Meshfree Likelihood Interpolation

• The paper introduces a polynomial surrogate model that approximates the log-likelihood, reducing expensive MCMC evaluations by efficiently targeting high-probability regions.
• Meshfree likelihood interpolation is a numerical technique that constructs adaptive, mesh-independent surrogates using scattered data rather than structured grids.
• The method integrates seamlessly with MCMC workflows by using early training data to update interpolation nodes, achieving convergence acceleration of 2–4 times with negligible bias.

Meshfree likelihood interpolation is a numerical methodology in which a likelihood function, typically expensive to evaluate at arbitrary points in parameter space, is efficiently approximated using scattered data interpolation. This approach enables substantial reductions in computational cost for inference problems where the likelihood is a smooth function of the parameters, eliminating the need for structured grids and often accelerating Monte Carlo or Bayesian sampling pipelines. Meshfree interpolation leverages polynomials, radial basis functions, and/or other mesh-independent kernel-based representations, with interpolation nodes adaptively concentrated in regions of interest (e.g., high posterior probability). The technique has been successfully applied to accelerate Markov Chain Monte Carlo (MCMC) in cosmological parameter estimation, and is broadly applicable to other domains involving expensive likelihood evaluations (Bouland et al., 2010).

1. Interpolation Methodology

The core of the meshfree likelihood interpolation approach, as originally detailed by the InterpMC patch for CosmoMC (Bouland et al., 2010), is the construction of a high-order polynomial surrogate for the log-likelihood function over the free parameter space. The surrogate takes the form of a multivariate polynomial:

$$\log \mathcal{L}(x_1,\ldots,x_P) \approx c_0 + \sum_{i=1}^P c_i x_i + \sum_{i \leq j} c_{ij} x_i x_j + \sum_{i \leq j \leq k} c_{ijk} x_i x_j x_k + \ldots$$

where $x_i$ are the parameters (normalized to zero mean, unit variance), and the expansion is truncated at order $n$ ($n=4$ is typical).

During the initial phase of an MCMC run (10–30% of steps), a set of points (comprising both accepted and rejected steps close to the maximum likelihood) is collected as the training set. The coefficients $\{c\}$ are determined via unweighted least-squares fitting, often using established numerical routines (e.g., algorithm AS 174).

By interpolating $\log \mathcal{L}$ rather than $\mathcal{L}$, improved numerical stability is achieved because cosmological likelihoods are approximately Gaussian near their peak and are well-approximated additively in the log domain.

2. Integration with MCMC

The polynomial surrogate is integrated into the MCMC workflow to replace expensive likelihood computations:

• When a new likelihood evaluation is requested, the algorithm first checks:
  • The candidate parameter point is within a predefined “cut” in $\log \mathcal{L}$ (to avoid making predictions for low-likelihood points); and
  • The difference in the interpolated value between $n$th and $(n-1)$th order polynomials is below a defined tolerance.
• If both conditions are met, the surrogate returns the interpolated value as the likelihood estimate, bypassing the costly evaluation (such as running CAMB for cosmology).
• If not, the true likelihood is computed, the point is added to the training cache, and the interpolation may be updated.

This procedure accelerates parameter estimation runs because, in typical converged chains, most proposals are made in high-probability regions, and thus a large fraction of computations can be replaced with polynomial evaluations.

The method is completely “meshfree” in the sense that interpolation is built adaptively from the points already sampled along the chain; no predetermined grid or mesh is required.

3. Quantitative Performance and Validation

Performance benchmarks reported in (Bouland et al., 2010) indicate:

• For realistic cosmological analyses (including up to 9 free parameters), the meshfree surrogate accelerates MCMC convergence by a factor of 2–4 for well-converged chains relative to standard CosmoMC runs.
• For parameter points within $\sim$8 units of the peak $\log \mathcal{L}$ (i.e., within a likelihood factor of $\sim$3000), the polynomial interpolation has an error in $\log \mathcal{L}$ of approximately 0.01–0.05 (i.e., $\sim$2–4% in $\log \mathcal{L}$).
• Marginalized posteriors—1D and 2D likelihood contours—are visually and statistically indistinguishable between interpolated and direct-evaluation chains, confirming that the surrogate introduces negligible bias when properly applied within its intended region.

This efficiency arises because the majority of likelihood evaluations, particularly in the equilibrium phase of the Markov chain, are spent in the high-likelihood “core” where the polynomial fit is most accurate.

4. Applicability, Assumptions, and Limitations

The meshfree likelihood interpolation strategy is generalizable to any Bayesian or likelihood-based inference problem with the following characteristics:

• The likelihood surface is sufficiently smooth in the relevant region of parameter space.
• Evaluations of the likelihood function are computationally intensive and dominate inference cost.

Potential limitations include:

• For likelihoods with sharp features, discontinuities, or other pathologies, high-order polynomial interpolation may be unstable or inaccurate.
• The number of polynomial coefficients grows combinatorially with both the polynomial degree $n$ and number of parameters $P$, limiting practicality in very high-dimensional spaces.
• The surrogate is only used within the “cut” region (near the maximum), ensuring stability at the expense of not accelerating evaluations far out in the wings.

Points outside the “trusted” interpolation region are always evaluated via the full likelihood, ensuring that rare excursions of the Markov chain do not misrepresent the true probability landscape.

5. Prospective Technical Extensions

Possible technical improvements and extensions discussed in (Bouland et al., 2010) include:

• Intelligent term selection via variable selection or regularization to reduce the number of polynomial coefficients, or inclusion of only those strongly correlated terms, potentially enabling even faster surrogates or higher effective dimensionality.
• Transformation to a basis with diagonal second-order covariance (e.g., via PCA or Fisher matrix analysis), which could suppress cross-terms and simplify the interpolation, especially in anisotropic likelihood landscapes.
• Allowing adaptive polynomial order (higher order in some directions, lower in others) to match the local curvature of the likelihood.
• Storing and reusing interpolation data from prior runs when chains are extended or parameter boundaries are changed.
• Applying the meshfree surrogate not only in MCMC but also in other statistical tasks such as evidence evaluation or alternative sampling algorithms.

These directions highlight opportunities to broaden the applicability, reduce costs, or improve accuracy in diverse likelihood-based inference tasks.

6. Relevance to Related Methodology and Fields

The polynomial meshfree surrogate outlined in (Bouland et al., 2010) differs from precomputed grid-based interpolants or machine-learned emulators by being constructed dynamically (“on the fly”) from points naturally explored by the Markov chain, with no a priori assumptions about the optimal training set.

Relative to classical grid-based surrogates, this avoids the curse of dimensionality and naturally focuses computational effort in the high-posterior region.

Meshfree interpolation in this context is closely related to kriging and RBF-based surrogates, but the decision to use high-order polynomials leverages the particular structure and smoothness of cosmological likelihoods, and the normalization of parameters prior to fitting aids both accuracy and numerical stability.

The underlying principles are transferable to other areas with smooth, high-dimensional, and expensive likelihood evaluations, including statistical mechanics, systems biology, engineering design, and more.

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In summary, meshfree likelihood interpolation via polynomial surrogates—exemplified by the InterpMC algorithm—realizes significant efficiency gains in parameter estimation by exploiting the local smoothness and structure of the log-likelihood, dynamically constructing surrogates from the MCMC trajectory itself, and carefully validating the region in which the interpolant is invoked. This approach has established practical value in contemporary cosmological inference and offers a paradigm of adaptability for other computational fields where similar inference bottlenecks arise (Bouland et al., 2010).