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Review :: statistics!!

Peter Ralph

12 March 2018 – Advanced Biological Statistics

Overview

  • Big-picture methods
  • (Bayesian) statistics :: concepts
  • Estimation :: using Stan
  • Types of model
  • Probability distributions
  • Model evaluation tools

Big-picture methods

  1. Hierarchical models (random effects)

  2. Shrinkage (towards a group mean) // sharing power

  3. Sparsifying priors (the horseshoe)

  4. Robustness to outliers, and the importance of properly modeling noise

(Bayesian) statistics :: concepts

  1. Likelihood

  2. Posterior = prior \(\times\) likelihood

  3. Strong versus weak/uninformative priors

  4. Credible intervals

Estimation :: using Stan

  1. Markov chain Monte Carlo (random walk)

  2. Convergence, and mixing

  3. Multiple optima

  4. Reparameterization to improve the posterior

  5. Optimization (hill-climbing)

Types of model

  1. Binary data: coin flips, where the probability depends on something

    • beta-binomial
    • logistic regression

  2. Count data: where the mean depends on something

    • Poisson regression

  3. Metric data in groups: group means relate to each other

    • Robust ANOVA
    • shrinkage of group means

  4. Metric data with metric predictors: regression and relatives

    • Robust regression
    • many predictors: sparsifying prior on coefficients

More types of model

  1. Mixture models: deconvolution

    • nonnegative matrix factorization

  2. Dimension reduction: visualization

    • t-SNE

  3. Time series

    • autoregressive
    • latent (unobserved) process

  4. Spatial statistics

    • spatial autocovariance

Probability distributions

  1. Beta, and Dirichlet (renormalized Exponentials)

  2. Beta-binomial

  3. Gamma (and Exponential)

  4. Poisson: many rare events

  5. Cauchy: has outliers

  6. Student’s \(t\)

  7. Scale mixtures of Normal (\(\sigma\) is random)

  8. Multivariate Normal (/Gaussian)

Model evaluation tools

  1. Model debugging by simulation

  2. Model prediction by simulation (posterior predictive sampling)

  3. Goodness-of-fit:

    • by simulation
    • by crossvalidation

  4. Model selection by crossvalidation

thanks!!