| ID | Result | Provenance | Proof | Proof status | Coverage | Source locator | |
|---|---|---|---|---|---|---|---|
| 2 | PROP-03-2 | prp-nosuff | derived-here | inline, ch. 3 | proved-here | fixed-known-discrimination factorization and statistic qualification are displayed inline | elementary factorization and definition of a statistic; no originality claim |
| 4 | PROP-04-2 | prp-location | derived-here | inline, ch. 4 | proved-here | one-line location-orbit algebra | elementary inline algebra; no originality claim |
| 5 | PROP-04-3 | prp-scale | derived-here | inline, ch. 4 | proved-here | scale-orbit algebra | elementary inline algebra; no originality claim. Consistent with San Martin (2016, Table 8.1) |
| 10 | PROP-06-2 | prp-plugin | derived-here | inline, ch. 6 | proved-here | implicit-function and delta-method derivation | first-order implicit-function/delta-method derivation; no magnitude claim from scoring-sample P alone |
| 11 | PROP-07-1 | prp-msem-jensen | derived-here | inline, ch. 7 | proved-here | Jensen argument | elementary concavity argument; no originality claim |
| 12 | PROP-08-1 | prp-mean-weight | derived-here | inline, ch. 8 | proved-here | convexity argument | elementary convexity; corresponds to Appendix C draft (C.23)-(C.24); no originality claim |
| 15 | PROP-09-1 | prp-cstar | restated | Appendix A A.3.2 | proved-here | fixed-item discrimination-multiplier ordering under unique increasing branches | vignettes/theory-reliability.Rmd Corollary 1; the non-monotonicity caveat is the package’s own wording and is preserved |
| 16 | PROP-10-1 | prp-joint-posterior | adapted | inline, ch. 10 | proved-here | displayed posterior factorization for the book model | ch. 2 sec. 2.1 eqs. 2.1-2.2 give the person/item conditional-independence factorizations; adapted here to the four-level centered-item model |
| 17 | PROP-10-2 | prp-centered-item | derived-here | inline, ch. 10 | proved-here | centering-matrix covariance algebra | elementary linear algebra on the centering matrix; checked numerically; no originality claim |
| 19 | PROP-11-3 | prp-underdispersion | adapted | inline, ch. 11 | proved-here | law-of-total-variance proof and constant-error specialization | sec. 2.1 gives the Gaussian ensemble under-dispersion and moment repair; the general law-of-total-variance identity and constant-error specialization are proved inline |
| 20 | PROP-12-3 | prp-rank-inconsistency | adapted | inline, ch. 12 | proved-here | Rasch and fixed-known-2PL MLR exception is proved; generic warning is sourced | sec. 1 gives the generic warning and Appendix A the stochastic-order exception; Shen and Louis (1998) Theorem 2 corroborates the exception; the Rasch/known-2PL monotone-likelihood-ratio verification is inline |
| 21 | PROP-13-1 | prp-sumscore | derived-here | inline, ch. 13 | proved-here | finite-grid Poisson-binomial exhibits with frozen verification | exact Poisson-binomial recursion plus 4,001-point fixed-grid quadrature on [-12,12]; the structural dependence of score shape on G and item design is not claimed as original |
| 26 | PROP-15-5 | prp-discrete | restated | Appendix A A.1.2 | proved-here | immediate from the stick-breaking representation | immediate from the stick-breaking form; Ferguson sec. 4 gives an independent construction |
| 27 | PROP-16-1 | prp-semi-location | adapted | inline, ch. 16 | proved-here | shift-orbit proof | sec. 1 identification problem, adapted with the explicit shift orbit from Appendix F sec. F.2.2 |
| 28 | PROP-16-2 | prp-semi-scale | adapted | inline, ch. 16 | proved-here | Rasch/2PL scale-orbit argument | secs. 8.4-8.7 and Table 8.1 for the Rasch/2PL scale distinction; Appendix F sec. F.2.3 and inline push-forward extend it to G |
| 29 | PROP-16-3 | prp-dpm-constraint | adapted | Appendix A A.2.2 | proved-here | finite positive base-variance assumptions and realized-mean variance identity | sec. 1 supports the ordinary-base-centering statement; the chapter explicitly does not generalize it to every constrained or transformed nonparametric prior |
