A Probabilistic Approach to Inference with Limited Information in Sensor Networks

1
A Probabilistic Approach to Inference with
Limited Information in Sensor Networks

• Rahul BiswasSebastian ThrunLeonidas Guibas
• Stanford University

Modified and presented by Kyoungwon Suh
2
Outline

• Short tutorial MCMC Metropolis algorithm
• Motivation for this work
• Problem definition
• Bayesian network-based Model
• Probabilistic solution based on MCMC
• Future work and Conclusion

3
Markov chain Monte Carlo (MCMC)

• The set X and the distribution p(x) we wish to
  sample from
• Key Idea
• Suppose it is hard to sample p(x).
• If we have MC with its stationary Distr p(x),
  then use the Markov chain to do a random walk to
  draw random samples from p(x)
• ltQgt How to construct such MC?
• ltA1gt MCMC Metropolis algorithm

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Stationary distribution

• Consider the Markov chain given
• The stationary distribution is
• Some samples

Empirical Distribution
1,1,2,3,2,1,2,3,3,2 1,2,2,1,1,2,3,3,3,3 1,1,1,2,3,
2,2,1,1,1 1,2,3,3,3,2,1,2,2,3 1,1,2,2,2,3,3,2,1,1
1,2,2,2,3,3,3,2,2,2
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Metropolis algorithm

• How to pick a suitable Markov chain for our
  distribution?
• We define a Markov chain with the following
  process
• Sample a candidate point x(t1) from a proposal
  distribution q(x(t1) x(t)) which is symmetric
  q(xy)q(yx)
• Compute the importance ratio (Usually, it it easy
  to obtain!)
• With probability min(r,1) transition to x(t1) ,
  otherwise stay in the same state

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Metropolis intuition

• Why does the Metropolis algorithm work?
• Proposal distribution can propose anything it
  likes (as long as it can jump back with the same
  probability)
• Proposal is always accepted if its jumping to a
  more likely state
• Proposal accepted with the importance ratio if
  its jumping to a less likely state
• The acceptance policy, combined with the
  reversibility of the proposal distribution, makes
  sure that the algorithm explores states in
  proportion to p(x)!

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Traditional Algorithm Design
output
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In a Perfect World
output
sensor data
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In the Real World
sensor data
sensor data may be noisy, missing, and expensive
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What to Do?
probabilistic output
sensor data
Our Algorithm
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Scenario

• Friendly agent at known location LF

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Sensors

• N sensors at known locations LS1LSN

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Enemy Agents

• M enemies at unknown locations LE1LEM

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Query

• Is LF in convex hull of LE1LEM?

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Sensor Model

• Sensors stochasticallydetect presence ofenemy
  agents

ideal observation
sensor observation
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Problems in Answering Query

• May not be sensor near enemy agent
• Sensor near agent may not have sensed
• Sensor may malfunction
• may report enemy when none is present
• may not report enemy when one is present

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Our Approach

• Capture relations in a Bayesian network
• Recast query as a posterior estimate

observed
sensor measurements
enemy agent locations
query
unobserved
D
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Sensor Measurement Variables

• Ri is known to us
• Binary variable either yes or no

observed
sensor measurements
enemy agent locations
query
unobserved
D
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Enemy Agent Location Variables

• LEi is not known
• Continuous variables

observed
sensor measurements
enemy agent locations
query
unobserved
D
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Query Variable

• D is not known
• Binary variable either yes or no

observed
sensor measurements
enemy agent locations
query
unobserved
D
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Prior Probability P(LEi)

• Uniform distribution across region

sensor measurements
enemy agent locations
query
D
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Conditional Probability P(RjLE)

• As specified by sensor model

ideal observation
sensor observation
sensor measurements
enemy agent locations
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Conditional Probability P(DLE)

• Let CH1 CHS be the convex hull of LE
• D is true iff

counterclockwise
clockwise
enemy agent locations
query
D
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Query Reformulation

• Want to estimate P(DR)
• Produces probabilistic answer
• D can be replaced by any query
• Extent and Area of Point Set
• Maximally Distant Point

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MCMC for Inference

• MCMC Markov Chain Monte Carlo
• Approximate inference algorithm
• Allows us to feed in the R variables
• Generates a probability for D
• Uses traditional convex hull methods

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How MCMC Works

• Draw T samples LE1LET from P(LER)
• How to generate samples from P(LER)?

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How MCMC Works (continued)

• Draw samples from Markov Chain
• Desired stationary distribution P(LER)
• Start with random LE11 LE1T
• At each step, move LEti to new location
• If more likely, accept
• min(1, P(LEt1,R1,..Rp)/ P(LEt,R1,..Rp))1
• If less likely, sometimes accept
• 0 lt min(1, P(LEt1,R1,..Rp)/ P(LEt,R1,..Rp)) lt 1

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Experimental Results
enemy agents (five)
sensors
friendly agent
low
high
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Experimental Results

• P(DR)
• 51
• no sensorshave sensed

low
high
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Experimental Results

• P(DR)
• 44
• not useful region

low
high
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Experimental Results

• P(DR)
• 81
• useful region

low
high
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Experimental Results

• P(DR)
• 80
• not important

low
high
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Experimental Results

• P(DR)
• 86
• near edge

low
high
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Experimental Results

• P(DR)
• 93
• negative information

low
high
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Usefulness of More Sensors
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Future Work

• Intelligent Sensor Selection
• Actively choose sensors that will sense
• Possible to predict entropy reduction
• Distributed MCMC
• Run on Crossbow MICA2 Motes

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Conclusion

• Presented an interface between
• noisy, incomplete sensor data
• existing algorithms
• Used probabilistic model to analyze data
• Approximate inference via MCMC

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Acknowledgment

• Slides are borrowed from
• Teg Grenagers talk on MCMC (stanford)
• Rahul Biswas IPSN talk (stanford)

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• Backup slides .

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Markov chain Monte Carlo (MCMC)

• The set X and the distribution p(x) we wish to
  sample from
• Suppose that it is hard to sample p(x) but that
  it is possible to walk around in X using only
  local state transitions
• Insight we can use a random walk to help us
  draw random samples from p(x)

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Markov Chains for sampling

• Markov chain on a space X with transitions T is a
  random process (x(0), x(1),x(t),) that satisfy
• In order for a Markov chain to be useful for
  sampling p(x), the stationary distribution of the
  Markov chain must be p(x)
• If this is the case, we can start in an arbitrary
  state, use the Markov chain to do a random walk
  for a while, and stop and output the current
  state x(t)
• The resulting state will be sampled from p(x)!