Statistical and Scientific Inference

Summary

The book’s most conceptually ambitious chapter opens with the Portuguese man o’ war — a colony of specialized animals, none of which makes sense alone — as the image for Bayesian data analysis: “Each component can function and pass all diagnostic checks, but still not make sense as part of the whole.” It names the failure mode this produces, scientifically degenerate analysis (statistically valid, scientifically meaningless), gives four documented real examples, and proposes a tangle — theory → estimand → design → sample → estimator → estimate → target — as the structure a full scientific workflow needs, borrowing the visualization practice of formalized mathematical proofs.

Overview

The opening image (Ch. 10, p. 175)

“The Portuguese man o’ war (Physalia physalis) is neither Portuguese nor a warship. It looks like a jellyfish, but it’s not even an animal. Instead it’s a colony of individual animals, each with a specialized job. Some sting, others feed, and yet others raise a sail to ride the waves. None of these individuals makes sense in isolation. Their function emerges from structure. They fuse together in cooperation to sail and sting their way across the ocean. Like a pirate ship made of jelly.

Bayesian data analysis is similar. It comprises functionally integrated parts. Or at least it should. None of the parts makes much sense by itself. Each component can function and pass all diagnostic checks, but still not make sense as part of the whole.”

Four instances of components that pass locally but fail globally:

• “a wrong model that badly overfits can still pass a posterior predictive check for some test variables”;
• “in cross validation, the consistency relies on conditional independence and stationarity of the predictors”;
• “in causal inference, there are always untestable causal assumptions, no matter how many models we have fit”;
• “more generally, statistics relies on some extrapolation, for which some assumption is always needed.”

What this book has deliberately left out

“This book focuses on how workflow serves data analysis … But we have neglected equally important aspects of Bayesian statistics, including theory construction, design, measurement, and data collection (coming before the data analysis), as well as decision making and communication (coming after data analysis).”

“Instead of treating each step and its diagnostics as an isolated laboratory, a full scientific workflow requires justifications and diagnostics that span the network of these models, data, theories, goals, and summaries. Some of these links are necessarily subjective and give meaning and shared strength to the objective algorithms embedded in them.”

The stated aim: “We wish for a statistics that credibly serves scientific goals instead of the too-common substitution of statistical goals for scientific ones. We don’t know yet exactly how to achieve this.”

Main Content

Scientifically degenerate analyses

Scientifically degenerate (Ch. 10.1, p. 176)

An analysis that is “statistically valid but lack[s] the essential quality of scientific relevance.”

“One way to conduct data analysis is to find some variables seemingly related to a research question, explore some associations among them, and then tell a story about how these associations address the original question. The statistical workflow in such a plan can be immaculate without leading to any scientific insights from the inference. In the absence of transparent and auditable links between theories of phenomena and our statistical procedures and summaries, results can be reliable and repeatable but also completely wrong.”

“A mature scientific workflow attends not only to the internal coherence of statistical procedures, but also to the scientific coherence among generative models of scientific phenomena, queries about these models, measurements, estimators, and summaries of estimates.”

Four documented cases

1. Conditioning on a post-treatment variable. “A minimal example … is an analysis of a randomized experiment that stratifies by a variable that is causally downstream of the treatment. There is nothing necessarily wrong with the statistical model in this context. It is only that the estimator is incompatible with the inferential goal of estimating the total causal effect.” In the wild: “the widespread practice of conditioning on ‘attention checks’ in psychology” (Montgomery, Nyhan, and Torres 2018).

2. Table 2 fallacy — reporting control coefficients as causal effects (Westreich and Greenland 2013). “Suppose that one or more control variables are necessary to properly estimate a treatment effect. So … the estimator is properly matched to the scientific goal. However the coefficients from these control variables are reported as total causal effects, just like the target treatment effect. This can be scientifically degenerate, because an estimator designed to yield the causal effect of a treatment does not automatically also yield causal effects for control variables.” In Westreich and Greenland’s examples, “one must assume zero unobserved confounding of variables beyond the treatment and outcome.”

3. The hot hand (Gilovich, Vallone, and Tversky 1985 → Miller and Sanjurjo 2018). An early study “failed to find evidence for the hot hand and concluded that the belief was a fallacy arising from flaws in human reasoning. Despite some published criticisms … the view … persisted within academic psychology and economics until Miller and Sanjurjo (2018) showed that the streak estimator … was systematically biased by selection. That earlier paper had no derivation or evaluation of the statistical properties of its procedure; it was merely justified by intuition. Once a proper estimator was derived using a generative model, evidence of the hot hand was found in the original data.”

“Nearly thirty years of confusion could have been avoided had the original researchers simply conducted a simulation study of their own estimator.”

Worked out in full in Simulated-Data Experimentation as Virtual Replication.