| 31 | PROP-17-1 | prp-pm-bayes | restated | inline, ch. 17 | proved-here | posterior-risk decomposition | sec. 2, read directly; the decomposition including the irreducible posterior-variance term is theirs; source lambda_p is translated to registered v_p under TD-3 |
| 35 | PROP-18-2 | prp-cb-affine | adapted | inline, ch. 18 | proved-here | positive-affine invariance algebra | sec. 3 p. 459 states rank invariance; the standardized-moment and shape statements are the book’s elementary positive-affine extension |
| 37 | PROP-19-2 | prp-gr-edf | restated | inline, ch. 19 | proved-here | permutation preserves the discrete empirical CDF | sec. 2.4 p. 459; the permutation argument is exact, while the explicit tie-block completion and verification are supplied here |
| 38 | PROP-19-3 | prp-gr-regret | restated | inline, ch. 19 | proved-here | actual-permutation regret identity and conditional sorted form | sec. 4.2 eq. (14) p. 462 gives the sorted form after adopting posterior-mean ranks, which the source says agree with optimal ranks for a wide class; the book states the actual-permutation identity first and checks both cases in V8 |
| 40 | PROP-26-1 | prp-2pl-pattern-id | derived-here | inline, ch. 26 | proved-here | finite response-pattern normalization and explicit K by K+1 null-space construction of two observationally equivalent mixing distributions | finite response alphabet and a K by K+1 null-space construction with K=2^I; no originality claim; the free-item sharp analogue is explicitly not asserted |
| 1 | THM-03-1 | thm-suff | restated | inline, ch. 3 | proved-here | Rasch factorization and sufficiency argument are displayed inline | secs. 2.4.1 and 2.4.3; direct attribution to Debelak et al. (2022), not Andersen (1970) |
| 6 | THM-04-4F | thm-equiv-fixed | derived-here | inline, ch. 4 | proved-here | fixed-effects orbit proof | elementary orbit algebra; no originality claim; fixed-effects scope |
| 7 | THM-04-5R | thm-equiv-random | derived-here | inline, ch. 4 | proved-here | random-effects change-of-variable proof | change-of-variable proof; no originality claim; random-effects scope |
| 14 | THM-08-3 | thm-jensen-sandwich | adapted | inline, ch. 8 | proved-here | three-term Jensen adaptation is proved inline | vignettes/theory-reliability.Rmd Theorem 1 gives only the outer inequality; the placement of rho_Theta between the two and the organized three-term proof are this book’s Jensen adaptation |
| 18 | THM-11-1 | thm-eap-shrinkage | adapted | inline, ch. 11 | proved-here | normal-working-model completing-the-square proof | sec. 2.1 Gaussian compound model; adapted to the IRT normal-working likelihood and proved inline by completing the square |
| 25 | THM-15-4 | thm-antoniak | restated | Appendix A A.1.4; inline summary ch. 15 | proved-here | pmf, expectation, nonatomic qualification, and partition-model sufficiency are proved | sec. 4, printed pp. 1160-1163; the digamma closed form is one line from his sum and is verified numerically in V8 |
| 33 | THM-17-3 | thm-incompatible | adapted | inline, ch. 17 | proved-here | explicit nondegenerate two-posterior counterexample with frozen values | secs. 2.1-2.4 supply the three Bayes actions and Theorem 1’s midpoint-quantile ISEL solution; Paddock et al. abstract/sec. 2.1 supplies the general incompatibility motivation; the narrower existence statement and correctly computed normal example are this book’s and checked against a frozen table in V8 |
| 34 | THM-18-1 | thm-cb | adapted | inline, ch. 18 | proved-here | book specialization and existence boundary are derived from the sourced general rule | Theorem 1 and proof pp. 534-535 give the general rule; Remarks 1-2 give inflation and nonexistence conditions; V8 checks the independent-posterior form, its exact equality to the package expression through R’s P-1 sample-variance denominator, and the omitted cross-person covariance term |
