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0 0 0 0 1 0 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 1 1 1... COS 424/SML 302 Probability and Statistics Review February 6, 2019 49 / 69
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Consistency of the MLE example 0.00 0.25 0.50 0.75 1.00 0 1000 2000 3000 4000 5000 Index MLE of pi Run 1 10 2 3 4 5 6 7 8 9 COS 424/SML 302 Probability and Statistics Review February 6, 2019 50 / 69
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Important distributions that we will discuss a lot Let’s talk briefly about a few distributions. Bernoulli Multinomial Poisson Gaussian The question I hope to answer in this discussion is: When I analyze a data set, what is the most appropriate distribution to select to model specific features? COS 424/SML 302 Probability and Statistics Review February 6, 2019 51 / 69
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Important distributions to know: Bernoulli A Bernoulli distribution models a binary random variable Support : x ∈ { 0 , 1 } Parameter : π [0 , 1] (probability of heads, or bias ) Probability mass function : p ( x | π ) = π [ x =1] (1 - π ) [ x =0] MLE estimates of π : ˆ π = n 1 n 0 + n 1 , where n 1 is the number of 1s and n 0 is the number of 0s in n samples. Conjugate prior for π : beta distribution The binomial distribution models m draws from a Bernoulli distribution. COS 424/SML 302 Probability and Statistics Review February 6, 2019 52 / 69
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Modeling dominant hand using a Bernoulli distribution 0 10 20 30 40 0.00 0.25 0.50 0.75 1.00 Dominant hand (0=left; 1=right) count Histogram of handedness data MLE estimates tell us the right-handed bias is 0 . 85 in this class. COS 424/SML 302 Probability and Statistics Review February 6, 2019 53 / 69
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Important distributions to know: Multinomial Support : Vectors of counts of m independent draws from one of K categories [ x 1 , . . . , x K ] Parameter : θ , θ k 0, 1 = K k =1 θ k Probability mass function : p ( x | θ ) = m ! x 1 ! . . . x K ! θ x 1 1 . . . θ x K K MLE estimates of θ : ˆ θ = h m 1 m , . . . , m K m i , where m k is the count of observations in category k Conjugate prior for θ : Dirichlet distribution COS 424/SML 302 Probability and Statistics Review February 6, 2019 54 / 69
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Data that have been modeled using a Multinomial Rolls of a die Votes for candidates in an election Word frequencies in documents Movie/book/song ratings COS 424/SML 302 Probability and Statistics Review February 6, 2019 55 / 69
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Modeling birth month using a multinomial distribution 0.00 0.05 0.10 0 5 10 Birth Month density Histogram of Birth Month The MLE estimates say you are three times more likely to be born in September than in October. COS 424/SML 302 Probability and Statistics Review February 6, 2019 56 / 69
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Important distributions to know: Poisson The Poisson distribution naturally estimates count data Support : non-negative integers { 0 } S Z + Parameter : λ ∈ < + (mean and variance) Probability mass function : p ( x | λ ) = λ x e - λ x ! MLE estimates of λ (the empirical mean): ˆ λ = 1 n n X i =1 x i Conjugate prior for λ : gamma distribution COS 424/SML 302 Probability and Statistics Review February 6, 2019 57 / 69
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Data that have been modeled using a Poisson The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry [Bortkiewicz] The number of yeast cells used when brewing Guinness beer [Gosset]
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  • Spring '09
  • Probability theory

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