What is Elliptical Slice Sampling?
Elliptical Slice Sampling is a Markov Chain Monte Carlo method for latent variables with a multivariate Gaussian prior that searches an ellipse for a state passing a likelihood slice threshold.
Quick Facts
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How It Works
Rotate on a prior-preserving ellipse
For a centered prior f ~ N(0, Sigma), draw an auxiliary direction nu ~ N(0, Sigma) and define f(theta) = f cos(theta) + nu sin(theta). Rotation preserves the joint Gaussian law of (f, nu), so the prior term does not need to appear in an acceptance ratio. A nonzero prior mean must be subtracted before rotation and restored afterward.
Murray, Adams, and MacKay's original paper derives the update for latent Gaussian and Gaussian Process models. The Gaussian-prior factorization is the defining assumption, not an optional convenience.
Shrink an angular likelihood slice
Draw logY = log L(f) + log(U), choose an initial angle uniformly over a full revolution, and form an angle bracket of width 2*pi. If the rotated candidate has likelihood below the threshold, shrink the bracket on the rejected side and draw another angle until a valid point is found.
The current state remains inside the bracket, which guarantees termination under ordinary finite-likelihood conditions and is part of the invariance argument. Changing the bracket rule, rotating with the posterior covariance, or including the Gaussian prior again in log L defines a different and potentially invalid transition.
Measure likelihood work and posterior movement
ESS has no proposal step size to tune, but each retained state may require several likelihood evaluations. An informative or sharply constrained likelihood can produce small accepted angles and strong autocorrelation; a costly dense covariance factorization can dominate runtime before sampling starts.
Track likelihood evaluations, accepted-angle displacement, bulk and tail effective sample size, Monte Carlo error, multiple-chain agreement, and wall time. Spell out Elliptical Slice Sampling rather than using ESS alone when Effective Sample Size appears in the same report.
Key Characteristics
- Requires a multivariate Gaussian prior or an equivalent transformed representation
- Rotates the current state and an independent prior draw along an ellipse
- Applies a slice threshold only to the likelihood factor
- Shrinks an angular bracket until it finds an acceptable state
- Avoids proposal-scale and acceptance-rate tuning
- Can still mix slowly under concentrated likelihoods or separated modes
Common Use Cases
- Latent Gaussian Process classification and count models
- Spatial or temporal latent Gaussian fields
- Gaussian-prior Bayesian inverse problems
- Baseline inference for strongly correlated latent vectors
- Gradient-free updates inside larger blocked samplers
Example
Loading code...Frequently Asked Questions
How does Elliptical Slice Sampling differ from ordinary Slice Sampling?
Ordinary Slice Sampling searches a level set of the full target, often along coordinates or chosen directions. Elliptical Slice Sampling preserves a known Gaussian prior by rotating on an ellipse and applies the slice threshold only to the likelihood.
Does Elliptical Slice Sampling require tuning?
It has no random-walk step size or target acceptance rate, but it still requires a correct Gaussian prior mean and covariance representation. Numerical factorization, parameterization, blocking, and likelihood evaluation cost can materially affect performance.
Can Elliptical Slice Sampling be used with a non-Gaussian prior?
Not directly. The ellipse relies on rotational invariance of two independent Gaussian draws. A model may be transformed to a latent Gaussian representation, but the transformation and its likelihood contribution must be derived correctly.
Does every Elliptical Slice Sampling iteration produce an independent draw?
No. Each ellipse is anchored at the current state, so retained states form a Markov chain. Multiple chains, autocorrelation-based effective sample size, Monte Carlo error, and mode coverage remain necessary diagnostics.
When can Elliptical Slice Sampling be inefficient?
It can require many likelihood evaluations when the likelihood occupies a narrow part of the prior ellipse, and it can move by small angles under strong data constraints. Dense prior covariance operations and separated posterior modes can also dominate cost.