Reweighting and Error Bars

Reweighting estimates a new expectation from configurations sampled under a reference distribution. It is useful when the two ensembles overlap well. Its uncertainty depends on correlated fluctuations of a numerator and denominator, so treating them as independent estimates can give a badly wrong error bar.

Required background. Monte Carlo basics defines importance sampling; autocorrelation analysis determines appropriate blocking.

The self-normalized ratio

Suppose the target is proportional to $q(x)w(x)$ and samples come from normalized $q$. For nonnegative weights with finite nonzero expectation,

$$\mu=\frac{\mathbb E_q[wA]}{\mathbb E_q[w]}, \qquad \widehat\mu=\frac{\sum_iw_iA_i}{\sum_iw_i}.$$

Unknown common weight normalization cancels. At finite $N$, the ratio is generally biased; under regularity and finite-moment conditions, the leading bias is typically $O(N^{-1})$. Consistency requires enough overlap and an appropriate law of large numbers.

To avoid overflow, subtract a common maximum log weight before exponentiating. This preserves the ratio exactly in exact arithmetic, but cannot restore configurations that were never sampled.

The fluctuation that controls uncertainty

Let $Y=wA$ and $W=w$. Linearizing the ratio around its true means gives

$$\widehat\mu-\mu \approx\frac{\bar Y-\mu\bar W}{\mathbb E_qW}.$$

For independent draws, its leading variance is

$$\operatorname{Var}(\widehat\mu) \approx \frac{\operatorname{Var}_q[w(A-\mu)]} {N(\mathbb E_qw)^2}.$$

The covariance is already included in this expression. For a chain, sum autocovariances of the combined fluctuation $w(A-\mu)$, or use blocks long enough for that fluctuation. The autocorrelation time of $A$ alone may be irrelevant when rare large weights dominate.

A two-state example

Sample states zero and one equally under $q$. Let $w=(1,3)$ and $A=(0,1)$. The target mean is $3/4$. The independent ratio variance to leading order is $9/(64N)$: $w(A-\mu)$ takes values $-3/4,+3/4$, and $\mathbb E_qw=2$.

Computing separate numerator and denominator errors while omitting their covariance gives a different, incorrect result. Enumerating binomial counts of the two states also reveals finite-$N$ ratio bias. This small example is useful for verifying a jackknife implementation without trusting a large simulation.

Blocked jackknife and bootstrap

Partition a production series into $B$ equally long contiguous blocks. Keep each block’s numerator sum and denominator sum together. For leave-one-block-out ratios $\widehat\mu_{(-b)}$, let their average be $\overline\mu_{\rm jack}$. The jackknife variance estimate is

$$\widehat{\sigma}_{\rm jack}^2 =\frac{B-1}{B} \sum_b(\widehat\mu_{(-b)}-\overline\mu_{\rm jack})^2.$$

Recompute the full nonlinear ratio on every deletion. For a block bootstrap, resample whole blocks and recompute the ratio on each resample. Resampling individual observations from a correlated chain understates uncertainty. Use enough blocks to estimate the error, and verify stability as block length changes.

If leave-one-out denominators approach zero or resamples become highly skewed, the usual Gaussian interval may be unjustified. This is a diagnostic to report, not a reason to discard inconvenient resamples.

Overlap and effective weighted counts

For nonnegative weights, $N_{\rm eff,w}=(\sum_iw_i)^2/\sum_iw_i^2$ is a useful weight-concentration diagnostic. It is not a complete variance formula and does not include Markov correlation or dependence of $A$ on the weights. Inspect largest normalized weights and stability under independent reference runs.

Signed or complex weights require sign/phase analysis; a positive-weight effective-count formula does not certify their ratio uncertainty.

Exercises

If one sample carries essentially all normalized weight, what happens to the weighted effective count and the reliability of a nominally large run?

Solution

The diagnostic tends to one, regardless of the total stored sample count. The result is controlled by a single configuration and usually cannot support a stable uncertainty estimate. More samples help only if they actually explore the target-relevant region; moving the reference ensemble closer may be necessary.

References

• Flyvbjerg, H., and H. G. Petersen, “Error estimates on averages of correlated data,” The Journal of Chemical Physics 91, 461–466 (1989), doi:10.1063/1.457480 — blocking correlated measurements.
• Hastings, W. K., “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika 57, 97–109 (1970), doi:10.1093/biomet/57.1.97 — stationary sampling and proposal correction.