PML-Homework 2 Solved

Exercise 1
Use the following lemma to generate 500 samples from a 𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙(30, 0.5) distribution in pyro. Plot the resulting distribution using sns.distplot function.

Lemma Let 𝑋1, … , 𝑋𝑛 be independent 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖(𝑝) random variables, then 𝑋 = 𝑋1 + … + 𝑋𝑛 is a random variable with

𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙(𝑛, 𝑝) distribution.

Exercise 2
1.   (theory) Consider the Gamma distribution1 Gamma(𝛼, 𝛽𝛽) with p.d.f.𝛼 𝑥𝛼−1𝑒−𝛽𝑥

Γ(𝛼)

and the Poisson distribution Poisson(𝜆) with p.m.f.

𝜆𝑘𝑒−𝜆

                                                                                                                                   𝑘!    .

Given the generative model

𝜆 ∼ Gamma(𝛼, 𝛽) 𝑥 ∼ Poisson(𝜆)

prove that the Gamma distribution is a conjugate prior2 for the Poisson likelihood.

2.  (code) Set the parameters of the Gamma distribution to 𝛼 = 1 and 𝛽 = 2. Write the pyro code to sample 𝜆 and 𝑥, extract 300 samples from the distributions of both random variables and plot their histograms.

3.  (code) Suppose you observe some data 𝑥 = [3, 10, 2, 5, 6, 7]. Plot 300 samples from the posterior distribution of 𝜆 as the number of observed data points from 𝑥 increases:

𝑥 = [3, 10] 𝑥 = [3, 10, 2, 5] 𝑥 = [3, 10, 2, 5, 6, 7].

1Section "Probability distributions" in notebook 01.

2Section "Conjugate priors" in notebook 02.