Special Topics In Scientific Computing

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Presentation transcript:

Special Topics In Scientific Computing Pattern Recognition & Data Mining Lect2: Probability and One Example of Model Fitting

Ref:

Probability Theory Apples and Oranges

Probability Theory Marginal Probability Conditional Probability Joint Probability

Probability Theory Sum Rule Product Rule

The Rules of Probability Sum Rule Product Rule

Bayes’ Theorem posterior  likelihood × prior

Probability Densities

Transformed Densities

Expectations Conditional Expectation (discrete) Approximate Expectation (discrete and continuous)

Variances and Covariances

The Gaussian Distribution

Gaussian Mean and Variance

The Multivariate Gaussian

Gaussian Parameter Estimation Likelihood function

Maximum (Log) Likelihood

Model Fitting

Polynomial Curve Fitting

Sum-of-Squares Error Function

0th Order Polynomial

1st Order Polynomial

3rd Order Polynomial

9th Order Polynomial

Over-fitting Root-Mean-Square (RMS) Error:

Polynomial Coefficients

Data Set Size: 9th Order Polynomial

Data Set Size: 9th Order Polynomial

Regularization Penalize large coefficient values

Regularization:

Regularization:

Regularization: vs.

Polynomial Coefficients

Curve fitting: Statistical View

Maximum Likelihood Determine by minimizing sum-of-squares error, .

Predictive Distribution

Predictive Distribution

MAP: A Step towards Bayes Determine by minimizing regularized sum-of-squares error, .

Bayesian Curve Fitting

Bayesian Predictive Distribution