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Birth-Death-Sampling (BDS) Models

Overview

The bds_model.py module implements phylogenetic models based on birth-death-sampling (BDS) processes. These models describe how lineages speciate (birth), go extinct (death), and are sampled over evolutionary time. They are particularly suited for modeling time-calibrated trees under epidemiological or macroevolutionary scenarios.

The module includes: - Time-homogeneous and state-dependent BDS likelihoods - Analytical expressions for survival probabilities (q(t)) and extinction probabilities (p₀(t)) - Simulation utilities for stochastic tree generation - Extensions with linear state-dependent birth rates

Core Quantities

Let \(b\), \(d\), and \(s\) be the birth, death, and sampling rates. The Stadler q-function:

\[ q(t) = 2(1 - c^2) + e^{-ct}(1 - c)^2 + e^{ct}(1 + c)^2 \]

with:

\[ c = \frac{-(b - d - s)}{\sqrt{(b - d - s)^2 + 4bs}} \]

governs the contribution of each branch to the tree likelihood【123†source】.

The extinction probability \(p_0(t)\) is:

\[ p_0(t) = \frac{b + d + s + c \cdot \frac{e^{-ct}(1 - c) - (1 + c)}{e^{-ct}(1 - c) + (1 + c)}}{2b} \]

Tree Likelihood: Homogeneous Model

For a tree with \(E\) edges and \(T\) tips, the log-likelihood is:

\[ \log \mathcal{L} = \sum_{e \in \text{edges}} \log \frac{q(t_e^\text{child})}{q(t_e^\text{parent})} + (T-1)\log b + T \log s \]

Implemented by:

BirthDeathSamplingTreeWise.log_likelihood(treecal)

State-Dependent BDS

Allows birth, death, and sampling to vary by hidden discrete state. Given per-edge parent state probabilities \(p_k\), the likelihood term is:

\[ \sum_{e \in \text{edges}} \log \left( \sum_k p_k \cdot \frac{q_k(t_\text{child})}{q_k(t_\text{parent})} \cdot r_k \right) \]

where \(r_k\) is \(b_k\) or \(s_k\) depending on whether the child is internal or a tip.

Implemented by:

StateDependentBirthDeathSampling.log_likelihood(treecal, ancestor_states)

Simulation: BirthDeathSamplingSimulator

Stochastically grows a tree forward in time using specified BDS rates. Events: - Birth: splits lineage into two children - Death: terminates lineage - Sampling: marks lineage as a tip

Simulation continues until a target number of tips is sampled【123†source】.

Extensions: Linear Marker Birth Model

The LinearMarkerBirthModel allows the birth rate to depend linearly on hidden marker states:

\[ b_n = \exp\left(\beta_0 + \sum_{m,k>0} \beta_{mk} \cdot p_{n,m,k}\right) \]

This enables modeling trait-dependent diversification. The log-likelihood aggregates over edges:

\[ \sum_e \log \left( \frac{q_{\text{child}}}{q_{\text{parent}}} \right) + \sum_{\text{birth events}} \log b + \sum_{\text{tips}} \log s \]

API Summary

Likelihood Classes

  • BirthDeathSamplingTreeWise: homogeneous log-likelihood
  • StateDependentBirthDeathSampling: per-state likelihood with posterior weighting
  • LinearMarkerBirthModel: linear trait-dependent birth rate

Simulation

  • BirthDeathSamplingSimulator: forward-in-time stochastic simulation
  • BirthDeathSamplingNodeData: holds per-node rates

Utilities

  • stadler_q(t, b, d, s): scalar q function
  • stadler_q_matrix: vectorized q(t) over nodes and states
  • stadler_p0: extinction probability

References

  • Stadler, T. (2010). Sampling-through-time in birth–death trees. JTB.
  • Maddison, W. P., et al. (2007). Estimating a binary character’s effect on speciation.
  • Rabosky, D. L. (2014). Automatic detection of key innovations.

Source

Defined in bds_model.py【123†source】.