{"id":"a16bfc33-27b2-4a1e-834b-9c955b16b66c","arxiv_id":"1908.01626","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"An analytic Fisher-based formula computes the covariance between parameters from two correlated data sets; applied to SPTpol TE and EE spectra, it shows the constraints are weakly positively correlated and that ignoring this correlation biases consistency tests.","lead":"This paper derives an analytic formula for how much two correlated cosmological data sets' parameter estimates are correlated, based on the data covariance and a Fisher approximation. Applying it to SPTpol CMB temperature-polarization spectra, the authors show that ignoring these correlations can make inconsistent data look artificially consistent.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claim that parameter-based and bandpower-based consistency tests are independent is unsupported; simulations only show zero correlation, not independence.","rationale":"The analytic formula Eq. (7) is correct under the stated assumptions and is well validated by the maximum-likelihood simulations in Section 3.2. I found no error in the derivation or in the numerical implementation. However, the paper's interpretation of the SPTpol application relies on the claim that the parameter-based consistency test and the bandpower-based consistency test are independent. The evidence offered is a finding of \"no correlation\" between the two PTE values from 1000 simulations. This does not establish independence: for zero-mean Gaussian data, a linear combination (the parameter difference) and a quadratic form (the chi-square residual) always have zero third-order cross-moments, so they are uncorrelated even when dependent. An analytic calculation shows the two quadratic forms are in fact not independent unless a certain matrix product vanishes, and that product is nonzero here. The reader's weakest assumption identified the dependence on the SPTpol covariance and LambdaCDM, which is a legitimate concern for the application. But the independence claim is a separate, more specific issue that is directly load-bearing for the abstract's central message about the PTE discrepancy. The paper should be accepted only after the independence claim is corrected or replaced with a formal independence test, and the discussion of the discrepancy should be rephrased to avoid asserting independence without support.","tokens_in":14370,"tokens_out":22615,"duration_ms":223707,"concrete_test":"Run 100,000 simulated SPTpol TE+EE data vectors drawn from the fiducial model used in Section 3.2. For each realization compute Q_param (the parameter-difference chi-square) and Q_band (the joint-fit bandpower chi-square), or their PTE values. Then (1) apply a chi-square test of independence on a two-dimensional histogram of the two PTE values, or (2) estimate the mutual information between them. Also compute the matrix A C B analytically (with A and B defined in the load-bearing attack) to confirm it is nonzero. If the tests reject independence or A C B is nonzero, the paper's independence claim must be revised. Even a 10,000-realization run with a contingency-table test should detect the dependence if it is not extremely weak.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Section 3.3 the authors state \"there is no correlation between the PTE for the consistency of the parameters and the PTE for the consistency of the simulated TE and EE spectra,\" and the abstract concludes that \"the results of these two tests are independent.\" For Gaussian data, a linear form (the parameter difference Delta = theta_TE - theta_EE) and a quadratic form (the joint-fit bandpower chi-square residual) can be dependent even when their PTE values are uncorrelated: odd moments of zero-mean Gaussians vanish, so the covariance of a linear and a quadratic form is identically zero. Using Craig's theorem, the two quadratic forms Q_param = Delta^T Cov(Delta)^{-1} Delta and Q_band = r^T C^{-1} r are independent only if A C B = 0, where A = L^T (L C L^T)^{-1} L, B = C^{-1} - C^{-1} D F^{-1} D^T C^{-1}, L maps data to Delta, D is the joint design matrix, and F is the joint Fisher matrix. With L D = 0, the product reduces to A C B = L^T (L C L^T)^{-1} L, which is nonzero for any nontrivial Delta. Hence the two tests are formally dependent. The 1000-realization \"no correlation\" check in the paper is insensitive to this dependence because correlation between a linear and a quadratic form in zero-mean Gaussians vanishes regardless of dependence. Thus the abstract's independence claim is not established and is likely false; the interpretation that the PTE 0.017/0.53 discrepancy is \"simply statistical fluctuations\" should be re-quantified using the joint distribution rather than independence.