not like simple interest and so are multiplicative In hydrology the

Not like simple interest and so are multiplicative in

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not like simple interest, and so are multiplicative) In hydrology, the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals COS 424/SML 302 Probability and Statistics Review February 6, 2019 64 / 69
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Data modeled using a Gaussian distribution Measurement errors in experiments: using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors In standardized testing, results are modeled using a Gaussian. E.g., the SAT’s traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100. Percentile ranks (“percentiles” or “quantiles”), normal curve equivalents, and z-scores. Bell curve grading assigns relative grades based on a normal distribution of scores COS 424/SML 302 Probability and Statistics Review February 6, 2019 65 / 69
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Modeling height data using a Gaussian distribution 0.00 0.05 0.10 60 65 70 75 Height (inches) density Histogram of height data in inches ˆ μ MLE = 67 . 8in (5 ft 7.8 in), ˆ σ 2 MLE = 15 . 4 What would be a better way to model these data? COS 424/SML 302 Probability and Statistics Review February 6, 2019 66 / 69
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Model pitfalls What’s wrong with modeling height with a Gaussian distribution? Assigns positive probability to numbers < 0 and > 100 Ignores important biological covariates (i.e., male/female) The data do not look like they came from a Gaussian “All models are wrong. Some models are useful.” (G. Box) COS 424/SML 302 Probability and Statistics Review February 6, 2019 67 / 69
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More interesting statistical models We will extend these distributions to more sophisticated models using the graphical model framework. We will see the following models in this class: Naive Bayes classification Linear regression and logistic regression Generalized linear models Hidden variables, mixture models, and the EM algorithm Factor analysis and principal component analysis Sequential models COS 424/SML 302 Probability and Statistics Review February 6, 2019 68 / 69
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Additional resources Machine Learning: A Probabilistic Approach (Chapter 2) Michael Lavine, Introduction to Statistical Thought (an introductory statistical textbook with plenty of R examples, and it’s online too) Chris Bishop, Pattern Recognition and Machine Learning (Ch 1 & 2) (video) Sam Roweis: Machine Learning, Probability and Graphical Models, Part 1 (video) Michael Jordan: Bayesian or Frequentist: Which Are You? wikipedia (much of the material in today’s lecture is available on wikipedia) COS 424/SML 302 Probability and Statistics Review February 6, 2019 69 / 69
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  • Spring '09
  • Probability theory

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