8 Day 8 (February 12)

8.1 Announcements

  • Please upload activity 2 to Canvas by end of the day on Sunday.
  • Donuts today
    • About 20-30 min at end of class to meet your peers!
  • Read and re-read pgs. 10-14 of Spatio-temporal statistics with R.
  • Questions/clarifications from journals
    • “Wouldn’t having a clean sampling design framework in place prior to collecting/analyzing data alleviate a lot of the model application guess work and confusion we’ve been discussing in class? During my MS, I was taught that developing a clean conceptual framework and sampling design a priori is the appropriate way to accommodate/account for process noise and variance.”
    • “This was touched on a bit in class today, but I feel that a big criticism of Bayesian and hierarchical models are that you can do really anything you want to them to get a desired result.”
    • “Working through hierarchical modelling (pages 25–30) changed how I think about “data” in a very basic way. I used to treat a map or an index value as if it were the reality. Now I see it more like a noisy clue about something deeper.”
    • One thing I am still working through is how distributions are selected at each level of the hierarchical modeling framework and how they align with the structure of the data and the modeling assumptions.

8.2 Hierarchical models

  • During this course we will implement many models using the hierarchical framework
    • Hierarchical models are pretty common (e.g., mixed models, kriging, most Bayesian models)
    • Today is a crash course on hierarchical and Bayesian statistical models
    • Study technical note 1.1 on pg. 13 of Spatio-temporal statistics with R
  • The Bayesian hierarchical modeling framework

\[\text{Data model:} \;\;[\mathbf{z}|\mathbf{y},\boldsymbol{\theta}_{D}]\] \[\text{Process model:} \;\;[\mathbf{y}|\boldsymbol{\theta}_{P}]\] \[\text{Parameter model:} \;\;[\boldsymbol{\theta}]\]

  • Given a Bayesian hierarchical model we want the following:
    • The posterior distribution of the parameters \([\boldsymbol{\theta}|\mathbf{z}]\)
    • The posterior predictive distribution \([\mathbf{z}_{\text{pred}}|\mathbf{z}]\)
  • Using Bayes’ theorem… \[[\boldsymbol{\theta}|\mathbf{z}]=\int\frac{[\mathbf{z}|\mathbf{y},\boldsymbol{\theta}][\mathbf{y}|\boldsymbol{\theta}][\boldsymbol{\theta}]}{\int\int\mathbf{[z}|\mathbf{y},\boldsymbol{\theta}][\mathbf{y}|\boldsymbol{\theta}][\boldsymbol{\theta}]d\mathbf{y}d\mathbf{\boldsymbol{\theta}}}d\mathbf{y}\] \[[\mathbf{z}_{\text{pred}}|\mathbf{z}]=\int\int\mathbf{[z}_{\text{pred}}|\mathbf{y},\boldsymbol{\theta}][\mathbf{y}|\boldsymbol{\theta}][\boldsymbol{\theta}|\mathbf{z}]d\mathbf{y}d\mathbf{\boldsymbol{\theta}}\]
  • White board whooping crane example of BHM.