Schedule

• Introduction; review of topics from probability; properties of the sample mean for a simple random sample from a finite populaiton

Wed, Jan 23

• In class, we will work on: Overview and review
  • Discuss syllabus, overview of course
  • Review of prerequisites
    • Topics from probability: pdf (also posted on resources page)
    • Practice examples for probability: pdf
    • Common probability distributions: pdf (also posted on resources page; you don’t need to memorize this material)
    • Topics from calculus (not discussed today, please review after class): pdf (also posted on resources page)
• After class, please do the following:
  • Register for GitHub here if you haven’t already; I will ask you to provide your GitHub user name in the questionairre below.
  • Fill out a brief questionnairre
  • Fill out this brief poll about when my office hours should be held
  • Sign up for our class at Piazza (anonymous question and answer forum): https://piazza.com/mtholyoke/spring2019/stat343
  • Reading
    • Review the prerequisite material from probability and calculus. If there are any topics from probability or calculus that you don’t have memorized, please start memorizing them. This will make the rest of this class go much more smoothly for you. If you have any questions on this material, please don’t hesitate to ask me in office hours or on Piazza. I am happy to help with this material.
    • Over the next few classes we will cover material from Sections 7.1 - 7.3 of the text. Give those sections in the book a brief skim.
  • Homework 0 due Wed, Jan 30

Fri, Jan 25

• In class, we will work on: Sections 7.1, 7.2, 7.3.1, and 7.3.2 (mostly lecture)
  • Last two probability examples from last class
  • Briefly talk through the “common probability distributions” handout
  • Motivating example for Chapter 7: pdf
  • Definitions from Chapter 7 (some on motivating example handout): population parameter, statistic, estimator, estimate, simple random sample, sampling distribution, bias, variance, mean squared error
  • Start on some derivations of the expected value, variance, and MSE of the sample mean based on a simple random sample from a finite population.
  • If time (unlikely), start on estimating the population variance based on a simple random sample from a finite population.
  • Lecture notes for Sections 7.1, 7.2, 7.3.1, and 7.3.2: pdf
  • Set up git on RStudio (if you took Stat 340 with me, you can leave early)
• After class, please do the following:
  • Start memorizing any definitions from today’s class that you don’t know yet.
  • Problem Set 0 due Wed, Jan 30

Section 7.3.3 (see Stat 344SM, survey sampling, for the rest of Chapter 7 and much more); start on Chapter 8

Mon, Jan 28

• In Class, we will work on: Sections 7.1, 7.2, 7.3.1, and 7.3.2 (mostly lecture, some worked examples)
  • Finish stuff from Sections 7.2, 7.3.1, and 7.3.2 that we didn’t get to on Friday.
• After class, please do the following:
  • Problem Set 0 due Wed, Jan 30

Wed, Jan 30

• In Class, we will work on: Section 5.3 and part of 7.3.3 (mostly lab)
  • Central Limit Theorem, more on the sampling distribution of the sample mean
  • Lab 1 on GitHub
  • Topics from R:
    • for loops: pdf
    • dplyr::sample_n
• After class, please do the following:
  • Problem Set 0 due Fri, Feb 1

Fri, Feb 1

• Problem Set 0 due today!!
• In Class, we will work on: Section 7.3.3
  • Mean Squared Error = Bias\(^2\) + Variance. Lecture notes: pdf
  • Finish lab from last time
• After class, please do the following:
  • Problem Set 1 due Fri, Feb 15

Mon, Feb 4

• In Class, we will work on: Section 7.3.3
  • A first look at confidence intervals for a population mean based on a simple random sample. pdf (not on the lecture notes but discussed in class are the coverage probability and nominal coverage probability of a confidence interval)
  • R topics: Logical operations on vectors, mutate and summarize pdf
  • Lab 2: confidence intervals for simple random sampling
• After class, please do the following:
  • Problem Set 1 due Fri, Feb 15

Wed, Feb 6

• In Class, we will work on: Finish Section 7.3.3
  • Finish confidence intervals lab from last class
  • If you finish early, free time or work on homework
• After class, please do the following:
  • Problem Set 1 due Fri, Feb 15

Fri, Feb 8

• In Class, we will work on: Start on Sections 8.1 through 8.5
  • Introductory worksheet, friend or foe: pdf
  • Some plots to go with the introductory worksheet: pdf
    • Rmd file in case you want to see code for making the plots: Rmd
  • Defined the likelihood function, log-likelihood function, maximum likelihood estimator, maximum likelihood estimate
  • Argued that the parameter value that maximizes the log-likelihood also maximizes the likelihood, since log is an increasing function.
  • Reviewed some calculus related to finding a maximum of a univariate function
• After class, please do the following:
  • Problem Set 1 due Fri, Feb 15

Mon, Feb 11

• In Class, we will work on: Sections 8.1 through 8.5
  • Finish first example of maximum likelihood from last class
  • Second example: pdf Solutions: pdf. See also R part and solutions posted as lab 3 on GitHub.

Wed, Feb 13

• In Class, we will work on: Sections 8.1 through 8.5
  • Maximum likelihood for models with more than 1 parameter.
  • Example of normal distribution. Lecture notes: pdf
  • Example of simple linear regression, variance known
    • Handout: pdf
    • Solutions: pdf. Solutions to R part posted on GitHub.