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript derives an analytic expression, Eq. (7) with the response matrix defined in Eq. (8), for the covariance between maximum-likelihood parameter vectors estimated from two correlated data sets. The derivation assumes a Gaussian likelihood with a parameter-independent data covariance and a linear response of the mean vector about a fiducial model. The authors validate the formula by comparing it with the distribution of best-fit parameters from 1000 simulated SPTpol realizations (Fig. 2). They then apply the method to the SPTpol TE and EE power spectra, reporting weak positive correlations between TE-only and EE-only Lambda CDM parameters (9% for H0, 25% for log(As), 32% for ns) even though the TE-EE bandpower correlations are predominantly negative. The TE-EE parameter differences are consistent with zero (PTE 0.53), in contrast to the SPTpol bandpower-level consistency PTE of 0.017. Using simulations, the authors claim that these two consistency tests are independent. The paper further analyzes the cosmic-variance-limited case as a function of multipole range and demonstrates that neglecting parameter correlations biases chi-square low, making parameters appear more consistent.","tokens_in":14703,"tokens_out":8409,"duration_ms":84011,"significance":"The methodological contribution is sound and useful. Equation (7) is a clean extension of Fisher forecasting that avoids expensive simulations for cross-data-set parameter covariances, and the validation against maximum-likelihood simulations is convincing. The authors are explicit about the assumptions and test sensitivity to the fiducial cosmology, finding percent-level changes in the correlations. The concrete SPTpol predictions are falsifiable, and the paper's recommendation to account for parameter correlations in future internal-consistency checks is well supported by the Figure 6 histograms. The main weakness is an overinterpretation in the application: the claim that the parameter- and bandpower-based consistency tests are 'independent' is not established by the simulations and is contradicted by a standard Gaussian-form argument; this needs correction before publication.","major_comments":[{"comment":"The claim that the two consistency tests are independent is not supported. The parameter difference Delta = theta_TE - theta_EE is a linear function of the data, while the SPTpol bandpower chi-square is a quadratic form in the data. For zero-mean Gaussian data, a linear form and a quadratic form always have zero covariance, so the reported absence of correlation between the two PTE values over 1000 simulations carries no information about independence. A direct application of Craig's theorem to the quadratic forms Q_param = Delta^T Cov(Delta)^{-1} Delta and Q_band = r^T C^{-1} r shows that they are independent only if A C B = 0, where A projects onto Delta and B projects onto the residual space; with L D = 0, one obtains A C B = L^T (L C L^T)^{-1} L, which is nonzero for any non-trivial Delta. Hence the tests are formally dependent. The authors should either remove 'independent' from the abstract and Section 5, or provide a joint-distribution analysis to quantify the expected scatter between the two PTE values. The conclusion that the 0.017 versus 0.53 difference can arise from statistical fluctuations may still be correct, but it is not established by the independence claim.","section":"Abstract and Section 3.3"}],"minor_comments":[{"comment":"The quoted PTE of 0.53 is computed with the joint-fit SPTpol cosmology as the fiducial model, and the Planck-fiducial rerun changes the chi-square from 4.16 to 6.2 (PTE 0.29). The abstract presents 0.53 without this caveat; the fiducial dependence should be stated wherever the PTE is quoted.","section":"Section 3.3"},{"comment":"The load-bearing assumption that the SPTpol bandpower covariance matrix correctly describes the scatter in the TE and EE spectra is stated in Section 3.3, but it is not repeated in the abstract or conclusions; given that all derived correlations and PTEs depend on this assumption, it should be flagged more prominently.","section":"Section 3.3"},{"comment":"The note after Eq. (8) that the covariance 'cannot be calculated using a single Fisher matrix containing two sets of varying parameters' is terse; a joint fit with two parameter vectors yields the joint posterior covariance, which is a different quantity from the cross-covariance of the individual single-data-set estimators, and the sentence should be reworded to avoid implying a joint fit is impossible.","section":"Section 2"},{"comment":"There are typographical errors: 'minimum mulipole moment' should be 'minimum multipole moment', and the Figure 5 caption has 'minumum' for 'minimum'.","section":"Section 4.1"},{"comment":"The sentence 'We find that there is no correlation between the PTE for the consistency of the parameters and the PTE for the consistency of the simulated TE and EE spectra with Lambda CDM predictions' should be rephrased as 'no linear correlation' to avoid implying the stronger property of independence, consistent with the major comment above.","section":"Section 3.3"}],"recommendation":"major_revision","confidential_remarks":"The analytic method is sound, but the overclaim of independence between the two consistency tests is a headline result that needs to be corrected or substantially qualified. If the authors can replace the independence claim with a joint-distribution analysis, or soften the language to 'uncorrelated in the simulations' and remove 'independent' from the abstract, the paper would be suitable for acceptance. The manuscript is within the scope of the journal and otherwise well-presented."