12 Section 12. Extended topics

2021-12-02

12.2 Notes

12.2.1 Lecture 12.1 Frequency evaluation, hypothesis testing and variable selection

  • Bayesian vs. Frequentist
    • Bayesian theory has epistemic and aleatory probabilities
    • Frequency evaluations focus on frequency properties given aleatoric repetition of an observation and modeling
  • on “null hypothesis testing”:
    • often inappropriate to test the probability that a value is 0
      • for continuous data, the probability of a single value is always 0
      • “region of practical equivalence” (ROPE) is another option
    • best to focus on describing the full posterior
      • e.g. amount of the posterior greater than or less than an important value
      • e.g. where most of the posterior density is (89% or 95% HDI)
  • be careful about only looking at marginal posteriors, too
    • joint posterior distributions may be informative
    • e.g. height and weight variables in beta-blocker model are highly correlated; both marginals overlap 0, but joint does not
  • most common statistical tests are linear models
classical test Bayesian equivalent in ‘rstanarm’
t-test mean of data stan_glm(y ~ 1)
paired t-test mean of diffs stan_glm((y1 - y2) ~ 1)
Pearson correlation linear model stan_glm(y ~ 1 + x)
two-sample t-test group means stan_glm(y ~ 1 + gid)
ANOVA hierarchical model stan_glm(y ~ 1 + (1|gid))

12.2.2 Lecture 12.2 Overview of modeling data collection, BDA3 Ch 8, linear models, BDA Ch 14-18, lasso, horseshoe and Gaussian processes, BDA3 Ch 21

  • LASSO and Bayesian LASSO
    • Bayesian LASSO uses Laplace distribution as a prior
    • is equivalent to L1 penalty in MLE LASSO, but because we still integrate over the entire posterior, it does not have the same “sparsifying” effect
    • therefore, Bayesian LASSO is empirically worse than MLE LASSO
    • final thought: best to separate the process of prior selection, posterior inference, and decision analysis
    • regularized horseshoe prior a better choice if you have prior information that only some of the covariates are informative

projpred selection vs LASSO