7.4 Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) is a powerful sampling method that improves on traditional MCMC techniques. It uses concepts from physics to explore parameter space more efficiently, especially in high-dimensional and complex models.

HMC leverages gradient information and Hamiltonian dynamics to propose better updates, reducing random walk behavior. This leads to faster convergence, better scaling to high dimensions, and improved exploration of correlated parameters in Bayesian inference.

Foundations of HMC

• Hamiltonian Monte Carlo (HMC) revolutionizes Bayesian inference by leveraging physical system dynamics to improve sampling efficiency
• HMC addresses limitations of traditional Markov Chain Monte Carlo (MCMC) methods, particularly in high-dimensional and complex posterior distributions
• Integrates concepts from statistical physics and differential geometry to explore parameter space more effectively

Motivation for HMC

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• Overcomes random walk behavior of Metropolis-Hastings algorithm, enabling faster convergence to target distribution
• Utilizes gradient information to propose more informed updates, reducing autocorrelation between samples
• Scales better to high-dimensional problems, crucial for complex Bayesian models (hierarchical models)
• Improves exploration of the posterior distribution, especially in cases with strong parameter correlations

HMC vs traditional MCMC

• HMC incorporates gradient information, while traditional MCMC relies solely on likelihood evaluations
• Produces longer-range proposals, allowing for more efficient exploration of the parameter space
• Exhibits reduced random walk behavior, leading to faster mixing and convergence
• Requires tuning of fewer parameters compared to adaptive MCMC methods (Metropolis-adjusted Langevin algorithm)
• Achieves higher effective sample sizes per computation time in many scenarios

Hamiltonian dynamics basics

• Borrows concepts from classical mechanics, specifically the Hamiltonian formulation of particle motion
• Describes system evolution using position (q) and momentum (p) variables
• Hamiltonian (H) represents total energy, sum of potential energy U(q) and kinetic energy K(p)
• Hamilton's equations govern the time evolution of the system:
  • $\frac{dq}{dt} = \frac{\partial H}{\partial p}$
  • $\frac{dp}{dt} = -\frac{\partial H}{\partial q}$
• Conserves total energy and volume in phase space, key properties for maintaining detailed balance in MCMC

HMC algorithm components

• HMC integrates Hamiltonian dynamics into the MCMC framework to generate proposal states
• Combines deterministic updates based on Hamiltonian dynamics with stochastic elements for exploration
• Requires careful implementation of numerical integration and parameter tuning for optimal performance

Leapfrog integrator

• Symplectic integrator used to approximate Hamiltonian dynamics in discrete time steps
• Preserves important properties of Hamiltonian systems (energy conservation, time-reversibility)
• Consists of three steps: half-step momentum update, full-step position update, half-step momentum update
• Leapfrog update equations:
  • $p_{t+\epsilon/2} = p_t - \frac{\epsilon}{2}\frac{\partial U}{\partial q}(q_t)$
  • $q_{t+\epsilon} = q_t + \epsilon M^{-1}p_{t+\epsilon/2}$
  • $p_{t+\epsilon} = p_{t+\epsilon/2} - \frac{\epsilon}{2}\frac{\partial U}{\partial q}(q_{t+\epsilon})$
• Epsilon (ε) represents the step size, crucial for algorithm stability and efficiency

Momentum variables

• Auxiliary variables introduced to enable Hamiltonian dynamics in the extended state space
• Typically drawn from a multivariate Gaussian distribution with zero mean and covariance M (mass matrix)
• Facilitate exploration of the parameter space by providing "velocity" to the Markov chain
• Momentum resampling at each iteration injects stochasticity into the otherwise deterministic Hamiltonian dynamics
• Choice of mass matrix M affects the algorithm's performance and can be tuned for optimal sampling

Potential energy function

• Derived from the negative log-posterior of the target distribution
• Represents the "landscape" that the HMC sampler navigates in parameter space
• Gradient of potential energy provides the force guiding the Hamiltonian dynamics
• Computed as U(q) = -log(π(q)), where π(q) is the target posterior distribution
• Requires efficient computation of log-posterior and its gradient for HMC to be practical

Kinetic energy function

• Represents the energy associated with the momentum variables
• Typically defined as K(p) = (1/2)p^T M^(-1) p, where M is the mass matrix
• Quadratic form ensures momentum variables follow a Gaussian distribution
• Balances the potential energy, allowing the sampler to explore regions of lower probability
• Choice of kinetic energy function impacts the algorithm's behavior and can be modified for specific problems