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Josh, here's my take on 1908.01626.\n\nThe paper is worth a serious referee. The new thing is Equation (7): an analytic expression for the covariance between maximum-likelihood parameters from two correlated data subsets, built from the usual Fisher ingredients and the cross-data covariance. That is genuinely useful—it replaces expensive simulations for many internal-consistency checks, and it's validated against 1000 ML fits. I think the derivation is clean and the assumptions (Gaussian likelihood, linear response, known covariance) are stated plainly. The SPTpol application is also done carefully: the PTE of 0.53 for TE vs EE parameters, the positive parameter correlations despite negative bandpower correlations, and the demonstration in Figure 6 that ignoring the cross-correlation biases chi-square low are all solid results. Credit where due: the authors check the sensitivity to the fiducial cosmology and explicitly flag the two assumptions (LambdaCDM true, SPTpol covariance correct).\n\nNow the soft spot, and it's more than cosmetic. The paper claims, in the abstract and Section 3.3, that the parameter-based and bandpower-based consistency tests are 'independent' because the PTE values show no correlation in simulations. That inference is not valid. For Gaussian data, the parameter difference is a linear form and the bandpower residual chi-square is a quadratic form in the same data vector. Two such statistics can be dependent even when their correlation is exactly zero—odd moments of a zero-mean Gaussian vanish. Craig's theorem gives the condition for independence, and it fails here: with A = L^T(L C L^T)^{-1} L and B = C^{-1} - C^{-1}D F^{-1}D^T C^{-1}, A C B is nonzero. So the two tests are formally dependent, and the 'no correlation' check in the paper is insensitive to that dependence. The claim should be softened to 'no linear correlation' or, better, re-quantified using the joint distribution. This does not sink the analytic method or the PTE 0.53, but it does mean the explanation of the 0.017/0.53 discrepancy as 'simply statistical fluctuations' is not fully established.\n\nThere are minor caveats: tau is fixed, and the data-level conclusions rest on the SPTpol covariance matrix, but the authors are upfront about both.\n\nBottom line: send it to a referee. The derivation and the main consistency results deserve to be in the literature. The independence claim needs to be fixed—either retracted or replaced with a proper dependence check—but that's a revision, not a rejection.","headline":"A useful analytic cross-covariance formula and a solid SPTpol application, but the claim that the two PTE tests are independent is not supported and likely false.","tokens_in":15252,"tokens_out":3839,"would_cite":true,"duration_ms":34297,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives an analytic formula for the covariance between maximum-likelihood parameters from two correlated data sets and applies it to SPTpol TE and EE CMB spectra, finding weak positive correlations and a TE-EE parameter PTE of…","keywords":["cosmic microwave background","parameter covariance","Fisher matrix","SPTpol","TE and EE power spectra","internal consistency check","LambdaCDM","maximum likelihood estimation"],"falsifier":"Compute the TE-only and EE-only parameter-difference covariance using a bandpower covariance matrix estimated from end-to-end simulations that include non-Gaussian lensing and a deliberately perturbed TE-EE cross-block (for example, inflated by 50%); if the resulting PTE for the parameter differences moves substantially away from 0.53, or if the PTE distribution from repeated simulations is no longer uniform, the analytic claim fails.","tokens_in":14146,"feed_emoji":"🔭","tokens_out":11857,"duration_ms":102649,"temperature":0.7,"pith_summary":"This paper derives an analytic formula for the covariance between maximum-likelihood parameters estimated from two correlated data sets when the data covariance is known. The formula extends Fisher analysis: the parameter-level cross-covariance between data sets X and Y is obtained by mapping the data covariance block through theory derivatives and inverse Fisher matrices. Applied to SPTpol's TE and EE CMB spectra, it yields weak positive parameter correlations (9% for $H_0$, 25% for $\\log A_s$, 32% for $n_s$) even though the TE and EE power spectra are negatively correlated. The TE-only and EE-only parameter differences are consistent with zero (PTE 0.53), and the paper shows this parameter test is statistically independent of the bandpower-level consistency test with PTE 0.017, so the gap can be a statistical fluctuation. A practical corollary is that ignoring the parameter correlations in TT-TE and TE-EE comparisons biases $\\chi^2$ low and should be avoided.","feed_headline":"SPTpol TE and EE parameters agree (PTE 0.53); tension is a fluctuation","feed_subtitle":"A new analytic formula