| 13 | PROP-08-2 | prp-separation | restated | Appendix A A.3.1 | partial-derivation | separation identity is proved; strata convention remains source-only | sec. 5.5; read directly at intake, C-002 and C-003 confirm the manuscript attribution |
| 30 | PROP-16-4 | prp-partial-id | restated | Appendix A A.2.1 | partial-derivation | pattern factorization and algebraic identity are proved; source inversion theorem remains source-only | Theorem 5 and expression (11), not Theorem 3 as the Appendix F draft states (see C-014); Theorem 6 gives the asymptotic conditions; no equivalence with a selected pair’s total-variation separation is claimed |
| 32 | PROP-17-2 | prp-rank-of-mean | restated | Appendix A A.4.1 | partial-derivation | value distinction is proved; stochastic-order conclusion invokes Shen-Louis Theorem 2 | sec. 2.1 p. 457 gives the expected-rank action and its generally noninteger values; sec. 3 Theorem 2 p. 459 gives induced-order agreement under stochastic ordering; Goldstein & Spiegelhalter (1996) supplies the rank-uncertainty context |
| 39 | PROP-20-1 | prp-rank-reliability | adapted | inline, ch. 20 | partial-derivation | Gaussian source statement plus independently computed matched-reliability Rasch grid | sec. 2 gives the Gaussian statement under its model conditions; the Rasch calculation uses full-support midpoint-quantile quadrature, common-wbar interpolation, a frozen 400-versus-800-node convergence receipt, and a unit-variance logistic heavy-tail working shape in code/R/02-rank-concordance.R and tables/F-rank-reliability-*.rds |
| 23 | THM-15-2 | thm-stick | restated | Appendix A A.1.1 | partial-derivation | weight sum, expected weights, and remainder are proved; DP finite-dimensional law remains source-only | sec. 1 construction; Theorems 3.4 and 4.3 for the Dirichlet marginals and conjugacy |
| 24 | THM-15-3 | thm-crp | restated | Appendix A A.1.3 | partial-derivation | partition rule is derived; martingale convergence and DP limit remain source-only | abstract and sec. 1; the reduction of the restaurant metaphor to this theorem is this book’s presentation and answers R2-4b |
| 36 | THM-19-1 | thm-gr | restated | Appendix A A.4.2 | partial-derivation | loss targets and discrete/continuous EDF distinction are derived; source construction remains attributed | sec. 2.1 p. 457 gives expected ranks; sec. 2.3 pp. 457-458 gives ISEL and midpoint quantiles; sec. 2.4 pp. 458-459 gives the three-step construction and optimal assignment; uniform randomization within exact tie blocks is the book’s exchangeability-preserving completion |
| 3 | DEF-04-1 | def-ident | restated | source locator only | source-only | definition is restated from the registered locator | sec. 8.2.2 (p. 131) |
| 22 | DEF-15-1 | def-dp | restated | source locator only | source-only | Ferguson finite-dimensional definition; no Appendix A proof is claimed | secs. 1 and 3; alpha is a MEASURE, whose total mass is the concentration and whose normalization is the base measure |
| 8 | THM-05-1 | thm-neyman-scott | restated | source locator only | source-only | Haberman/Andersen theorem is stated with conditions; published proof is not reproduced | sec. 9.2.1 (pp. ~152-154), reporting Andersen (1973, pp. 66-69) |
| 9 | THM-06-1 | thm-warm | restated | source locator only | source-only | Warm theorem and assumptions are restated at the registered locator | eqs. (6)-(10) and theorem, pp. 430-432; Appendix assumptions and separate variance proof, pp. 444-448 |
Appendix A — Proofs
Chapter 2 states the rule this appendix implements: proofs run inline where they take less than a page and move here otherwise, and nothing is asserted without either an argument or a source locator. The two alternatives must not be blurred. Reproducing a useful special-case derivation is not the same as reproducing the source theorem that licenses it.