Implementation of HMC

• Successful implementation of HMC requires careful consideration of several key parameters and components
• Balancing exploration and computational efficiency is crucial for optimal performance
• Adaptive techniques can help automate parameter tuning and improve robustness across different problems

Step size selection

• Determines the discretization of Hamiltonian dynamics in the leapfrog integrator
• Crucial for maintaining numerical stability and achieving desired acceptance rates
• Too large step sizes lead to inaccurate approximations and high rejection rates
• Too small step sizes result in inefficient exploration and increased computational cost
• Adaptive methods (dual averaging) can automatically tune step size to target a specific acceptance rate
• Optimal step size often depends on the scale and geometry of the target distribution

Number of steps

• Defines the length of the Hamiltonian trajectory before a Metropolis acceptance step
• Affects the distance traveled in parameter space and the correlation between samples
• Too few steps may result in random walk behavior, similar to standard Metropolis-Hastings
• Too many steps increase computational cost without proportional gains in effective sample size
• Can be randomly drawn from a distribution (uniform, Poisson) to improve mixing
• Adaptive methods (No-U-Turn Sampler) dynamically determine trajectory length based on the geometry of the target distribution

Mass matrix tuning

• Impacts the relative scaling of momentum variables and their coupling with position variables
• Diagonal mass matrix scales each dimension independently, suitable for uncorrelated parameters
• Full mass matrix can account for parameter correlations, improving efficiency in strongly correlated spaces
• Adapting the mass matrix during warmup can significantly improve sampling efficiency
• Common approaches include using the empirical covariance of warmup samples or stochastic optimization techniques

Acceptance probability calculation

• Ensures detailed balance is maintained, guaranteeing convergence to the target distribution
• Computed using the Metropolis-Hastings ratio, accounting for both the target distribution and the Hamiltonian
• Acceptance probability: min(1, exp(H(q, p) - H(q', p')))
• High acceptance rates (65-80%) typically indicate good algorithm performance
• Low acceptance rates suggest issues with step size, mass matrix, or other tuning parameters
• Monitoring acceptance rates provides valuable diagnostics for HMC performance

Advanced HMC techniques

• Ongoing research in HMC has led to several advanced techniques that address limitations and improve efficiency
• These methods often focus on automating parameter tuning and adapting to complex geometries
• Advanced techniques can significantly enhance HMC's applicability to a wider range of Bayesian inference problems

No-U-Turn Sampler (NUTS)

• Adaptive extension of HMC that automatically tunes the number of leapfrog steps
• Eliminates the need for manual tuning of trajectory length, a common challenge in standard HMC
• Uses a recursive algorithm to build a set of candidate points, stopping when the trajectory begins to turn back on itself
• Implements sophisticated slice sampling within the expanded trajectory for efficient state selection
• Achieves better exploration in complex posterior geometries compared to fixed-length HMC
• Widely adopted in probabilistic programming languages (Stan, PyMC) due to its robustness and ease of use

Riemannian manifold HMC

• Generalizes HMC to account for the local geometry of the parameter space
• Utilizes a position-dependent metric tensor to adapt the mass matrix at each point
• Improves sampling efficiency in highly correlated and ill-scaled distributions
• Particularly effective for hierarchical models and problems with varying scales across parameters
• Requires computation of higher-order derivatives, increasing computational cost
• Softened and sparse variations reduce computational burden while retaining benefits

Parallel tempering HMC

• Combines HMC with parallel tempering to improve sampling from multimodal distributions
• Runs multiple Markov chains at different "temperatures," allowing easier traversal between modes
• Periodically swaps states between chains to facilitate global exploration
• Leverages HMC's efficiency for local exploration within each mode
• Effective for problems with isolated modes or complex energy landscapes
• Requires careful tuning of temperature ladder and exchange frequencies for optimal performance

HMC in practice

• Implementing HMC effectively requires consideration of practical aspects beyond the core algorithm
• Proper software tools, diagnostics, and awareness of common issues are crucial for successful application
• Understanding these practical considerations helps in troubleshooting and optimizing HMC for specific problems

Software implementations

• Stan provides a robust, user-friendly implementation of NUTS with automatic differentiation
• PyMC3 and PyMC4 offer HMC and NUTS within a Python ecosystem, suitable for integration with data science workflows
• TensorFlow Probability includes HMC implementations optimized for GPU acceleration
• JAGS (Just Another Gibbs Sampler) supports HMC for certain model specifications
• Custom implementations in languages like R, Python, or Julia allow for greater flexibility and experimentation
• Choice of software depends on model complexity, performance requirements, and integration with existing workflows

