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Learning bayesian data analysis with Statistical Rethinking

28 Lecture 19 - 2019

28.1 Gaussian process regression: continuous categories

Traditional clusters are discrete, unordered. Every category is equally different from all others.

But take, for example, income. There is no reason that every increase or decrease in 1 dollar should be equivalently weighted. Other examples include age, phylogenetic distance, social network distance, etc. There are no obvious cut points, but we know that similar values are more similar.

Gaussian process regression considers infinite numbers of categories. (Need pooling)

28.1.1 Example: spatial autocorrelation of oceanic tool complexity and population size

  1. Construct a distance matrix of distances between the islands. We could improve this by considering sailing distance for example.
  2. Recall the model: tool complexity is distributed as poisson. Innovation rate, population size, rate of diminishing returns and loss rate.
  3. To add a Gaussian process, add a factor k in front for each population
    • k=0, exactly as expected by model
    • k=-0.5, 60% (exp(-0.5) = 0.6)
    • k=-0.5, 130% (exp(0.25) = 1.3)
  4. Gaussian process prior: multivariate Gaussian 10x10 covariance (distance in this case)
  5. Modelling covariance: covariance between islands i and j is the max covariance multiplied by the rate of decline with distance and the squared distance, added to the variance within self (also called a jitter function so multiple observations of the same island have a different covariance). This is the Gaussian, the bell curve.
  6. Linear (Dij) vs squared (Dij^2). Linear assumes that the rate of decline if fastest at the start. Usually not true so we assume a squared term.,

Result is lower covariance than predicted but much slower decay.

28.2 Phylogenetic relationship

Phylogenetic relationships are sort of a proxy for unobserved confounds.

Some ways of incorporating phylogenetic information into a GLM:

  • Brownian motion model (PGLS)
  • Ornstein-Uhlenbeck (OU) processes
  • …

All use a covariance matrix to represent phylogeny and are in principle a Gaussian process regression

28.2.1 Example: primates

Body size -> Group size, Body size -> Brain size, Brain size -> Group size

G ~ MVNormal(mu, sigma)

mu = alpha + beta body + beta body S = sigma ^ 2 * I

I being an identity matrix, therefore this simplest version (with no covariance), is just the diagonals with sigma squared

Brownian motion is the oldest and most conservative assumption. The decline in covariance is linear since the date the species diverged. Alternatively, use a non-linear relationship between covariance and distance with a Gaussian process to consider infinite different functions.