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Probabilistic Reasoning for Modeling Unreliable Data

This presentation discusses the use of probabilistic reasoning and Bayesian inference for modeling uncertainty in unreliable data. Topics include modeling probability distributions, clustering data using expectation maximization, and a simple example.

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Probabilistic Reasoning for Modeling Unreliable Data

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  1. Probabilistic Reasoning for Modeling Unreliable Data Ron Tal York University

  2. Agenda • Modeling Uncertainty • Bayesian Reasoning • Modeling Probability Distributions • Clustering Data Using Expectation Maximization • Simple Example • Discussion

  3. Modeling Uncertainty • Why is it necessary? • The only certainty in this world is uncertainty • Often we cannot afford or are not capable of explicitly enumerating all variables absolutely • Sometimes uncertainty is caused by a limit of the reliability of the technology • Making decisions with unreliable data

  4. Modeling Uncertainty (cont.) • Three competing paradigms: • Non-monotonous Calculus • Fuzzy Logic • Probability Theory • Since we cannot construct a deterministic solution to many problems, we model sources of uncertainty as probability distribution

  5. Bayesian Reasoning • At the core of probabilistic frameworks is Bayesian Inference • Let’s define a few concepts: • - The probability of witnessing evidence E according to hypothesis H • - The probability of hypothesis H given the evidence E • - Probability of Hprior to observing E • -

  6. Bayesian Reasoning: Bayes’ theorem • States that: • Our life becomes simpler Easy to determine! We already know! We don’t always care! Hard to Determine!

  7. Bayesian Reasoning: Bayes’ theorem • If we prefer, it can also be written as The joint probability

  8. Maximum Likelihood • Represents support for a hypothesis in terms of the probabilities of all observations: • Lets us estimate best parameters of a model!

  9. Maximum Likelihood (cont.) • To determine parameters of a model, we maximize the negative log likelihood:

  10. Modeling Probability Distributions • There are many popular ways to represent probabilities • Perhaps the most popular is the Normal or Gaussian Distribution

  11. Modeling Probability Distributions (cont.) • Why? • Frequently occurs in nature • Nice mathematical properties • Maximum Likelihood Estimator is simple Least-Squares

  12. Modeling Probability Distributions (cont.) • Mixture of Gaussians • Provides good approximation of complex distributions • Mathematically treated as a sum of Gaussians • Flexible • Simple Least-Squares won’t do

  13. Modeling Probability Distributions (cont.) • Non-Parametric • Usually a histogram • Simple to use • For some cases parametric models won’t do • Unflexible

  14. Clustering Data with EM • EM is an iterative algorithm for parameter fitting • Simple premise • Make a guess of the parameters • Find new parameters that maximize the likelihood of the model • Repeat until convergence

  15. Clustering Data with EM (cont.) • Very popular for GMM fitting of noisy data • Guaranteed to converge (at some point)

  16. Results: Qualitative Evaluation • Result: 5 pix drift, overall motion is 38 pix • Single frame motion varies from 0.1 pix to 13 pix

  17. Discussion: Performance • Reliability of algorithm requires a larger number of generated samples and a GMM with a large number of components • Under these circumstances, the EM algorithm is bound to be slow • Correspondence of a single feature-point is insufficient for robust object tracking

  18. Discussion: Future Work • Migration from MATLAB to OpenCV • Modification of GMM estimation to incorporate additional information and reduce the number of observations needed • Additional experiments • A tracker based on clustering of multiple feature-points using an affine motion model

  19. Questions

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