The index therefore records a claim-level proof status. proved-here means the displayed claim is established in this book; partial-derivation means this appendix establishes only the named component and leaves the source theorem at its locator; source-only means the published result is invoked under its stated conditions and no stand-alone proof is claimed here. This appendix improves checkability, but it does not promise to replace every cited proof.
A.1 Index
Two things are worth reading off this table. Every result tagged derived-here has a proved-here argument. Restated or adapted results can be proved-here, partial-derivation, or source-only; none of those labels is an originality claim.
A.2 A.1 The Dirichlet process
Chapter 15 defines the process and states three structural results from their sources. The arguments are short enough to give in full, and giving them makes Part V self-contained.
A.2.1 A.1.1 Stick-breaking
Source result (Theorem 15.1, restated from Sethuraman 1994). Let \(V_1, V_2, \dots\) be i.i.d. \(\mathrm{Beta}(1,\alpha)\), let \(\phi_1, \phi_2, \dots\) be i.i.d. from the normalized base measure \(\bar G_0\) and independent of the \(V\)’s, and set
\[p_n = V_n \prod_{\ell < n} (1 - V_\ell), \qquad P = \sum_{n=1}^{\infty} p_n \delta_{\phi_n} .\]
Then \(P \sim \mathrm{DP}(\alpha \bar G_0)\).
Proof status: partial derivation. The two weight facts used later are proved below. The finite-dimensional Dirichlet law of \(P\) is Sethuraman’s theorem and remains source-only.
Proof of the two facts the chapter uses. The weights sum to one almost surely. Writing \(R_N = \prod_{\ell \le N}(1 - V_\ell)\) for the unbroken remainder after \(N\) breaks, induction on the definition gives \(\sum_{n \le N} p_n = 1 - R_N\). Since the \(V_\ell\) are i.i.d. with \(\operatorname{E}[\log(1-V_\ell)] = \int_0^1 \log(1-v)\,\alpha(1-v)^{\alpha-1}\,dv = -1/\alpha\), which is a strictly negative finite constant, \(\log R_N\) is a random walk with negative drift and \(R_N \to 0\) almost surely. Hence \(\sum_n p_n = 1\).
The mean weight sequence is geometric. \(\operatorname{E}[V_n] = 1/(1+\alpha)\) and \(\operatorname{E}[1 - V_\ell] = \alpha/(1+\alpha)\), and independence across \(\ell\) gives
\[\operatorname{E}[p_n] = \frac{1}{1+\alpha}\left(\frac{\alpha}{1+\alpha}\right)^{n-1} = \frac{\alpha^{n-1}}{(1+\alpha)^n} .\]
So the expected weights decay geometrically at rate \(\alpha/(1+\alpha)\), and the expected mass left after \(N\) atoms is \((\alpha/(1+\alpha))^N\), which is the truncation bound Chapter 15 uses. V8 checks all three statements numerically (THM-15-2, THM-15-2b, THM-15-2c).
That the resulting \(P\) has Dirichlet finite-dimensional distributions is Sethuraman’s theorem and is not reproved here; the locator is in the index. □
A.2.2 A.1.2 A draw is discrete with probability one
Claim (Proposition 15.1, restated; immediate from Theorem 15.1).
