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Probability, Statistics, and Uncertainty Modeling

(CTU-MATH317.AU1)
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Skills You’ll Get

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1

Modeling Uncertainty with Probability Distributions

  • Representing Data
  • Summarizing and Visualizing Data
  • The Basics of Probability and Probability Distributions
  • Hypothesis Testing
  • Basic Problems in Machine Learning
  • Sample Spaces and Events
  • The Counting Approach to Probabilities
  • Set-Wise View of Events
2

Probabilistic Reasoning with Bayes’ Theorem

  • Conditional Probabilities and Independence
  • The Bayes Rule
  • The Basics of Probability Distributions
  • Distribution Independence and Conditionals
  • Summarizing Distributions
  • Compound Distributions
  • Functions of Random Variables (*)
3

Parameter Estimation with Maximum Likelihood

  • Maximum Likelihood Estimation
  • Reconstructing Common Distributions from Data
  • Mixture of Distributions: The EM Algorithm
  • Kernel Density Estimation
  • Reducing Reconstruction Variance
  • The Bias-Variance Trade-Off
  • Popular Distributions Used as Conjugate Priors (*)
4

Statistical Inference and Hypothesis Testing

  • The Basics of Regression
  • Two Perspectives on Linear Regression
  • Solutions to Linear Regression
  • Handling Categorical Predictors
  • Overfitting and Regularization
  • A Probabilistic View of Regularization
  • Evaluating Linear Regression
  • Nonlinear Regression
  • Generative Probabilistic Models
  • Loss-Based Formulations: A Probabilistic View
  • Beyond Classification: Ordered Logit Model
5

Advanced Applications of Bayes’ Theorem

  • The Central Limit Theorem
  • Sampling Distribution and Standard Error
  • The Basics of Hypothesis Testing
  • Hypothesis Tests For Differences in Means
  • χ2-Hypothesis Tests
  • Analysis of Variance (ANOVA)
  • Machine Learning Applications of Hypothesis Testing
  • Markov Chains
  • Machine Learning Applications of Markov Chains
  • Markov Chains to Generative Models
  • Hidden Markov Models
  • Applications of Hidden Markov Models
  • Jensen’s Inequality
  • Markov and Chebyshev Inequalities
  • Approximations for Sums of Random Variables
  • Tail Inequalities Versus Approximation Estimates

1

Modeling Uncertainty with Probability Distributions

  • Preparing Data for Regression and Visualization
  • Performing Hypothesis Testing
  • Modeling Sensor Noise in Robotics
2

Probabilistic Reasoning with Bayes’ Theorem

  • Implementing the Bayes Classifier
  • Training a Naïve Bayes Model
  • Creating a Naïve Bayes Spam Classifier
3

Parameter Estimation with Maximum Likelihood

  • Implementing the Bias-Variance Trade-Off
  • Working with Conjugate Priors and Estimating Parameters
4

Statistical Inference and Hypothesis Testing

  • Training a Linear Regression Model
  • Implementing Lasso Regression
  • Implementing Non-Linear Transformations of Predictors
  • Implementing Multinomial Logistic Regression
  • Training a Logistic Regression Model
5

Advanced Applications of Bayes’ Theorem

  • Using Sampling to Convert Bimodal Data to a Normal Distribution
  • Evaluating AI Model Accuracy with Statistical Tests
  • Testing Hypotheses: Type I and II Errors
  • Calculating and Interpreting Confidence Intervals
  • Using an HMM Model
  • Applying Markov and Chebyshev Inequalities
  • Applying Chernoff Bounds and Hoeffding Inequalities

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