4. Reproductive skew in population biology. “Many papers have been written comparing reproductive skew by sex and species, often with scarce data. Nearly all of these papers have used ad hoc estimators of skew that exhibit systematic bias at observed sample sizes” (Ross, Jaeggi, et al. 2020). “Only recently has the literature moved away from intuitive-but-degenerate estimators to estimators based upon and transparently derived from generative models of population dynamics” (Ross, Hooper, Smith, et al. 2023).

Also named: a “structurally analogous flaw” in routine DNA methylation analysis (Singer and Pachter 2015); social network permutation procedures that “fail to solve the problem that inspired them” (Hart et al. 2022); and baseline adjustments that “routinely cause more problems than they solve” (Glymour et al. 2005).

What the degenerate analyses have in common

“The degenerate hot hand and methylation analyses are reproducible and replicable. They look like science, but they lack any formal connection between a generative model of the phenomenon and the statistical approach.”

“What these real examples have in common is a failure to transparently connect scientific goals to statistical procedures.”

The prompt a scientific workflow would supply: “For each stratified variable, what is the scientific justification for its inclusion?” And crucially: “These assumptions do not have to be correct in order to be useful. They just have to be transparent and be used correctly in the design of the statistical model and the reporting of results.”

The proof analogy

Borrowing from formalized mathematics (Ch. 10.1, pp. 177-178)

“Consider the analogy of a mathematical proof. A proof has a target, the thing to be proved, and a set of axiomatic assumptions. In between, there may be many steps (lemmas), some taken in parallel, that converge on the target. Each step depends upon previous steps for its validity and requires its own justification. Linking all of these elements is a language, a set of procedures for combining intermediate results and assumptions into new results.”

The tooling that exists for proofs, and what it provides:

• Lean (Lean Focused Research Organization 2024) “can assist with the formal definition of each step, as well as the justifications required to move among them. The framework demands logical documentation for each definition and claim.”
• Blueprint (Massot 2024) “can visualize the proof as a directed network and facilitate navigation and inspection. This assists in constructing and completing the proof. And just as important, it helps others examine and check the proof.”

“Scientific data analysis is never really proven to be internally coherent. But the structural nature of the problem is similar. We require a representation of the analysis that helps us design, complete, validate, and transparently expose it to critique and modification.”

Figure 10.1 — the dependency graph, not a workflow

“In this figure, the arrows do not indicate any order of operation or action. Instead they indicate logical and computational dependence.”

flowchart LR
    T["Generative theories"] --> Q["Estimand<br/>(the question)"]
    T --> D["Research design"]
    Q --> D
    D --> S["Sample"]
    Q --> E["Estimator<br/>(the statistical model)"]
    T --> E
    S --> E
    S --> P["Estimate<br/>(posterior distribution)"]
    E --> P
    P --> R["Reported target<br/>(prediction / causal effect)"]
    Q --> R
    TP["Target population /<br/>target context"] --> R

Spelled out: “Statistical estimates, such as posterior distributions, depend upon the sample and the estimator, the statistical model. The justification for the estimator depends upon a specific set of questions, the estimand, which itself depends upon some set of generative theories for its coherence and relevance. And ultimately any reported summary targets of inference, whether predictions or causal effects, depend upon the estimates and a target population that is consistent with the estimand.” (Deffner et al. 2024 present a fully-coded example spanning theory development, estimation, and reporting.)

Scurvy, traced through the graph (Ch. 10.1, p. 178)

Node	Content
Theory	”we suspect that scurvy is caused by vitamin C deficiency” — vaguely expressed
Estimand	”What is the effect of consuming one lemon per week over the course of two months on symptoms of scurvy among sailors in the Royal Navy?”
Design	”theory and estimand together justify a research design, perhaps giving one lemon per week to all sailors on randomly selected vessels and recording scurvy symptoms upon return to port”
Sample	”The coherence and relevance of the sample depend upon the design. But often there will be unanticipated aspects of the sample that were not part of the design”
Estimator	must handle those unanticipated features: missing data (“some sailors slip away without health exams”) and measurement error (“who is doing the health examinations anyway?”). “These real features can be treated in the statistical model for better characterization of uncertainty. At the minimum we require a clear justification for decisions regarding missing data”
Estimate	assuming effective randomization, “a simple model of treatments (lemon ships versus no-lemon ships) produces … the posterior distribution of the symptoms in each treatment”
Target	”This distribution together with the estimand and a target population of sailors over a target duration of service (the target context) justify an estimate of the causal effect”

Note where the estimator’s dependence on theory enters even under randomization: “the estimator depends upon aspects of the generative theory and sample, if only to justify why confounding is or is not assumed.”

The graph as a peer-review instrument

“A reader could begin with a paper’s summary of an inference, ‘Vitamin C cures scurvy,’ and follow the graph backward to identify or mark as absent logical and computational dependence that the conclusion requires. An analyst who provides an annotated graph with links to code and data makes it easier for peers to appreciate, critique, and build upon their work.”

