How much independent information is in ten thousand frames

A heat bath slows the bead and keeps kicking it

Under strong friction, momentum relaxes rapidly and we follow only position $\mathbf r=(x,y)$ . The negative potential gradient pulls downhill. The bath adds random kicks. Isotropic friction is $\zeta$ and diffusion is $D=k_{\mathrm B}T/\zeta$ . Changing friction requires changing the noise to retain the same temperature.

Each independent component of $d\mathbf W_t$ has zero mean and variance $dt$ . A small numerical step combines downhill drift with a Gaussian displacement. The square boundary reflects excursions. Finite steps approximate the continuous process, so reducing $\Delta t$ is a necessary convergence check.

d\mathbf r_t=-\frac{\nabla U(\mathbf r_t)}{\zeta}\,dt+\sqrt{\frac{2k_{\mathrm B}T}{\zeta}}\,d\mathbf W_t

This is an isothermal, overdamped model without inertia. Its time belongs to the specified dynamics; adding a sampling bias changes those dynamics.

Detailed balance constrains probability flow

Temporarily divide space into cells. Let $P_{ij}$ be the probability of moving from cell $i$ to $j$ in one step, and $\pi_i$ its equilibrium population. Detailed balance equates the average flows in both directions for every pair. It excludes persistent circulating probability currents and is stronger than an unchanged total distribution.

Equal energy and temperature do not require equal speed. With the corresponding thermal noise, greater friction can trace the same statistical motion on a stretched clock. Chapter 7 uses this observation to construct equal free energies with different transition speeds.

\pi_iP_{ij}=\pi_jP_{ji}

Finite trajectories still need equilibration and exchange checks.

Correct an error estimate for similar frames

Choose an observable $A_n$ , such as horizontal position or the indicator for occupying the right valley. Its normalized covariance at lag $\ell$ is $\rho_A(\ell)$ . When correlations have a summable tail, the variance of its mean is multiplied by statistical inefficiency $g_A$ .

Worked example: suppose a stationary series has $\rho_A(\ell)=0.9^\ell$ . Summing the geometric series gives $g_A=19$ . Ten thousand frames then provide about $526$ independent observations. Treating them as independent understates the standard error by approximately $\sqrt{19}=4.36$ .

Effective sample size belongs to an observable. Horizontal position may exchange repeatedly even when a hidden coordinate stays trapped. A fast observable alone can give an overly favorable assessment.

g_A=1+2\sum_{\ell=1}^{\infty}\rho_A(\ell),\qquad N_{\mathrm{eff},A}\simeq\frac{N}{g_A},\qquad \operatorname{Var}(\bar A)\simeq\frac{g_A\,\operatorname{Var}(A)}{N}

These approximations require a stationary, sufficiently long record. Finite estimates truncate a noisy correlation tail and should be compared with blocks and independent seeds.

Source: PLUMED Masterclass 21.2 · Michael R. Shirts and John D. Chodera

A trapped trajectory can have a small error bar

A bead rapidly explores the bottom of the left valley. Its local mean stabilizes, and the error bar shrinks. If it never visits the right valley, that valley’s population remains unknown. A correlation estimator cannot measure a transition absent from the record.

Compare contiguous blocks, early and late intervals, and runs started in different valleys. Different long-run distributions from different starts call for more exchange or another sampling method. An error estimate that reaches a plateau as block length grows is useful evidence. Failure to find a plateau is also a result to retain.

Try the reasoning in an experiment

Slow the motion and compare equal recordings

Predict how greater friction changes equilibrium populations and transitions per unit time.
Change only friction and compare trajectories over equal model time. Compare the two survival curves at times related by the friction ratio and test for a stretched clock.
Use right-valley occupancy as an observable. Compare denser recording with a longer observation and explain which one can add independent exchange information.

Exercises and explanations

Record ten times more often without increasing simulation time. Should the standard error fall by about a factor of three?
Read the explanation
Usually it will not. The additional frames are more correlated. The continuous path contains little additional independent information. Longer observation or more independent exchanges is more useful. Recording too sparsely can discard information, so inspect correlation decay first.
Two seeds show tiny within-valley fluctuations but remain in different valleys. Can a small bootstrap error establish convergence?
Read the explanation
Resampling only rearranges the regions already visited. The disagreement reveals unknown relative valley populations. Improve exchange or design sampling that covers both valleys before interpreting the uncertainty.

Sources and reproduced scope

Worked examples and schematic figures are original teaching material. The numerical research slice compares specified estimator equations on our seeded two-coordinate records with independent quadrature and PyMBAR. The final chapter reads Figure 1 of the MBAR paper and states which experimental data remain outside that calculation.

Model definitions, scripts and numerical records

PLUMED Masterclass 21.2, Statistical errors in MD (documentation v2.9)
Background and Exercises 2, 7–9: marginal probabilities, correlated samples, blocks and bootstrap. The tutorial reports PLUMED 2.7.0 for its original exercises. Our examples use an independently defined two-coordinate model.
Michael R. Shirts and John D. Chodera, Statistically optimal analysis of samples from multiple equilibrium states, J. Chem. Phys. 129, 124105 (2008)
Frozen arXiv v3: Sections II–IV, Eqs. (9), (11)–(14), Section V and Figure 1, Appendices A–C. We implement the equilibrium estimating equations on our synthetic model. Figure 1 is read as an experimental example; its DNA-hairpin traces and plotted values are not reproduced.