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Bayesian Decision Theory: Maximizing A-posteriori Probability and Minimizing Conditional Risk

Learn about the statistical and non-statistical approaches of Bayesian Decision Theory, including pattern classification, distance-based methods, parameter estimation, non-parametric learning, and classifier fusion.

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Bayesian Decision Theory: Maximizing A-posteriori Probability and Minimizing Conditional Risk

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  1. ECE 471/571 – Lecture 2 Bayesian Decision Theory

  2. Pattern Classification Statistical Approach Non-Statistical Approach Supervised Unsupervised Decision-tree Basic concepts: Baysian decision rule (MPP, LR, Discri.) Basic concepts: Distance Agglomerative method Syntactic approach Parameter estimate (ML, BL) k-means Non-Parametric learning (kNN) Winner-takes-all LDF (Perceptron) Kohonen maps NN (BP) Mean-shift Support Vector Machine Deep Learning (DL) Dimensionality Reduction FLD, PCA Performance Evaluation ROC curve (TP, TN, FN, FP) cross validation Stochastic Methods local opt (GD) global opt (SA, GA) Classifier Fusion majority voting NB, BKS

  3. Bayes’ formula (Bayes’ rule) From domain knowledge a-priori probability (prior probability) Conditional probability density (pdf – probability density function) (likelihood) a-posteriori probability (posterior probability) normalization constant (evidence)

  4. pdf examples • Gaussian distribution • Bell curve • Normal distribution • Uniform distribution • Rayleigh distribution

  5. Bayes decision rule x Maximum a-posteriori Probability (MAP):

  6. Conditional probability of error

  7. Decision regions • The effect of any decision rule is to partition the feature space into cdecision regions

  8. Overall probability of error Or unconditional risk, unconditional probability of error 8

  9. The conditional risk • Given x, the conditional risk of taking action ai is: • lij is the loss when decide x belongs to class i while it should be j lij

  10. Likelihood ratio - two category classification Likelihood ratio

  11. Zero-One loss

  12. Recap • Bayes decision rule  maximum a-posteriori probability • Conditional risk • Likelihood ratio • Decision regions  How to calculate the overall probability of error

  13. Recap Maximum a-posteriori Probability Likelihood ratio Overall probability of error

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