Stat-3503/Stat-8109 Airoldi/Fall-21problem set no. 1 | due monday 10/25 before lecture startslearning objectives. compute likelihoods, both for a generic sample, i.e., (x1; :::; xn),and for a specific sample, i.e., (2; 3; 6; 4; 8; 5; 6; 2; 3; 6; 5); write some short programs in Rto generate fake data sets from a given model and plot the corresponding likelihoods.problem 1. set-up: you are interested in studying the writing style of a popular TimeMagazine contributor, FZ. you collect a simple random sample of his articles and count howmany times he uses the word however in each of the articles in your sample, (x1; :::; xn).In this set-up, xi is the number of times the word however appeared in the i-th article.question 1.1. (10 points) define the population of interest, the population quantity ofinterest, and the sampling units.question 1.2. (10 points) what are potentially useful estimands for studying writing style?(hint: you are interested in comparing FZ writing style to that of other contributors.)question 1.3. (10 points) model: let Xi denote the quantity that captures the numberof times the word however appears in the i-th article. let’s assume that the quantitiesX1; :::Xn are independent and identically distributed (IID) according to a Poisson distribution with unknown parameter λ,p(Xi = xi j λ) = Poisson(xi j λ) for i = 1; :::; n:using the 2-by-2 table of what’s variable/constant versus what’s observed/unknown, declarewhat’s the technical nature (random variable, latent variable, known constant or unknownconstant) of the quantities involved the set-up/model above: X1; ::Xn, x1; :::xn, λ and n.question 1.4. (10 points) write the data generating process for the model above.question 1.5. (10 points) define the likelihood L(λ) = p(· j ·) for this model and set-up atthe highest level of abstraction.question 1.6. (10 points) write the likelihood L(λ) for a generic sample of n articles,(x1; :::; xn).question 1.7. (10 points) write the log-likelihood ‘(λ) for a generic sample of n articles,(x1; :::; xn).question 1.8. (10 points) write the log-likelihood ‘(λ) for the following specific sample of 7articles (12; 4; 5; 3; 7; 5; 6).1Stat-3503/Stat-8109 Airoldi/Fall-21question 1.9. (10 points) plot the log-likelihood ‘(λ) in R for the same specific sample of7 articles (12; 4; 5; 3; 7; 5; 6). What is the maximum value of λ (approximately)?question 1.10. (10 points) draw a graphical representation of this model, which explicitlyshows the random quantities and the unknown constants only.edo says. mmmh … something is amiss. the articles FZ writes have different lengths. ifwe model the word occurrences in each article as IID Poisson random variables with rate λ,we are implicitly assuming that the articles have the same length. why? (10 points; extracredit) and if that is true, what is the implied common length? (10 points; extra credit)problem 2. set-up: you collect another random sample of articles penned by FZ andcount how many times he uses the word however in each of the articles in your sample,(x1; :::; xn). you also count the length of each article in your sample, (y1; :::; yn). In thisset-up, xi is the number of times the word however appeared in the i-th article, as before,and yi is the total number of words in the i-th article.question 2.1. (10 points) model: let Xi denote the quantity that captures the numberof times the word however appears in the i-th article. let’s assume that the quantitiesX1; :::Xn are independent and identically distributed (IID) according to a Poisson distribution with unknown parameter ν · 1000 yi , p(Xi = xi j yi; ν; 1000) = Poisson(xi j ν ·1000)fori = 1; :::; n: yiusing the 2-by-2 table of what’s variable/constant versus what’s observed/unknown, declarewhat’s the technical nature (random variable, latent variable, known constant or unknownconstant) of the quantities involved the set-up/model above: X1; ::Xn, x1; :::xn, y1; :::yn, νand n.question 2.2. (10 points) what is the interpretation of 1000 yi in this model? explain.question 2.3. (10 points) what is the interpretation of ν in this model? explain.question 2.4. (10 points) write the data generating process for the model above.question 2.5. (10 points) define the likelihood L(ν) = p(· j ·) for this model and set-up atthe highest level of abstraction.question 2.6. (10 points) write the likelihood L(ν) for a generic sample of n articles,(x1; :::; xn), and n article lengths, (y1; :::; yn).2Stat-3503/Stat-8109 Airoldi/Fall-21question 2.7. (10 points) write the log-likelihood ‘(ν) for a generic sample of n articles,(x1; :::; xn), and n article lengths, (y1; :::; yn).question 2.8. (10 points) Simulate the number of occurrences of the word however for 5articles using the data generating process. Assume ν = 10 and coresponding article lengthsy = (1730; 947; 1830; 1210; 1100). Record the number of occurrences of however in eacharticle.question 2.9. (10 points) write the log-likelihood ‘(ν) for the following the specific sampleof occurrences you generated in the previous question and their corresponding 5 articlelengths (1730; 947; 1830; 1210; 1100).question 2.10. (10 points) Plot the log-likelihood from the previous question in R. Doesthe maximum occur near ν = 10?question 2.11. (10 points) draw a graphical representation of this model, which explicitlyshows the random quantities and the unknown constants only.edo says. that was a more reasonable model. but FZ writes about different topics. ourmodel is not capturing that. is FZ more prone to offering his