The virtuous tangle

Tangle (Cartwright et al. 2023; Ch. 10.2, p. 179)

“We are making a distinction between statistical workflow and its virtuous result, a tangle of concepts that form a coherent and reliable scientific project.

The graph of this analysis does not imply a workflow. Instead it conveys logical and computational dependence, not necessarily indicating a temporal order. Sometimes an analysis begins with a sample and an initial estimand. Other times the question comes first, and we search for a sample that can address it, however imperfect. Either way, the graph indicates which additional assumptions and calculations are needed to transparently justify and qualify any inference made from the sample and which estimands are compatible.”

Figure 10.2 — how synthetic data creates the tangle

“Two commonplace workflows in which the tangle arises from revision and counter-flow are the use of model diagnostics and model development using synthetic data.

When we design a statistical model, leading to the estimator in the diagram, we should use features of the generative theory, the estimand, and design and collection of the data. And then in order to test and create transparent confidence in the estimator, we can use these features again to simulate synthetic data, a sample which obeys the generative theory and sampling design. Then we can produce synthetic estimates and evaluate the performance of the estimator.”

The consequence — diagnostics become ambiguous, and that is the point:

“This kind of workflow produces a tangle among many components of the analysis. When problems are found, they could be caused by any or all of the components. Ordinary diagnostics for stochastic algorithms work similarly. When diagnostics reveal poor mixing, the cause could be the estimator or rather the generative theory or a mismatch between the estimand and the other components.”

Each workflow component thickens the tangle: “the prior predictive check helps in developing the estimator, but it can also reveal important aspects of the generative model that were omitted. A posterior predictive check can similarly reveal inadequacies in the generative theory. But in light of specific scientific questions, not all flaws in posterior predictions are problematic.”

“A virtuously entangled scientific workflow must survive assault from many sides. But it also gains strength from the tangle.”

The irreducible subjectivity

“Many decisions about the adequacy of checks are necessarily subjective … a model can pass a posterior predictive check on one criterion but fail on another. For example, a model could capture the average treatment effect but fail to describe individual variation in outcomes. Whether or not such a model passes the posterior check depends upon our goals but also sometimes upon our subjective judgment of adequacy for those goals.”

Explicit and indirect strategies for justification

Explicit strategies (Ch. 10.2, p. 180)

“In mathematical proofs, the logical strategies that are used in each step must be explicit. And the Curry-Howard correspondence between computer programs and proofs extends this idea to algorithms. In data analysis workflows, this general approach of being explicit about the logic would also be of value.”

The question the arrows should answer: “These arrows say that the estimator is compatible with the generative model (theory) and some question or goal (estimand). But which assumptions are required for this compatibility to hold?”

What “explicit” demands: “some calculus (set of assumptions and rules) that licenses operations on models and assumptions and data. For example, a structural causal model and an estimand can be used with the back-door criterion to justify a set of variables to stratify an estimate by.”

And the standard is transparency, not correctness: “the key issue isn’t whether the calculus is correct but rather that it be explicit, so that others can trace the logic of the workflow and make up their own minds about its correctness and generality.”

Also: “since reasons communicate assumptions in ways that assumptions do not, the augmentation of the workflow graph with explicit strategies that license the connections can also strengthen the tangle.”

Indirect strategies — justification by simulation

“Another way to justify a workflow is to appeal to simulation tests of efficacy.

For example, many analyses in population genetics make use of variance ratios. Sometimes it is not clear that these ratios are statistically sufficient, and we have good reason to think they might not be. But the approach may still lead to valid inferences, under certain constraints. This can be demonstrated with simulation studies. Speidel et al. (2025) actually drop data to get tighter confidence regions, and this is justified not with any logical proof but rather with intuition and simulation.”

What distinguishes indirect from explicit justification: “if we derive an adjustment set for a linear regression, we might still test that the software works correctly. What is different in indirect justification is that the core logic of the approach is being (partly) validated through workflow steps that are usually seen only as error checks.”

Connections

• The tangle is the scientific-scale version of Figure 2.1: where Figure 2.1 maps steps within data analysis, Figure 10.1 maps dependencies from theory to reported claim, with all of Figure 2.1 sitting inside the “estimator” node.
• “Degenerate analysis” names precisely the failure that Causal Inference as Generalization and Poststratification are designed to prevent: an estimator mismatched to its estimand.
• The subjectivity acknowledged here is the same one Specifying the Data Model and the Prior declines to argue about, preferring “transparency, consensus, impartiality…”

See Also

• Software Assisted Workflow — what tooling would have to do to support the tangle
• Simulated-Data Experimentation as Virtual Replication — the hot hand worked out in full
• The Replication Crisis and Multiple Levels of Variation — the consequences of degenerate analyses at scale
• Model Selection and Overfitting — the severe-tests argument this chapter generalizes