Proof. By A.1.1 the random measure is \(\sum_n p_n \delta_{\phi_n}\) with \(\sum_n p_n = 1\) almost surely. A countable convex combination of point masses is a discrete measure. The conclusion holds however smooth \(\bar G_0\) is: continuity of the base measure controls where the atoms fall, not whether there are atoms. That is the fact Chapter 16 needs when it warns that a DPM’s mixing distribution is discrete even under a Gaussian base. □
A.2.3 A.1.3 The Pólya urn and the CRP
Source result (Theorem 15.2, restated from Blackwell and MacQueen 1973). If \(\theta_1, \theta_2, \dots\) follow the Pólya urn scheme
\[\theta_{n+1} \mid \theta_1, \dots, \theta_n \;\sim\; \frac{\alpha}{\alpha + n}\,\bar G_0 + \frac{1}{\alpha + n}\sum_{i \le n} \delta_{\theta_i},\]
then the empirical measure converges almost surely to a discrete random measure \(P\), and \(P \sim \mathrm{DP}(\alpha \bar G_0)\) with the \(\theta\)’s i.i.d. \(P\) given \(P\).
Condition for the chapter’s distinct-value reading. The seating partition exists for any base measure. Identifying “a new table” with “a new observed value” additionally requires \(\bar G_0\) to be nonatomic; otherwise a fresh base-measure draw can coincide with a value already observed.
Proof of the partition statement the book uses. Read the scheme as a seating rule. Draw \(n+1\) joins an existing cluster of size \(m_k\) with probability \(m_k/(\alpha+n)\) and starts a new one with probability \(\alpha/(\alpha+n)\). The induced distribution on partitions is therefore exchangeable and depends on the labels only through the cluster sizes — the Chinese restaurant process. Marginalizing the urn over the \(\phi\)’s in A.1.1 gives the same partition law, which is the correspondence Chapter 15 uses to move between the two representations. The almost-sure convergence and the identification of the limit as a DP are Blackwell and MacQueen’s martingale argument, whose locator is in the index. □
A.2.4 A.1.4 Antoniak’s law for the number of occupied clusters
Claim (used in Chapter 15 and Chapter 16). Under the CRP with concentration \(\alpha\), the number \(K_J\) of occupied tables among \(J\) draws has
\[ \Pr(K_J=k\mid\alpha)= \frac{|s(J,k)|\alpha^k}{(\alpha)_J}, \qquad (\alpha)_J=\alpha(\alpha+1)\cdots(\alpha+J-1), \]
and
\[\operatorname{E}[K_J \mid \alpha] = \sum_{j=0}^{J-1} \frac{\alpha}{\alpha + j} = \alpha\left[\psi(\alpha + J) - \psi(\alpha)\right],\]
where \(\psi\) is the digamma function. If \(\bar G_0\) is nonatomic, occupied tables and distinct observed values coincide. Moreover, for fixed \(J\), \(K_J\) is sufficient for \(\alpha\) in the induced partition model.
Proof. Let \(B_j\) indicate that draw \(j\) starts a new table, \(j = 1, \dots, J\), with \(B_1 = 1\). By the seating rule of A.1.3, \(\Pr(B_{j+1} = 1 \mid \text{first } j) = \alpha/(\alpha + j)\), which does not depend on the realized partition. So the \(B\)’s are independent Bernoulli variables. Their probability-generating function is
\[ \operatorname{E}[t^{K_J}\mid\alpha] =\prod_{j=0}^{J-1}\frac{\alpha t+j}{\alpha+j} =\frac{\sum_{k=1}^{J}|s(J,k)|(\alpha t)^k}{(\alpha)_J}, \]
where the last equality is the rising-factorial expansion for the unsigned Stirling numbers of the first kind. Extracting the coefficient of \(t^k\) gives the pmf. Also \(K_J=\sum_j B_j\) gives the expectation, and \(\psi(x+1)-\psi(x)=1/x\) gives the digamma form.
