Bayesian Semiparametric Item Response Modelling for Person-Specific Latent Traits
Theory, Identification, and the Literature Behind the Estimators
About This Book
This is the theory-and-literature companion to Targeting Toward Inferential Goals in Bayesian Item Response Models for Estimating Person-Specific Latent Traits. The manuscript was submitted under a Rasch-only title; its revision extends selected questions to the two-parameter model, and the title here follows that broader umbrella. The umbrella is Rasch-centred, not a claim of parallel depth across the two models: Parts I–VII form the theory spine, and Part VIII is a focused 2PL extension of the mechanisms for which free discriminations materially change the argument. The book sets out the mathematics that paper rests on, traces each result to the work that established it, and says plainly where the paper’s own statements need correcting.
Four documents divide the work, and it helps to know which is authority for what.
| Question it answers | Authority for | |
|---|---|---|
| This book | What is the mathematics, and who established it? | Definitions, classical results, identification, the literature genealogy, decision-theoretic justifications |
| The simulation volume (Lee 2026a) | What did we run, and what did it show? | The design grid, realized values, Monte Carlo error, gate verdicts, results |
| The case-study volume (Lee 2026b) | What changes on real tests? | Thirteen Item Response Warehouse cases, both item models, consequence and transfer — with no correctness claims of its own |
| The manuscript | What is the contribution? | The paper’s claims, and nothing else |
Around these sit the software the models are fitted with — DPMirt (Lee 2026d), DPprior (Lee 2026e), and IRTsimrel (Lee 2026h) — the two methods papers the packages implement (Lee 2026i, 2026c), and two corpus studies of the Item Response Warehouse that measure what real tests look like: one of reliability across 889 datasets (Lee 2026g) and one of latent-distribution shape across 504 (Lee 2026f). Where a chapter leans on any of them, it says so in its provenance section.
The book has three evidence zones, not a blanket simulation-independence rule.
| Zone | Role | Boundary rule |
|---|---|---|
| Parts I–VII | Rasch-centred theory spine | Definitions, mechanisms, sourced results, and derivations lead. A realized companion result may enter only as a registered, explicitly attributed inset; it does not become a premise of the theory. |
| Part VIII | Focused 2PL extension | Develops the information, reliability, and identification changes created by free discriminations. Its realized simulation and case-study passages are explicitly labelled empirical insets and remain bounded by their source designs. |
| Part IX | Evidence synthesis | Reads the companion volumes’ realized results back into the theory. Frozen-source assertions and confirmatory/exploratory labels govern every imported result. |
This zoning admits the empirical crossings that are already useful in earlier chapters without pretending that Parts I–VIII would stand unchanged if the studies had never been run. The cross-book register records each crossing and the claim boundary it must keep.
Who this is for, and three ways in
A practitioner deciding how to score a short test: 1 The Estimation Problem for the problem, 8 Reliability in the Rasch Model for what reliability means and which coefficient answers which question, 17 Three Goals, Three Losses for why one set of scores cannot serve every purpose, 20 Ranking and Classification for ranking and cut-scores, 25 Reliability Under the Two-Parameter Model for what changes when items are allowed to differ in discrimination, and 27 What the Simulation Settled and 28 What Real Tests Look Like for what the simulation and the real-test corpus actually found.
A methodologist assessing the work: Parts II through VI in order, followed by the focused 2PL sensitivity route in Part VIII (24 Discrimination and Information through 26 Identification and the DPM Under the Two-Parameter Model). The claim-level index in Appendix A — Proofs distinguishes proved-here, partial-derivation, and source-only; nothing is asserted without an argument or a source locator.
A reviewer checking a specific point: Appendix C — Corrections to the Submitted Manuscript lists every place the submitted manuscript is wrong, imprecise, or merely unclear, with the section that fixes it.
How to read a numbered result
Every definition, theorem, proposition, lemma, and corollary carries a provenance tag:
Theorem 8.3 (Jensen gap) · restated from Lee (2026, Prop. 2)
Restated — established in the cited source, reproduced here, possibly in different notation. Adapted — established there for a different setting, modified here, with the modification stated. Derived here; no originality claim — proved in this text without claiming that the result is new to the literature.
An explicit novelty or priority claim would require a separate prior-art search. None of the current derived-here tags makes such a claim.
Restated and adapted numbered results carry a page-, section-, equation-, or theorem-level locator, and V1 requires a locator-read source for every one of them. Independent literature is primary-read; this programme’s own directly inspected books, manuscripts, packages, code, and tables are programme-artifact-read. The latter tier establishes fidelity to the named artifact, not independent corroboration.
Narrative citations are held to a weaker standard, and the tier says which is which. A source tagged primary-held was available but has not been read at a locator for this book. Such a source may be named in the text — to point at a literature or credit a lineage — but it carries no numbered result and no equation-level attribution. The gap is disclosed, not hidden, and Appendix D — Annotated Bibliography separately annotates 74 independent reads and 8 programme-artifact reads.
What the sources are, and what they are not
The recursive source inventory spans nine reference libraries: 609 PDFs and 563 Markdown sidecars representing 660 indexed assets. The bibliography contains 377 entries: 74 primary-read, 8 programme-artifact-read, 292 primary-held, and 3 secondary.
Parts II through IV were seeded by six appendix drafts written during the manuscript’s first revision. Those drafts were machine-generated by four different models and never reconciled with one another. They carry no citation authority here. Every statement taken from them has been re-derived or re-verified against the source named in the chapter’s provenance section, and the audit that established which of their claims were unsourced — 64 of 106 numbered results under the corrected paired-block audit — is recorded in the build log. Saying this is not modesty; it is the same standard the book applies to everything else it cites.
Reproducing this book
Rscript code/R/14-build-all.RThis command harvests and validates the bibliography, regenerates tables and figures, renders the book, and runs 9 verifiers (V1–V9) together with 4 claim-level checkers, including 63 numerical checks in V8. Chapters read only frozen tables under tables/ and figures under book/figures/. They compute nothing.
What that machinery does and does not guarantee is worth stating exactly, because an earlier edition of this page overstated it. Every table and figure is generated by a script in code/R/ and regenerated on every build, so a displayed artifact cannot drift from the code that made it. Prose, however, is written by hand. Where a sentence in Part IX quotes a companion volume, the binding is not the rendering pipeline but manifest/evidence-claim-register.csv, which records 54 imported claims with their chapter, line, source edition, snapshot locator and field. 11 of those are recomputed from the frozen store at build time and fail the build on divergence; the rest are registered for audit without being executed. Four claim-level checkers sit outside V1–V9 gives the full contract. The discipline exists because a hand-copied number goes stale silently, and this project has been bitten by exactly that — including once in the edition before this one.
The registers that make the provenance checkable are under manifest/: every numbered result, every notation symbol, every declared restatement from the companion volume, every reviewer comment and the section that answers it, and the running list of corrections to the manuscript.