B.5.TC:11 - SoTA-Echoing

For comparing what different mathematical accounts let a practitioner compute, adopt the operation-based comparison illustrated by Fong and Spivak’s Seven Sketches in Compositionality (2019), §2.5.2–§2.5.3. Reachability and cost use different operations for combining steps and choosing among alternatives. Against comparison by shared route terminology, this makes the lost answer and composition condition visible in :4.3 and :5.1. The finite compatibility example is a construction here. Reopen its application when the quantity or allowed composition changes. Author manuscript.

For physical formulations that may agree, adopt the constructive comparison in Sussman and Wisdom’s Structure and Interpretation of Classical Mechanics, second edition (2015), §1.6: derive the equations under the stated force and potential assumptions. This answers more than a numerical fit and often costs less than a new simulation. Against selecting a formulation solely by familiarity, preserve the advantage of coordinates adapted to constraints. Sections :4.3–:4.4 and :5.2 separate that choice from numerical approximation. Reopen when interactions, constraints or the requested result change. Publisher’s text.

For comparing causal uses, adopt intervention construction from Pearl’s Causal inference in statistics: An overview (2009), §3.2.1, and adapt it here to expose the particular consequence on which two accounts disagree. Agreement in the observed distribution, a serious comparator for prediction, can leave that intervention consequence undecided. This changes :4.4 and :5.3. C.28 supplies the broader causal method. Reopen when the causal premises or intended intervention change. Author’s paper.

For prediction, a serious alternative to choosing a single account is combining predictive distributions. Yao, Vehtari, Simpson and Gelman’s stacking method (2018) chooses weights through predictive performance when the candidate set need not contain the data-generating model. Adapt that question-relative choice into :4.5’s permission to retain combinations. Reject using such predictive weights as a verdict about causal mechanism or mathematical equivalence; those are different comparison questions. The statistical method remains with its domain, including its scoring and validation conditions. Reopen when prediction under another distribution or a different use is required. Authors’ paper.

This pattern combines these constructive contributions into a common comparison method. Its first result is the comparison obtained under the stated conditions.