For sufficiency, the CRP probability of a particular partition with block sizes \(n_1,\ldots,n_k\) is
\[ \frac{\alpha^k}{(\alpha)_J}\prod_{h=1}^{k}(n_h-1)!. \]
Its dependence on \(\alpha\) is only through \(k=K_J\), so the factorization criterion makes \(K_J\) sufficient for \(\alpha\) in this partition model. □
The independence is the step worth noticing: it is what makes \(\operatorname{E}[K_J]\) available in closed form, and it is why the elicitation of Chapter 15 can invert the relation to choose a prior on \(\alpha\) from a belief about \(K\). V8 checks the identity and the elicitation against it (THM-15-4, THM-15-4b, THM-15-4c).
A.3 A.2 Identification of \(G\) from a finite test
A.3.1 A.2.1 What a finite test identifies
Claim (Proposition 16.4, restated from San Martín et al. 2011, Theorem 5). Under the source’s anchor-item and availability conditions for the semiparametric Rasch model, the observable law is generated by the item parameters and the following \(I+1\) displayed basic integral evaluations of \(G\).
Argument. The observable is the distribution of the response vector \(\mathbf{U} \in \{0,1\}^I\), which is \(2^I\) probabilities summing to one. Under local independence and the Rasch form, the probability of any pattern with total score \(r\) is
\[\Pr(\mathbf{U} = \mathbf{u}) = \int \prod_{i} \frac{e^{u_i(\theta - \beta_i)}} {1 + e^{\theta - \beta_i}} \, dG(\theta),\]
Writing \(r=\sum_i u_i\), the pattern probability equals \(\exp(-\sum_i u_i\beta_i)\mu_r\), so the whole observable law is a function of the item parameters and of the \(I+1\) evaluations
\[\mu_r = \int \frac{e^{r\theta}}{\prod_i (1 + e^{\theta - \beta_i})}\, dG(\theta), \qquad r = 0, 1, \dots, I,\]
one for each attainable total score. These evaluations are not unconstrained independent coordinates. Summing all pattern probabilities gives the identity
\[ \sum_{r=0}^{I}e_r(e^{-\beta_1},\ldots,e^{-\beta_I})\mu_r=1, \]
where \(e_r\) is the \(r\)th elementary symmetric polynomial. Two latent distributions producing the same displayed evaluations are observationally equivalent. San Martín et al. invert the observable map under their source conditions; this argument establishes the factorization and identity, not an independent-dimension or universal “no more” claim.
The full inversion under the anchor restriction is their Theorem 5 and remains source-only. Their Theorem 3 concerns the Rasch Poisson counts model and gives full identification of \(\beta\) and \(G\) with at least two probes; conflating the two is the manuscript error recorded as C-014. □
A.3.2 A.2.2 A centred base measure does not centre the realized DPM
Claim (Proposition 16.3, adapted from San Martín et al. 2011). If \(G \sim \mathrm{DP}(\alpha G_0)\), \(\int \theta \, dG_0(\theta) = 0\), and \(0<\operatorname{Var}_{G_0}(\theta)<\infty\), then the realized mean \(M_G=\int\theta\,dG(\theta)\) is nondegenerate, with
\[ \operatorname{E}(M_G)=0, \qquad \operatorname{Var}(M_G)=\frac{\operatorname{Var}_{G_0}(\theta)}{1+\alpha}. \]
Proof. By A.1.1, \(\int \theta\,dG(\theta) = \sum_n p_n \phi_n\) with the \(\phi_n\) i.i.d. from \(\bar G_0\) and the \(p_n\) the stick-breaking weights, independent of them. Conditional on the weights, this is a weighted sum of mean-zero draws, so its conditional mean is zero and its conditional variance is \(\sigma^2_{G_0} \sum_n p_n^2\), which is finite and strictly positive under the stated assumption because the weights sum to one. The realized mean is therefore a non-degenerate mean-zero random variable, not the constant zero.
