The method as implemented

Notation follows Invernizzi and Parrinello. \(\beta = 1/k_BT\), \(\gamma\) is the bias factor, and \(\mathbf{s}\) the collective variables.

The probability estimate

OPES estimates the unbiased distribution by weighted kernel density estimation,

\[P_n(\mathbf{s}) = \frac{\sum_k^n w_k\, G(\mathbf{s}, \mathbf{s}_k)}{\sum_k^n w_k}, \qquad w_k = e^{\beta V_{k-1}(\mathbf{s}_k)}\]

with Gaussian kernels of fixed height \(h = \prod_i (\sigma_i\sqrt{2\pi})^{-1}\). This is OnlineKDE.getLogPDF(); the weight is the bias energy read from the OPES force group before the new kernel is deposited, so it really is \(V_{k-1}\).

Bandwidth

Bandwidths shrink as the effective sample size \(N_{\text{eff}} = (\sum_k w_k)^2 / \sum_k w_k^2\) grows, by Silverman's rule:

\[\sigma_i^{(n)} = \sigma_i^{(0)}\left[N_{\text{eff}}^{(n)}(d+2)/4\right]^{-1/(d+4)}\]

Normalization

\(Z_n\) normalizes over the CV space explored so far, and is approximated by a sum over the compressed kernel centers — OnlineKDE.getLogMeanDensity().

\[Z_n = \frac{1}{|\Omega_n|}\int_{\Omega_n} P_n(\mathbf{s})\, d\mathbf{s}\]

The bias

\[V_n(\mathbf{s}) = \left(1 - \tfrac{1}{\gamma}\right)\frac{1}{\beta} \log\left(\frac{P_n(\mathbf{s})}{Z_n} + \epsilon\right)\]

with \(\epsilon = e^{-\beta\Delta E/(1-1/\gamma)}\) limiting the bias to the barrier \(\Delta E\). This is OPES.getBias().

OPES-explore

The explore variant estimates the sampled distribution instead, with uniform weights,

\[p^{\text{WT}}_n(\mathbf{s}) = \frac{1}{n}\sum_k^n G(\mathbf{s}, \mathbf{s}_k), \qquad V_n(\mathbf{s}) = (\gamma-1)\frac{1}{\beta} \log\left(\frac{p^{\text{WT}}_n(\mathbf{s})}{Z_n} + \epsilon\right)\]

It explores faster and converges more slowly. getBias() reads the unweighted estimate in explore mode and the reweighted one otherwise. Because \(p^{\text{WT}}\) is about \(\sqrt{\gamma}\) times wider than \(P\), explore kernels start at \(\sqrt{\gamma}\,\sigma^{(0)}\), with a fixed biasWidth still read as the unbiased \(\sigma^{(0)}\).

Free energy

getFreeEnergy() always uses the importance-sampling estimate, \(F_n = -\beta^{-1}\log P_n\). In standard OPES the direct and reweighted routes are equivalent; in explore mode they differ until convergence, and the reweighted one converges better.

Kernel compression

Rather than storing a bias grid, kernels closer than compressionThreshold in Mahalanobis distance are merged, preserving total weight, mean and second moment. The number of compressed kernels is what makes \(|\Omega_n|\) estimable.

Warm-up bandwidth estimation

Both papers prescribe measuring \(\sigma^{(0)}\) from a short unbiased run. Passing warmupSteps automates this: no kernels are deposited until that many steps have elapsed, so the CV variance measured over that span is genuinely unbiased, and it is frozen (scaled by \(\gamma\), since kernels are built from a sampled-distribution variance by convention) once warm-up ends.