Diagnostics for HMC

• Trace plots visualize parameter trajectories, helping identify mixing issues or non-stationarity
• Autocorrelation plots assess sample independence and effective sample size
• R-hat statistic measures convergence across multiple chains, with values close to 1 indicating good mixing
• Energy plots and ∂H/∂t plots check for correct implementation and tuning of the Hamiltonian dynamics
• Effective sample size (ESS) per second of computation time evaluates overall sampling efficiency
• Divergences indicate areas of high curvature or ill-defined posteriors, requiring model reparameterization
• Monitoring step size, acceptance rate, and tree depth (for NUTS) provides insights into sampler behavior

Common pitfalls

• Improper parameterization leading to funnel geometries or extreme correlations
• Misspecified priors resulting in improper posteriors or unwanted shrinkage
• Numerical issues from log-sum-exp tricks or overflow/underflow in likelihood calculations
• Inefficient gradient computations slowing down overall performance
• Overreliance on default settings without considering problem-specific tuning
• Ignoring warmup samples in posterior summaries, leading to biased estimates
• Misinterpreting diagnostics or failing to check convergence thoroughly

Applications of HMC

• HMC's efficiency in high-dimensional spaces and complex geometries enables its application to a wide range of problems
• Particularly valuable in scenarios where traditional MCMC methods struggle or require excessive computation time
• Continues to expand its reach as new variants and implementations become available

High-dimensional problems

• Excels in sampling from posteriors with hundreds or thousands of parameters
• Effectively handles large-scale regression models with many predictors
• Enables inference in deep probabilistic models and Bayesian neural networks
• Facilitates analysis of high-dimensional time series and state-space models
• Improves efficiency in sampling from posteriors of complex spatial models
• Allows for simultaneous inference of many latent variables in mixture models

Hierarchical models

• Efficiently samples from multi-level models with varying effects across groups
• Handles complex variance structures and non-nested hierarchies
• Improves inference in longitudinal studies with individual-specific parameters
• Enables robust analysis of meta-analyses combining multiple studies or experiments
• Facilitates Bayesian modeling of educational and psychological data with clustered structures
• Enhances inference in ecological models with species-specific and location-specific effects

Gaussian processes

• Efficiently samples hyperparameters and latent functions in Gaussian process models
• Enables scalable inference for large-scale spatial and spatio-temporal models
• Improves sampling from posteriors in Gaussian process regression and classification tasks
• Facilitates Bayesian optimization using Gaussian process surrogates
• Enhances inference in models combining Gaussian processes with other probabilistic components
• Allows for efficient sampling in high-dimensional Gaussian process latent variable models

Limitations and challenges

• Despite its advantages, HMC faces certain limitations and challenges in specific scenarios
• Understanding these constraints helps in choosing appropriate sampling methods for different problems
• Ongoing research addresses many of these limitations, expanding HMC's applicability

Discrete parameters

• Standard HMC requires continuous parameter spaces, limiting direct application to discrete variables
• Workarounds include continuous relaxations or augmentation techniques (Bernoulli)
• Hamming Ball samplers extend HMC concepts to discrete spaces but with limited efficiency
• Combining HMC for continuous parameters with Gibbs sampling for discrete ones in hybrid approaches
• Developing specialized HMC variants for specific types of discrete parameters (ordinal, count data)
• Exploring connections with discrete optimization techniques to improve sampling in mixed discrete-continuous spaces

Multimodal distributions

• Basic HMC can struggle to move between isolated modes in the posterior
• Parallel tempering HMC and other tempered approaches improve exploration of multimodal landscapes
• Wormhole Hamiltonian Monte Carlo creates tunnels between modes for more efficient transitions
• Adaptive path sampling techniques guide HMC between known modes
• Combining HMC with variational inference to identify and explore multiple modes
• Developing diagnostics to detect multimodality and adapt sampling strategies accordingly

Computational cost considerations

• Gradient computations can be expensive for complex models or large datasets
• Requires efficient implementations of log-posterior and gradient calculations
• Stochastic gradient HMC variants reduce cost for large datasets but introduce additional noise
• GPU acceleration can significantly speed up computations, especially for neural network models
• Surrogate modeling approaches approximate expensive likelihoods to reduce computational burden
• Balancing number of chains, warmup length, and number of iterations to optimize computational resources
• Exploring connections with variational inference and other approximate methods for initialization or hybrid approaches