t-1) = P(Et|Xt)
    • That is: The agent’s observations depend only on the actual current state of the world
  • Assume stationary process:
    • Transition model P(Xt|Xt-1) and sensor model P(Et|Xt) are time-invariant (i.e., they are the same for all t)
    • That is, the changes in the world state are governed by laws that do not themselves change over time
complete joint distribution
Complete Joint Distribution
  • Given:
    • Transition model: P(Xt|Xt-1)
    • Sensor model: P(Et|Xt)
    • Prior probability: P(X0)
  • Then we can specify complete joint distribution:
example
Example

Raint-1

Raint

Raint+1

Umbrellat-1

Umbrellat

Umbrellat+1

This should look a lot like a finite state automaton (since it is one...)

inference tasks
Inference Tasks
  • Filtering or monitoring: P(Xt|e1,…,et)Compute the current belief state, given all evidence to date
  • Prediction: P(Xt+k|e1,…,et) Compute the probability of a future state
  • Smoothing: P(Xk|e1,…,et) Compute the probability of a past state (hindsight)
  • Most likely explanation: arg maxx1,..xtP(x1,…,xt|e1,…,et)Given a sequence of observations, find the sequence of states that is most likely to have generated those observations
examples
Examples
  • Filtering: What is the probability that it is raining today, given all of the umbrella observations up through today?
  • Prediction: What is the probability that it will rain the day after tomorrow, given all of the umbrella observations up through today?
  • Smoothing: What is the probability that it rained yesterday, given all of the umbrella observations through today?
  • Most likely explanation: If the umbrella appeared the first three days but not on the fourth, what is the most likely weather sequence to produce these umbrella sightings?
filtering
Filtering
  • We use recursive estimation to compute P(Xt+1 | e1:t+1) as a function of et+1 and P(Xt | e1:t)
  • We can write this as follows:
  • This leads to a recursive definition:
    • f1:t+1 =  FORWARD (f1:t, et+1)

QUIZLET: What is α?

filtering example
Filtering Example

Raint-1

Raint

Raint+1

Umbrellat-1

Umbrellat

Umbrellat+1

What is the probability of rain on Day 2, given a uniform prior of rain on Day 0, U1 = true, and U2 = true?

ad