Chapter 14 Probabilistic Model Formulation (with Time Warping and Genetic Stickiness)

Conceptual Description

  1. Gaussian Process (Topic Evolution):
    • The Gaussian process models how topic proportions naturally change over time, assuming a shared timeline for all individuals.
  2. Individual Time Warping:
    • The warping function W(t, ρ_d) stretches or compresses the timeline for each individual based on their genetic and/or environmental factors (modeled through the warping parameter ρ_d).
  3. Topic Distribution Update with Stickiness:
    • At each time point, an individual’s exposure to topic distributions depends on their warped time index.
    • New diagnoses update the topic distribution (θ{d,t}_) based on the relevant disease probabilities at that warped time and the genetic stickiness factor (γ_d).

Illustrative Example

  • Consider a topic representing cardiovascular diseases. Individuals with high warping factors will reach disease probabilities associated with this topic earlier compared to those with low warping factors.
  • An individual with high genetic stickiness will be less likely to shift away from this topic distribution even in the presence of new diagnoses unrelated to cardiovascular disease.

Notes

  • Expressing the interaction of warping and disease likelihoods in a closed-form mathematical expression within markdown can be difficult.
  • Computational implementation might involve procedural steps to determine the appropriate topic distribution for an individual at a given time, based on their warped timeline.
  • Consider incorporating a pseudocode-like representation to illustrate the procedural nature of the time warping and its influence.
  • Specify your choices for the mean function \(\mu(t)\) and any priors on Gaussian Process parameters.
  • Clarify how the time warping function W(t, ρ_d) influences disease probabilities π{v,k,t}_
  • Consider alternative inference techniques (e.g., variational inference) due to potential computational complexity.