Day 2. Introductory concepts for Bayesian analysis
The Bayesics
Duke University

env/bio 665 Bayesian inference for environmental models

The emergence of large data sets in environmental science has changed statistical analysis: more focus on data wrangling and algorithmic approaches. Bayesian analysis departs from alternative methods in its application of probability to all aspects of model fitting with side benefits of simplifying interpretation. The elements of a Bayesian analysis include distributions for prior, likelihood, and posterior. Hierarchical models emerge naturally in the Bayesian framework as a means for analyzing high-dimensional problems without requiring a change in approach. Graphs help to organize hierarchical modeling. Basic concepts are introduced using regression.

Logistics

From last time

Discussion reading: select two of these papers and come prepared to discuss them. Post questions for discussion on Canvas Discussions.

• Reproducibility (in The Atlantic and NY Times) is intimately linked to the notion of uncertainty and, thus, probability. Statistics are used both to communicate uncertainty and for forensic analyses of questionable results.

• Redefining statistical significance, motivated in part by the reproducibility crisis in psychology, classical and Bayesian statisticians compromise; if \(P\) values are to used, then \(P = 0.05\) is way too high, Nature.

• Why Big Data Could Be a Big Fail, Jordan on potential and limitations of Big Data (misleading title).

• Why environmental scientists are becoming Bayesians, why the proliferation of Bayes in environmental science, Ecol Letters.

Today’s plan

1. Breakout: Discussion of papers, approx 20 min

2. Foundation materials in Unit 1 and R

   • Regression example
   • Least squares versus maximum likelihood versus Bayes
   • Model graphs
   • Vectors and matrices (indexing, algebra, identity, inverse, vectorization)

For next time

Try problems in Intro to R, post to Canvas

Recall objectives:

1. Recognize and generate notation for a simple model
2. Identify the basic elements of a Bayesian model and its probability interpretation
3. Identify the deterministic vs stochastic variables in a model
4. Interpret a hierarchical model
   • construct a simple graphical model
   • assemble parameters, process, and data as a hierarchy
   • describe a regression model with notation and graph
5. Articulate advantages and disadvantages of observational and experimental evidence
6. Define Simpson’s Paradox and identify when it could be operating
7. Identify key advantages and limitations of artificial intelligence

From the readings

A few questions you might answer in group discussions of readings:

Redefining statistical significance

1. In a few words, define the following: P value, multiple testing, P-hacking, Bayes Factor.

2. Why is 0.005 suggested as an alternative to 0.05?

Why environmental scientists are becoming Bayesians

3. Using a model graph, explain how a hierarchical Bayes differs from a classical (frequentist) model and simple Bayes? In the graph, say what is random (stochastic) versus fixed and why that is important.

4. Indicate features in each category as present or not (or not sure).

approach	hyperprior	prior	likelihood	posterior
Least squares
Regression (e.g. lm() )
Maximum likelihood
Simple Bayes
Hierarchical Bayes