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The Workshop

  • The Watchmaker’s Philosophy
    • Why Build It Yourself?
    • The Watchmaker’s Way
    • The Gears of Understanding
    • On Mathematical Rigor
    • On Teaching Probability and Calculus
    • On Python Implementations
    • Your Journey

The Workbench (Prerequisites)

  • The Workbench (Prerequisites)
    • Likelihood-Based Probabilistic Inference
      • Why Likelihood?
      • The Likelihood Function
      • The Toolkit: Key Distributions
        • The Exponential Distribution: Coalescence Waiting Times
        • The Poisson Distribution: Mutations and the SFS
        • The Gamma Distribution: Ages and Rates
        • The Gaussian Distribution: Smoothness Priors
      • Maximum Likelihood Estimation (MLE)
        • Worked Example: Inferring Population Size from the SFS
        • Fisher Information and Confidence Intervals
      • Bayesian Inference
        • Conjugate Priors: When Bayesian Inference Has Closed-Form Solutions
      • Composite and Approximate Likelihoods
        • Worked Example: Composite Likelihood from Two Data Sources
      • The Other Paradigm: Neural Networks and Amortized Inference
        • The key idea
        • What amortized inference does well
        • What likelihood-based inference does well
        • Why this book focuses on the likelihood approach
      • Summary
    • Coalescent Theory
      • The Big Idea
      • The Wright-Fisher Model (Forward in Time)
      • Going Backwards: The Coalescent
        • The probability that two specific lineages coalesce in a given generation
        • Waiting time to coalescence
      • The Coalescent with \(n\) Samples
      • Expected Number of Lineages at Time \(t\)
      • Mutations on the Coalescent Tree
      • Summary
    • Ancestral Recombination Graphs
      • Why Trees Aren’t Enough
      • What Is Recombination?
        • What We’ve Established So Far
      • Recombination in the Coalescent
      • The Structure of an ARG: A Directed Acyclic Graph
      • Marginal Trees
      • The Tree Sequence Representation
      • Branch Lengths and the ARG
      • Why ARG Inference Is Hard
      • Summary
    • Hidden Markov Models
      • Why HMMs for ARG Inference?
      • A Warm-Up Example: Weather and Umbrellas
      • The Core Idea
      • Formal Definition
      • The Forward Algorithm
      • Scaling for Numerical Stability
      • Stochastic Traceback (Sampling)
      • The Li-Stephens Trick: Linear-Time Transitions
        • The Li-Stephens Transition Structure
        • The \(O(K)\) Forward Step
      • Summary
    • The Sequentially Markov Coalescent
      • The Problem with the Full Coalescent
      • What Does “Markov” Mean, and Why Does It Matter?
        • Intuitive Explanation
        • Formal Definition
        • Why Markov Matters for Computation
      • What Makes CwR Non-Markov?
        • The Mechanism
        • What Are Ghost Lineages? A Concrete Example
      • The SMC Approximation
        • Why Does This Restore the Markov Property?
        • How Good Is the Approximation?
      • The SMC Transition Probability
        • Deriving \(r_i\): The Recombination Probability
        • Deriving \(q_j\): The Re-joining Weights
      • PSMC: The Pairwise Case
      • The Cumulative Distribution Function
      • Why SMC Enables HMM Inference
      • Summary
    • The Diffusion Approximation
      • The Big Idea
      • From Wright-Fisher to Continuous Frequency
        • Mean and variance of \(\Delta x\)
        • The diffusion timescale
        • Code: WF trajectories converging to SDE paths
      • Stochastic Differential Equations
        • Euler-Maruyama simulation
      • From SDEs to PDEs: The Fokker-Planck Equation
        • The two terms: diffusion and advection
      • Boundary Conditions
        • Absorbing boundaries
        • Why \(x(1-x)\) vanishes at boundaries
        • The flux condition
        • Reflecting boundaries and mutation
      • Stationary Distributions
        • The neutral case
        • With mutation: the Beta distribution
        • With selection: exponential tilting
      • Numerical Solutions: Finite Differences for PDEs
        • Discretizing \(x\) on a grid
        • Finite-difference approximations
        • The method of lines
        • Crank-Nicolson time stepping
        • The curse of dimensionality
        • Code: 1D diffusion solver
      • Connection to the Site Frequency Spectrum
        • The binomial bridge
        • How dadi and moments differ
      • Summary
    • Ordinary Differential Equations
      • The Big Idea
      • What Is an ODE?
      • Euler’s Method
      • The Runge-Kutta Family
        • RK2: The Midpoint Method
        • RK4: The Classic Method
        • RK45: Adaptive Step Size (Dormand-Prince)
      • Systems of Coupled ODEs
      • Stiffness and Implicit Methods
      • The Matrix Exponential
      • Summary
    • Markov Chain Monte Carlo
      • The Big Idea: Why Sample?
      • Bayesian Inference in 60 Seconds
      • Markov Chains
        • Stationary Distribution
      • The Metropolis-Hastings Algorithm
      • Gibbs Sampling
      • Convergence Diagnostics
      • Practical Considerations
        • Proposal Tuning
        • Data-Informed Proposals
        • Parallel Tempering
        • When MCMC Is Not Enough
      • MCMC in Population Genetics: Three Applications
        • ARGweaver: Gibbs Sampling over ARGs
        • SINGER: MH with Data-Informed Proposals
        • PHLASH: Beyond MCMC
      • Summary

Timepieces

  • Timepieces
    • Verification Status
      • Timepiece I: PSMC
        • The Mechanism at a Glance
        • Why Just Two Sequences?
        • Chapters
      • Timepiece II: SMC++
        • The Mechanism at a Glance
        • Chapters
      • Timepiece III: The Li & Stephens HMM
        • The Mechanism at a Glance
        • Chapters
      • Timepiece IV: msprime
        • The Mechanism at a Glance
        • Chapters
      • Timepiece V: ARGweaver
        • The Mechanism at a Glance
        • Chapters
      • Timepiece VI: tsinfer
        • The Mechanism at a Glance
        • Chapters
      • Timepiece VII: SINGER
        • The Mechanism at a Glance
        • Chapters
      • Timepiece VII: Threads
        • The Mechanism at a Glance
        • Chapters
      • Timepiece IX: tsdate
        • The Mechanism at a Glance
        • Where tsinfer Ends and tsdate Begins
        • Chapters
      • Timepiece X: moments
        • The Mechanism at a Glance
        • Chapters
      • Timepiece XI: dadi
        • The Mechanism at a Glance
        • dadi vs. moments
        • Chapters
      • Timepiece XII: momi2
        • The Mechanism at a Glance
        • Chapters
      • Timepiece XIII: Gamma-SMC
        • The Mechanism at a Glance
        • PSMC vs. Gamma-SMC
        • Chapters
      • Timepiece XIV: PHLASH
        • The Mechanism at a Glance
        • Chapters
      • Timepiece XV: CLUES
        • The Mechanism at a Glance
        • Why Detect Selection?
        • Chapters
      • Timepiece XVI: SLiM
        • The Mechanism at a Glance
        • Chapters
      • Timepiece XVII: Relate
        • The Mechanism at a Glance
        • Where tsinfer and SINGER End and Relate Begins
        • Chapters
      • Timepiece XVIII: discoal
        • The Mechanism at a Glance
        • Chapters
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