Statistical Inference and Error Budgets

Lattice inference begins with correlated histories and ends with a claim assembled from many shared inputs. Between them lie equilibration choices, autocorrelation windows, nonlinear estimators, covariance matrices, spectral and continuum fits, scale setting, operator renormalization, model alternatives, and reproducibility checks. Treating any one stage as independent can understate uncertainty or count it twice; Sokal 1997, pp. 131–192 develops the correlation-aware Monte Carlo foundation, while Wolff 2004, pp. 143–153 gives a practical autocorrelation-error method.

Find the missing inference layer

Symptom or question	Start here	Evidence required
Error bars assume independent configurations	Autocorrelation times and effective sample size	Observable-specific window, tail, and replica checks
Ratios, scales, or channels share data	Estimators, covariance, and resampling	Correct resampling unit and joint covariance propagation
Fit changes with window or covariance treatment	Correlated fits, model selection, and stability tests	Residual, holdout, alternative-model, and synthetic-closure tests
Burn-in or late drift is uncertain	Equilibration, stationarity, and chain diagnostics	Multiple starts, slow observables, distributions, and independent streams
Analyst choices may depend on the result	Blinding and independent reproduction	Frozen decisions, controlled unblinding, alternative analysis
Scale, $Z$ factors, and continuum fits are coupled	Multi-stage correlated uncertainty	Dependency graph or nested/joint propagation
A final precision claim is needed	Complete lattice error budgets	Source, estimator, covariance, validation, and residual risk for every component

Correct transition-kernel construction belongs to the sampling chapter. This chapter starts with realized histories and asks whether the requested estimand and uncertainty are supported by those histories and the full downstream analysis.

Three statistical objects that must not be conflated

For an observable history $X_t$:

• the target distribution describes equilibrium variation of $X$;
• the transition kernel determines serial dependence and equilibration;
• the analysis map turns the correlated history and auxiliary inputs into a final estimator.

A kernel can have the correct invariant distribution but mix too slowly for a finite run. A stationary run can be analyzed with the wrong covariance. A statistically impeccable finite-ensemble estimate can still lack continuum, volume, or operator control.

The shared dependency graph follows these layers from immutable ensemble identity to the final claim. Dashed arrows mark shared inputs or tests whose correlations cannot be discarded.

A disciplined workflow

1. Preserve configuration order, stream identity, update units, rejected states, thermalization history, and all analysis inputs.
2. Predefine or justify the thermalization cut using multiple starts and slow observables.
3. Estimate autocorrelation for each material observable and derived direction, not only for a plaquette-like monitor.
4. Choose blocks, replicas, or nested resamples that preserve dependence through the entire nonlinear pipeline; Efron and Tibshirani 1993 supplies the general resampling framework.
5. Fit with the full justified covariance, inspect whitened residuals and identifiability, and vary windows and plausible models.
6. Propagate common scales, renormalization factors, tuning data, and interpolation inputs jointly.
7. Test closure and coverage on synthetic or exact fixtures; perform a held-out or independent analysis.
8. Assemble a claim-level uncertainty whose categories are mutually exclusive enough to avoid double counting.

What a final record contains

Component	Required record	Typical negative control
Ensemble identity	action, parameters, volume, stream, update count, seed lineage	swapped or duplicated configurations
Stationarity	cut rule, starts, drift tests, slow observables	injected exponential burn-in or late mean shift
Correlation	$\rho(t)$, window, tail model, $\tau_{\mathrm{int}}$, uncertainty	AR(1) chain with known $\tau_{\mathrm{int}}$
Resampling	unit, block length, replicas, nesting, transform order	treating correlated measurements as independent
Fit	data vector, covariance, conditioning, model, priors if any, window	hidden second state or rank-deficient covariance
Shared inputs	scale, $Z$, tuning, common ensembles, covariance handoff	independently resampling a shared factor
Alternatives	declared model and analysis variations	choose only the alternative nearest the preferred value
Final claim	component estimates, combination rule, correlations, validation	add one systematic twice or omit a slow tail

Chapter standard

A small statistical error does not compensate for failed stationarity, coverage, operator matching, or continuum control. Conversely, a conservative-looking total error is not defensible if its components have no estimator or are combined with unknown correlation. Every material component should state how it was estimated, propagated, validated, and bounded when not directly identifiable.

References

• Efron, B., and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman & Hall/CRC. DOI.
• Sokal, A. D. (1997). Monte Carlo methods in statistical mechanics: foundations and new algorithms. In C. DeWitt-Morette, P. Cartier, and A. Folacci (eds.), Functional Integration: Basics and Applications, pp. 131–192. Springer New York. DOI.
• Wolff, U. (2004). Monte Carlo errors with less errors. Computer Physics Communications, 156, 143–153. DOI.