Independence assumptions and model risk: when familiar formulas stop applying

Write the assumption as a testable statement

Independence is more specific than saying that two rows look unrelated. For events A and B it requires P(A and B)=P(A)P(B). The Wisconsin probability lecture develops this joint-probability definition. For a sequence, mutual independence is a statement about all relevant joint combinations, not just adjacent pairs. Record why the data-generating design makes the assumption plausible. Separate that rationale from empirical diagnostics, because a finite sample can fail to reveal dependence that still exists in the underlying process.

Distinguish new observations from copied information

Suppose a single fair binary observation Z is duplicated one hundred times in a file. Every row has the same marginal distribution as Z, but the average of the hundred rows equals Z exactly. It has variance one quarter, not one quarter divided by one hundred. The file contains many rows and only one independent piece of information. This simple example shows why row count is not automatically an effective sample size. Duplicates, repeated measurements and shared source records need explicit treatment before calculating uncertainty as though every row were independent.

Compare hypothetical collection mechanisms

The table identifies questions to investigate, not a universal correction factor. Different dependence structures affect different estimators. A method suitable for repeated measurements on one device may not be suitable for clustered households or a changing industrial process. The collection story should guide the model.

Inspect time order rather than only a histogram

A histogram can look similar after observations are reordered, while temporal dependence changes completely. The NIST autocorrelation reference considers relationships between a variable’s values at different lags and states an equally spaced observation assumption for its formulation. Preserve timestamps and spacing before calculating a lag statistic. Missing intervals, irregular sampling or aggregation can change the interpretation. A coefficient near zero at one lag does not establish independence at every lag or exclude nonlinear relationships.

Separate changing parameters from random fluctuation

A process can produce independent observations whose distributions change over time, or dependent observations with stable marginal distributions. Those are different departures from the simplest identical-independent model. If a sensor calibration changes halfway through a record, combining the periods without marking the change can create a misleading average or spread. Document known interventions and inspect the relevant time segments. Do not explain every shift as random noise merely because the analysis began with a fixed-parameter assumption. A model should be revisable when its substantive premises no longer match the collection process.

Understand what averaging theorems require

The MIT law-of-large-numbers material connects sample averages with expectations under stated conditions. Other versions of averaging results can accommodate forms of dependence, but they require their own hypotheses. The right response to a violated assumption is not to invoke the same theorem more confidently because the dataset is large. Identify a model or method justified for the dependence structure, or narrow the claim. Increasing the number of duplicated or systematically selected observations does not automatically solve the original information problem.

Keep diagnostics and validation claims proportionate

A diagnostic can reveal a pattern inconsistent with a simple model, but passing one diagnostic does not prove all assumptions. Some tests have limited sensitivity to particular alternatives or small samples. Repeatedly trying many diagnostics also changes how surprising one flagged result should appear. Keep the analysis plan and report what was actually checked. Avoid saying a process is certified, fair or secure merely because a plotted series looks irregular or one correlation estimate is small. Those claims require evidence beyond this article’s mathematical examples.

Model-assumption checklist

• Identify the observational unit and collection mechanism.
• Look for duplication and shared sources.
• Preserve grouping identifiers and time order.
• Distinguish independence from identical distributions.
• Record changes in equipment or procedure.
• Use diagnostics with stated limits.
• Match uncertainty formulas to the supported model.

A concise assumption register can list each premise, its justification and what evidence would challenge it. This is useful even for a small classroom project because it separates arithmetic certainty from uncertainty about the world. The goal is not to make every model maximally complicated; it is to prevent a convenient simplification from becoming an unqualified factual claim.

Sources and verification

• Independence, conditional probability and Bayes rule — University of Wisconsin–Madison — 2026-10-23
• Autocorrelation — NIST/SEMATECH e-Handbook — 2026-10-23
• Weak law of large numbers lecture introduction — MIT OpenCourseWare — 2026-10-23