CIS 730 Introduction to Artificial Intelligence Lecture 10 of 32

1
KDD Group Research SeminarFall, 2001
Presentation 2b of 11
Adaptive Importance Sampling on Bayesian
Networks (AIS-BN)
Friday, 05 October 2001 Julie A.
Stilson http//www.cis.ksu.edu/jas3466 Reference
Cheng, J. and Druzdzel, M (2000). AIS-BN An
Adaptive Importance Sampling Algorithm for
Evidential Reasoning in Large Bayesian Networks.
Journal of Artificial Intelligence Research, 13,
155-188.
2
Outline

• Basic Algorithm
• Definitions
• Updating importance function
• Example using Sprinkler-Rain
• Why Adaptive Importance Sampling?
• Heuristic initialization
• Sampling with unlikely evidence
• Different Importance Sampling Algorithms
• Forward Sampling (FS)
• Logic Sampling (LS)
• Self-Importance Sampling (SIS)
• Differences between SIS, AIS-BN
• Gathering results
• How RMSE values are collected
• Sample results for FS, AIS-BN

3
Definitions

• Importance Conditional Probability Tables
  (ICPTs)
• Probability tables that represent the learned
  importance function
• Initially, equal to the CPTs
• Updated after each updating interval (see below)
  
• Learning Rate
• The rate at which the true importance function
  is being learned
• Learning rate a (b / a) (k / kmax)
• A initial learning rate, b learning rate in
  last step, k number of updates that have
  been made, kmax total number of updates that
  will be made
• Frequency Table
• Stores the frequency with which each
  instantiation of each query node occurs
• Used to update importance function
• Updating Interval
• AIS-BN updates the importance function after
  this many samples
• If 1000 total samples are to be taken, and the
  updating interval is 100, then 10 total updates
  will be made

4
Basic Algorithm
k number of updates so far , m desired
number of samples , l updating interval for
(int i 1, i lt m, i) if (i mod l 0)
k Update importance function Prk(X\E)
based on total samples generate a sample
according to Prk(X\E), add to total
samples totalweight Pr(s,e) /
Prk(s) totalweight 0 T null for (int i
1 i lt m, i) generate a sample according to
Prkmax(X\E), add to total samples totalweight
Pr(s,e) / Prkmax(s) compute RMSE value of s
using totalweight
5
Updating Importance Function

• Theorem Xi in X, Xi not in Anc(E) gt Pr(Xi
  Pa(Xi), E) Pr(Xi Pa(Xi))
• Proved using d-connectivity
• Only ancestors of evidence nodes need to have
  their importance function learned
• The ICPT tables of all other nodes do not change
  throughout sampling
• Algorithm for Updating Importance Function
• Sample l points independently according to the
  current importance function, Prk(X\E)
• For every query node Xi that is an ancestor to
  evidence, estimate Pr(xi pa(Xi), e) based on
  the samples
• Update Prk(X\E) according to the following
  formula
• Pr(k1)(xi pa(Xi), e) Prk(xi pa(Xi), e)
  LRate (Pr(xi pa(Xi), e) Prk(xi pa(Xi),
  e)

6
Example Using Sprinkler-Rain

• Imagine Ground is evidence instantiated to Wet
• More probable that Sprinkler is on and that it is
  raining
• ICPT tables update the probabilities of the
  ancestors to evidence nodes to reflect this

7
Why Adaptive Importance Sampling?

• Heuristic Initialization Parents to Evidence
  Nodes
• Changes the probabilities of the parents to
  evidence to a uniform distribution when the
  probability of that evidence is sufficiently
  small
• Parents of evidence nodes are most affected by
  the instantiation of evidence
• Uniform distribution helps importance function be
  learned faster
• Heuristic Initialization Extremely Small
  Probabilities
• Extremely low probabilities would usually not be
  sampled much
• Slow to learn true importance function
• AIS-BN raises extremely low probabilities to a
  set threshold and lowers extremely high
  probabilities accordingly
• Sampling with Unlikely Evidence
• Importance function very different from CPTs with
  unlikely evidence
• Difficult to accurately sample without changing
  probability distributions
• AIS-BN performs better than other sampling
  algorithms with unlikely evidence

8
Different Importance Sampling Algorithms

• Forward Sampling / Likelihood Weighting (FS)
• Similar to AIS-BN, but importance function is not
  learned
• Performs well under most circumstances
• Doesnt do well when evidence is unlikely
• Logic Sampling (LS)
• Network is sampled randomly without regard to
  evidence
• Samples that dont match evidence are then
  discarded
• Simplest importance sampling algorithm
• Also performs poorly with unlikely evidence
• Inefficient when many nodes are evidence
• Self-Importance Sampling (SIS)
• Also updates an importance function
• Does not obtain samples from learned importance
  function
• Updates to importance function do not use
  sampling information
• For large numbers of samples, performs worse than
  FS

9
Gathering Results

• Relative Root Mean Square Error
• P(?i) is exact probability of sample
• P(?i) is estimated probability of
• sample from frequency table
• M arity, T number of samples
• RMSE Collection
• Relative RMSE computed
• for each sample
• Each RMSE value is stored in
• an output file printings.txt
• Graphing Results
• Open output file in Excel
• Graph results using Chart
• Example Chart
• ALARM network, 10000 samples
• Compares FS, AIS-BN