Fri, Feb 15

• In Class, we will work on: Sections 8.1 through 8.5
  • Wrap up examples from last class
    • Talk through the likelihood for the normal distribution model with 2 unknown parameters: html Rmd
    • Highlights from solutions for simple linear regression handout, posted for Wed, Feb 13
  • Review of Taylor’s Theorem. See last 2 pages of handout about calculus from first day, also posted on resources page.
  • Introduce methods for numerical maximization of functions. Slides: pdf
    • Note that on slide 5 I should have had lower case \(\ell\) (for the log-likelihood) rather than upper case \(L\).
  • Start Lab 5, on numeric optimization via Newton’s method; posted on GitHub

Mon, Feb 18

• In Class, we will work on:
  • Finish Lab 5

Wed, Feb 20

• In Class, we will work on:
  • Numeric maximum likelihood in Stan
    • Example pdf
    • Lab 6
  • Prepare for next class: Lab 7a

Fri, Feb 22

• In Class, we will work on:

  • Before class, please complete Lab 7a!! Note that in the calls to qbeta, you will need to change the shape2 parameter to b. Your revised code will look like this:

    qbeta(c(0.25, 0.75), shape1 = a, shape2 = b)
    qbeta(c(0.05, 0.95), shape1 = a, shape2 = b)

  • Bayesian Estimation: Section 8.6
    • Lecture notes: pdf
    • Lab 7b

Mon, Feb 25

• In Class, we will work on: Section 8.6
  • Examples of Bayesian inference with conjugate priors: pdf. Solutions: pdf.

Wed, Feb 27

• In Class, we will work on: Section 8.6
  • Lab 8

Fri, Mar 1

• In Class, we will work on:
  • Set up for MCMC: Limits of conjugate priors for handling normal distributions. Notes: pdf

Mon, Mar 4

• In Class, we will work on:
  • Justification of numerical integration using the law of large numbers. Lecture notes: pdf
  • MCMC for Bayesian inference using Stan.
    • Handout: pdf
    • Lab 9 posted on GitHub
  • MCMC for Bayesian inference.
    • Reading 1: the discussion of the law of large numbers and its application for numerical integration, discussed in Section 5.2 of Rice.
    • Reading 2 (optional?): I’ve posted a chapter from Introduction to Statistical Thought by Michael Lavine. This goes into more depth than we will, but it’s the most readable less-than-book-length introduction to this topic at an undergraduate level that I have found. It is linked to from the resources page: http://www.evanlray.com/stat343_s2019/resources.html

Wed, Mar 6

• In Class, we will work on:
  • Large-sample normal approximation to the posterior distribution.
    • Motivating examples from previous classes: pdf
    • Proof step 1: For large n, the prior’s effect on the posterior is irrelevant in comparison to the likelihood. Motivating (??) pictures: pdf
    • Proof step 2: Second order Taylor series approximation to log-likelihood. Lecture notes: pdf

Fri, Mar 8

• In Class, we will work on:
  • Wrap up of lab 9; solutions are posted on GitHub
  • Examples of large-sample normal approximations to the posterior
    • Binomial examples/motivation: pdf
    • Large sample CI example (see lab 10): pdf

Spring Break: safe travels!

Mon, Mar 18

• In Class, we will work on: ideas from Section 8.5.3, borrowed the idea of the likelihood ratio from Section 9.1.
  • Exact confidence intervals, based on exact form of sampling distribution of a statistic. Lecture notes: pdf
  • Develop some intuition for Fisher information: pdf

Wed, Mar 20

• In Class, we will work on:
  • Finish intuition handout from last class
  • Lots of lecture about Fisher Information and Observed Fisher Information, large-sample normal approximations to the sampling distribution of the MLE in terms of information; confidence intervals from the large sample normal approximation; Poisson example. Lecture notes: pdf
  • Handout to go with Poisson example: pdf

Fri, Mar 22

• In Class, we will work on:
  • Midterm on PS1 through PS5

Mon, Mar 25

• In Class, we will work on:
  • Practice with large sample approximations to sampling distribution of the MLE: pdf
  • Start Lab 12

Wed, Mar 27

• In Class, we will work on:
  • Some large-sample results about maximum likelihood estimators. Consistency, Cramer-Rao Lower Bound, efficiency, shrinkage estimators. Lecture notes: pdf
  • Finish Lab 12, Start Lab 11

Fri, Mar 29

• In Class, we will work on:
  • Shrinkage estimators. No lecture notes, but we talked about how the posterior mean for the mean of a normal distribution and the parameter theta for a binomial distribution can both be viewed as an appropriately weighted average of the prior mean and the data mean.
  • Finish old labs
  • Start an example about practice with MSE, which will be due for homework: pdf
  • Start Lab 13, which will be due for homework.

Mon, Apr 1

• In Class, we will work on:
  • Bootstrap estimation of a sampling distribution. Handout: pdf
  • Lab 14

Tue, Apr 3

• In Class, we will work on:
  • Bootstrap confidence intervals. Handout (NOTE: the version I distributed in class wasn’t quite right on the last page. Corrected version here.): pdf
  • Lab 15

Fri, Apr 5

• Class cancelled

Mon, Apr 8

• In Class, we will work on:
  • Start on hypothesis testing; Chapter 9
  • Handout: pdf
  • Lecture Notes: pdf

Wed, Apr 10

• In Class, we will work on:
  • Warm up about likelihood ratio tests: pdf
  • Revisit example of tests about mean of a normal distribution with known variance, simple hypotheses from last class
    • Wrap up discussion from last class
    • A Bayesian approach
  • Some things to avoid in hypothesis testing
    • ASA Statement on p-values: pdf
    • Moving Beyond p < 0.05: pdf. See especially
      • Page 2
      • First point from Hubbard et al., page 11
      • Everything else…
  • Bayesian approach to analysis of normal distributions example: pdf
  • Jump into some things to avoid, just so we know about them:
    • Type I and Type II errors, Power

Fri, Apr 12

• Class cancelled