Two consequences the book uses. Centring the base measure is not the same restriction as centring the realized distribution, so a location constraint imposed on the base does not identify the model; and \(\operatorname{E}[\sum_n p_n^2] = 1/(1+\alpha)\) gives the displayed unconditional variance. Thus the discrepancy shrinks as \(\alpha\) grows and is largest exactly where the prior is most flexible. □
A.4 A.3 Reliability, separation, and the design constant
A.4.1 A.3.1 Separation and strata
Claim (Proposition 8.2, restated from Wright and Masters 1982, § 5.5). With \(\bar w = \sigma^2/(\sigma^2 + \mathrm{MSEM})\), the person separation index and the number of strata are
\[S = \frac{\sigma}{\mathrm{RMSEM}} = \sqrt{\frac{\bar w}{1 - \bar w}}, \qquad H = \frac{4S + 1}{3} .\]
Proof of the separation identity. \(\bar w/(1-\bar w) = \sigma^2/\mathrm{MSEM}\), and taking square roots gives \(S = \sigma/\mathrm{RMSEM}\).
The strata formula is not derived here, and Wright and Masters do not derive it either. They state \(H = (4G+1)/3\) for both the item and the person index and give the convention it encodes — statistically distinct strata have centres three calibration or measurement errors apart — without showing where the constants come from. That convention is the substantive content, it is what the manuscript had used without a source, and it is recorded as C-003. This book restates the formula on their authority; a reader wanting the constants justified will not find it in the source, and inventing a derivation here would misrepresent what is known.
This is the identity that exposes C-001. Computing \(S\) from a rounded RMSEM rather than from the exact target reliability is what produces the manuscript’s \(1.3\), \(1.4\) and \(3.3\) where the exact column is \(1.0\), \(1.2\), \(1.5\), \(2.0\), \(3.0\). V8 checks both readings (PROP-08-2, C-001, C-001b). □
A.4.2 A.3.2 The design constant, and the condition it needs
Claim (Proposition 9.1, deferred from Chapter 9). Fix the item count, difficulties, baseline discriminations, and population distribution. Let \(c>0\) multiply every discrimination, and suppose both reliability functions have unique strictly increasing branches through the target. The discrimination multiplier required under the MSEM definition is at least the multiplier required under the average-information definition.
Proof. At a fixed value of the discrimination multiplier \(c\), let \(\mathcal{J}_c(\theta)\) be the information computed with both the \(c\)-scaled slopes and the resulting \(c\)-dependent response probabilities. MSEM is the mean of the conditional error variances, \(\operatorname{E}_\theta[1/\mathcal{J}_c(\theta)]\); the average-information summary is the reciprocal of the mean information, \(1/\operatorname{E}_\theta[\mathcal{J}_c(\theta)]\). Since \(x \mapsto 1/x\) is strictly convex on \((0,\infty)\), Jensen gives
\[\operatorname{E}_\theta\!\left[\frac{1}{\mathcal{J}_c(\theta)}\right] \;\ge\; \frac{1}{\operatorname{E}_\theta[\mathcal{J}_c(\theta)]},\]
with equality if and only if \(\mathcal{J}_c(\theta)\) is constant in \(\theta\). So the MSEM error is the larger and the MSEM reliability is no larger at every common \(c\). On the assumed unique increasing branches, the inverse image of a common target is therefore reached no earlier by \(\bar w(c)\) than by \(\tilde\rho(c)\), so \(c^\star_{\bar w}\ge c^\star_{\tilde\rho}\). The strictness of the inequality away from constant information is why the two definitions diverge exactly in the item-response setting, where information depends on \(\theta\) by construction (Chapter 7).
The uniqueness condition is not cosmetic: without a single strictly increasing branch the inverse image is not a point and “the required discrimination multiplier” is not well defined. V8 checks the inequality and, separately, that scaled 2PL information is not obtained by freezing the response probabilities (PROP-09-1, PROP-09-1b). □
A.5 A.4 Posterior summaries
A.5.1 A.4.1 Rank of the mean against mean of the rank
Claim (Proposition 17.2, restated from Shen and Louis 1998). The rank of the posterior mean and the posterior mean of the rank differ in value in general, and differ in ordering only outside stochastically ordered regimes.