own opinions when he writesabout politics than when he writes about other topics? let’s investigate.problem 3. set-up: you collect a random sample of articles penned by FZ and count howmany times he uses the certain word I in each of the articles in your sample, (x1; :::; xn).In this set-up, xi is the number of times the word I appeared in the i-th article.question 3.1. (10 points) model: let Xi denote the quantity that captures the number oftimes the word I appears in the i-th article. let Zi indicate whether the i-th article isabout politics, denoted by Zi = 1, or not, denoted by Zi = 0. let’s assume that the quantities X1; :::; Xn are independent of one another conditionally on the corresponding valuesof Z1; :::; Zn. let’s assume that the quantities Z1; :::; Zn are independent and identicallydistributed (IID) according to a Bernoulli distribution with parameter π, p(Zi j π) = Bernoulli(zi j π)fori = 1; :::; n:let’s further assume that the number of occurrences of the word I in an article aboutpolitics follows a Poisson distribution with unknown parameter λP olitics,p(Xi = xi j Zi = 1; λP olitics) = Poisson(xi j λP olitics)fori = 1; :::; n; and that the number of occurrences of the word I in an article about any other topic followsa Binomial distribution with size 1000 and unknown parameter θOther,p(Xi = xi j Zi = 0; 1000; θOther) = Binomial(xi j 1000; θOther) for i = 1; :::; n:3Stat-3503/Stat-8109 Airoldi/Fall-21using the 2-by-2 table of what’s variable/constant versus what’s observed/unknown, declarewhat’s the technical nature (random variable, latent variable, known constant or unknownconstant) of the quantities involved the set-up/model above: X1; ::Xn, x1; :::xn, Z1; ::Zn,z1; :::zn, π; λP olitics, θOther and n.question 3.2. (10 points) write the data generating process for the model above.question 3.3. (10 points) simulate 1000 values of Xi in R from the data generating processassuming pi = 0:3, λP olitics = 30 and θOther = 0:02. Plot the values of XijZi = 1 andXijZi = 0 as two histograms on the same plot. Color the histograms by the value of Zi sothe two populations can be distinguished.question 3.4. (10 points) write the likelihood for 1 article, Li(λP olitics; θOther) = p(Xi =xi j λP olitics; θOther).question 3.5. (10 points) write the likelihood L(λP olitics; θOther) for a generic sample of narticles, (x1; :::; xn).question 3.6. (10 points) write the log-likelihood ‘(λP olitics; θOther) for a generic sample ofn articles, (x1; :::; xn).question 3.7. (10 points) write the log-likelihood ‘(λP olitics; θOther) for the following specificsample of 8 articles (12; 4; 8; 3; 3; 10; 1; 9).question 3.8. (10 points) draw a graphical representation of this model, which explicitlyshows the random quantities and the unknown constants only.edo says. wait, but is it reasonable to assume that the rate λ is an unknown constant inall of our models? it seems like a stretch. (10 points; if you agree)problem 4. set-up: let’s go back to the simplest possible set-up for this exercise. youcollect a random sample of articles penned by FZ and count how many times he uses theword and in each of the articles in your sample, (x1; :::; xn). In this set-up, xi is the numberof times the word and appeared in the i-th article, as before.question 4.1. (10 points) model: let Xi denote the quantity that captures the number oftimes the word and appears in the i-th article. let’s assume that the quantities X1; :::Xnare independent and identically distributed (IID) according to a Poisson distribution withunknown parameter Λ,p(Xi = xi j Λ = λi) = Poisson(xi j λi) for i = 1; :::; n:4Stat-3503/Stat-8109 Airoldi/Fall-21in addition, let’s assume that the rate Λ is distributed according to a Gamma distributionwith unknown parameters α and θ,f(Λ = λi j α; θ) = Gamma(λi j α; θ):using the 2-by-2 table of what’s variable/constant versus what’s observed/unknown, declarewhat’s the technical nature (random variable, latent variable, known constant or unknownconstant) of the quantities involved the set-up/model above: X1; ::Xn, x1; :::xn, Λ, λ1; :::λn,α; θ and n.question 4.2. (10 points) write the data generating process for the model above.question 4.3. (10 points) in R simulate 1000 values from the data generating process.Assume α = 10 and θ = 1. Compute the mean and variance of the Xi.question 4.4. (10 points) in R simulate 1000 values assuming λi = 10 for all i (ignorethe Gamma distribution). Compute the mean and variance of the Xi now. How do theycompare to the mean and variance you calcualted in question 4.3?question 4.5. (10 points) write the likelihood for 1 article, Li(α; θ) = p(Xi = xi j α; θ).question 4.6. (10 points) write the log-likelihood ‘(α; θ) for a generic sample of n articles,(x1; :::; xn).question 4.7. (10 points) write the log-likelihood ‘(α; θ) for the following specific sampleof 8 articles (64; 61; 89; 55; 57; 76; 47; 55).question 4.8. (10 points) draw a graphical representation of this model, which explicitlyshows the random quantities and the unknown constants only.edo says. do you recognize the very special probability mass function you just obtainedfor p(Xi = xi j α; θ) = Li(α; θ)? (10 points; extra credit) excellent! you just proved a usefulresult: Gamma mixture of Poisson is a … .Generate samples from this distribution and verify graphically that you get the distributionlooks the same as that in 4.3 (you must use appropriate parameters you identified above).(10 points; extra credit)5

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