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Probability for Machine Learning
/Information & Computational Methods
/Approximate Inference

Approximate Inference

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Approximate Inference

Approximate inference methods provide practical solutions when exact probabilistic calculations are computationally intractable. These techniques trade mathematical precision for computational feasibility, enabling probabilistic reasoning in complex models.

The need for approximate inference arises because exact computation of posterior probabilities p(θ|x) often involves intractable integrals or summations, particularly in high-dimensional spaces. Approximate methods offer practical alternatives that scale to complex models and large datasets.

Sampling-Based Methods:

  • Markov Chain Monte Carlo (MCMC): Constructs a Markov chain whose stationary distribution is the target posterior, generating samples for approximating expectations. Popular algorithms include:
    • Metropolis-Hastings: Proposes moves and accepts/rejects based on probability ratios
    • Gibbs Sampling: Updates one variable at a time, conditioning on all others
    • Hamiltonian Monte Carlo: Uses gradient information for efficient exploration
  • Sequential Monte Carlo: Evolves a population of particles to approximate posterior distributions as data arrives sequentially, crucial for online learning and filtering problems.

Deterministic Approximations:

  • Variational Inference: Approximates the posterior with a simpler distribution by minimizing KL divergence, converting inference into optimization.
  • Expectation Propagation: Iteratively approximates local factors in a graphical model, creating a global approximation through message passing.
  • Laplace Approximation: Approximates the posterior with a Gaussian centered at the maximum a posteriori (MAP) estimate, using the Hessian to determine covariance.

Modern Developments:

  • Amortized Inference: Uses neural networks to directly predict approximate posterior parameters from data, enabling rapid inference for new observations.
  • Differentiable Sampling: Incorporates sampling operations into differentiable computational graphs for end-to-end learning.
  • Normalizing Flows: Transforms simple distributions into complex ones through sequences of invertible transformations, enabling highly flexible variational approximations.

Each approach offers different tradeoffs between accuracy, computational cost, ease of implementation, and applicability to different model types. The field continues to evolve rapidly, with hybrid methods increasingly combining strengths of different approaches.