Proof. For the value statement, the expected rank of unit \(k\) is \(\bar R_k = \sum_q \Pr(\theta_q \le \theta_k \mid \mathbf{u})\), which for independent posteriors is a sum of \(K\) probabilities and takes values in \([1, K]\) on a continuous scale, while \(\operatorname{rank}(\eta_k)\) is an integer in \(\{1, \dots, K\}\). They are different objects and coincide only by accident.
For the ordering statement, the tempting term-by-term comparison of the self and cross summands is false. The valid result is Shen and Louis’s Theorem 2: under their stochastic ordering condition, posterior means, expected ranks, and the relevant rank summaries induce the same order. That published theorem is invoked at its locator rather than replaced by the invalid summand argument. The converse fails: two normals with different variances are not stochastically ordered, their standardized arguments cross, and the expected-rank ordering can then disagree with the ordering of the means.
Chapter 12 supplies the case that matters for this book. In a common-form Rasch model the posterior depends on the data only through the total score and the family is ordered in it, so the regime is the stochastically ordered one and the orderings agree — which is why the correction recorded there withdrew an earlier claim that differential shrinkage reorders. □
A.5.2 A.4.2 The triple-goal construction
Claim (Theorem 19.1, restated from Shen and Louis 1998, §§ 2.1, 2.3, 2.4). The triple-goal estimator is built in three steps: form \(\bar G_K(t) = K^{-1}\sum_k \Pr(\theta_k \le t \mid \mathbf{u})\) and take its \(K\) midpoint quantiles \(\hat U_j = \bar G_K^{-1}\!\left(\frac{2j-1}{2K}\right)\); compute expected ranks \(\bar R_k\) and discretize them to integer ranks \(\hat R_k\); and set \(\hat\theta_k = \hat U_{\hat R_k}\).
What each step optimizes. Step 1: the unrestricted pointwise squared-error Bayes action for the ensemble EDF is the continuous posterior-mean function \(\bar G_K\). When the action is restricted to empirical CDFs with \(K\) equal masses, the ISEL-optimal discrete action is \(\hat G_K\), whose atoms are the midpoint quantiles \(\hat U_j\) of \(\bar G_K\). In general \(\hat G_K\ne\bar G_K\); discretization approximates the continuous Bayes action and does not carry it exactly. Step 2: the expected rank minimizes squared-error rank loss, by the same posterior-mean argument applied to \(R_k\); the source construction converts those real-valued actions to a permutation. Step 3 assigns the ordered atoms by that expected-rank permutation. Separately, the rearrangement inequality says the coordinate-SEL-minimizing permutation pairs the ordered atoms with the ordered posterior means. These are the same assignment only when the expected-rank order and posterior-mean order agree.
Two things follow, both used in Chapter 19. Every permitted step-3 assignment is a permutation of the atoms and therefore carries the discrete action \(\hat G_K\) exactly. It does not carry the continuous \(\bar G_K\) exactly. And \(\bar G_K(t)\) is a probability — an average of \(K\) posterior probabilities — not a trait estimate. □
A.6 A.5 What is deliberately not here
The proofs of Sethuraman’s finite-dimensional DP theorem, Blackwell and MacQueen’s martingale convergence, San Martín et al.’s inversion theorem, Ghosh’s general constrained-Bayes theorem, and Shen and Louis’s stochastic-order theorem are not reproduced. Each remains source-only at the locator in the claim-level inventory. What is established above is narrower: the weight identities, partition/Antoniak algebra, finite-test factorization and identity, realized-DP mean variance, reliability ordering at common discrimination scale, and the correct distinction among \(\bar G_K\), \(\hat G_K\), rank loss, and conditional coordinate-SEL assignment.