Demographic history is the history of evolution and development of populations. It includes such parameters as size of population, time of splits and migration rates.

Understand population history

Conservation biology studies

$\partial a \partial i$ uses local search optimization algorithms.

GADMA — Genetic Algorithm for Demographic Model Analysis

• Several simulation engines ($\partial a \partial i$, moments)
• Has common interface
• Effective optimization based on the genetic algorithm
• Handles up to 3 populations

Goal: minimize unknown function $f$ in as few evaluations as possible.

• Black-box optimization
• Expensive evaluations

On each iteration:

• Approximate objective function $f$ with a surrogate model $M$
• Choose next point as $argmax$ of the acquisition function $\alpha_M$ $$ \htmlClass{fragment}{ x_{n+1} = \argmax_{x\in\c{X}} \alpha_M(x) } $$

Definition. A Gaussian process is random function $f : X \to \R$ such that for any $x_1,..,x_n$, the vector $f(x_1),..,f(x_n)$ is multivariate Gaussian.

Every GP is characterized by a mean $\mu(\.)$ and a kernel $k(\.,\.)$. We have $$ \htmlClass{fragment}{ f(\v{x}) \~ \f{N}(\v{\mu}_{\v{x}},\m{K}_{\v{x}\v{x}}) } $$ where $\v\mu_{\v{x}} = \mu(\v{x})$ and $\m{K}_{\v{x}\v{x}'} = k(\v{x},\v{x}')$.

$$ \htmlData{class=fragment fade-out,fragment-index=9}{ \footnotesize \mathclap{ k_\nu(x,x') = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)} \del{\sqrt{2\nu} \frac{\norm{x-x'}}{\kappa}}^\nu K_\nu \del{\sqrt{2\nu} \frac{\norm{x-x'}}{\kappa}} } } \htmlData{class=fragment d-print-none,fragment-index=9}{ \footnotesize \mathclap{ k_\infty(x,x') = \sigma^2 \exp\del{-\frac{\norm{x-x'}^2}{2\kappa^2}} } } $$ $\sigma^2$: variance $\kappa$: length scale $\nu$: smoothness
$\nu\to\infty$: recovers squared exponential kernel

$\nu = 1/2$

Exponential

$\nu = 3/2$

Matern32

$\nu = 5/2$

Matern52

$\nu = \infty$

RBF

The predictive log probability when leaving out training case $(x_i, y_i)$ is: $$ \htmlClass{fragment}{ \footnotesize \mathclap{ \log p(y_i | X, y_{-i}, \theta) = -\frac{1}{2} \log \sigma_i^2 - \frac{(y_i - \mu_i)^2}{2\sigma^2} - \frac{1}{2}\log 2\pi, } } $$ where $\theta$ - parameters of Gaussian Process, $y_{-i} = Y \setminus \{y_i\}$, $\mu_i = \mu(x_i)$ and $\sigma_i = \sigma(x_i)$

Leave-One-Out log predictive probability: $$ \htmlClass{fragment}{ L_{LOO} = \sum_{i=1}^n \log p(y_i | X, y_{-i}, \theta) } $$

• Performance depends on kernel and acquisition function.
• Leave-one-out cross-validation:
  • poor for kernel selection based on large random dataset,
  • good for automatic kernel selection at the beginning.
• Bayesian optimization with automatic kernel choice showed faster convergence than the genetic algorithm for 4 and 5 populations.

• Combine Bayesian optimization and genetic algorithm.
• Stop criteria for Bayesian optimization.
• More experiments on real data.