shows the two SPTpol checks are independent; ignoring correlations understates inconsistency.","key_machinery":"The engine is Equation (7), a covariance-mapping identity: $\\langle(\\theta^X_{\\mathrm{ML}}-\\langle\\theta^X_{\\mathrm{ML}}\\rangle)(\\theta^Y_{\\mathrm{ML}}-\\langle\\theta^Y_{\\mathrm{ML}}\\rangle)^T\\rangle = (M^X)^T \\mathbf{C}^{XY} M^Y$, where $M^X = (\\mathbf{C}^{XX})^{-1}(\\partial\\mu^X/\\partial\\theta^X)(F^{XX})^{-1}$, $\\mathbf{C}^{XX}$ and $\\mathbf{C}^{XY}$ are data covariance blocks, $\\mu^X$ is the theory vector, and $F^{XX}$ is the parameter Fisher matrix. This identity converts an off-diagonal block of the data covariance into a parameter cross-covariance. The paper evaluates the derivative matrices by finite differences of the theory spectra computed with a Boltzmann solver and feeds in SPTpol's published bandpower covariance matrix.","core_discovery":"The central claim is that cross-data-set parameter covariances are computable analytically. For two data sub-sets $X$ and $Y$ sharing a Gaussian likelihood with known covariance, the covariance between their maximum-likelihood parameter vectors is $$\\langle(\\$\\theta$^X_{\\mathrm{ML}}-\\langle\\$\\theta$^X_{\\mathrm{ML}}\\rangle)(\\$\\theta$^Y_{\\mathrm{ML}}-\\langle\\$\\theta$^Y_{\\mathrm{ML}}\\rangle)^T\\rangle = (M^X)^T \\mathbf{C}^{XY} M^Y, \\qquad M^X = (\\mathbf{C}^{XX})^{-1}\\frac{\\partial \\mu^X}{\\partial\\$\\theta$^X}($F^{{XX}}$)^{-1}.$$ Applied to SPTpol, this yields weak positive correlations between TE-only and EE-only $\\Lambda$CDM parameters ($H_0$: 9%, $\\log A_s$: 25%, $n_s$: 32%) even though the TE and EE spectra are negatively correlated. The TE-EE parameter differences are consistent with zero ($\\chi^2=4.16$, 5 d.o.f., PTE 0.53), and 1000 simulations show the PTE of this parameter test is statistically independent of the bandpower-level PTE (0.017) that SPTpol reported, so the gap between 0.017 and 0.53 can be a statistical fluctuation. The paper also shows that dropping the parameter correlations in TT-TE and TE-EE consistency checks biases $\\chi^2$ low.","pith_inferences":["Editorial extension: The sign flip from negative data correlations to positive parameter correlations implies that parameter-correlation signs track the alignment of theory derivative vectors, so a derivative diagnostic could predict which parameter pairs are most sensitive to an ignored covariance.","Editorial extension: The same covariance-mapping formula could be applied to the tension between Planck and distance-ladder $H_0$ measurements if a covariance for shared calibrators were available; without that covariance, the formula at least bounds how large a shared systematic would need to be to erase the tension.","Editorial extension: The assumption that the data covariance does not depend on parameters is testable through second-order Fisher/derivative corrections; a parameter-dependent covariance would introduce extra terms into the covariance-mapping identity that the current derivation omits.","Editorial extension: The demonstrated independence of parameter-level and data-level consistency PTEs suggests that experiments reporting only one of the two tests are under-reporting their consistency information; both should be published, and readers should not convert one into the other."],"forward_implications":["Future CMB internal consistency checks (TT vs TE vs EE) should include parameter-level covariances; omitting them biases $\\chi^2$ low and can make inconsistent parameters look artificially consistent.","For SPTpol, the parameter-level PTE of 0.53 and the bandpower-level PTE of 0.017 are not contradictory; the two tests are statistically independent, so neither result undermines the other.","The weak correlations found for SPTpol are generic: for a cosmic-variance-limited experiment, TT-TE and TE-EE parameter correlations range from about 0% to 50%, while TT-EE correlations are generally below 10%, so high-resolution EE constraints will be largely independent of Planck TT constraints.","The analytic formula provides a fast check on simulations and can be applied to other correlated cosmological data sets with known covariance, such as sky-overlapping CMB experiments or supernova sub-samples with shared systematics."],"supporting_citations":[{"why":"Supplies the SPTpol TE and EE bandpowers, the bandpower covariance matrix, and the PTE = 0.017 bandpower consistency result against which the paper compares.","marker":"Henning et al. 2018"},{"why":"Provides the Gaussian-likelihood and linear-response assumptions that the derivation adopts as its starting point.","marker":"Huang et al. (2019)"},{"why":"Supplies the MCMC sampler used for the maximum-likelihood simulations that validate the analytic covariance.","marker":"Lewis & Bridle 2002"},{"why":"Supplies the Boltzmann code used to compute the theory spectra and finite-difference derivative matrices.","marker":"Lewis et al. 2000"},{"why":"Provides the alternate fiducial cosmology used to check the stability of the correlations and PTE.","marker":"Planck Collaboration VI 2018"},{"why":"Motivates the comparison by showing TE can constrain several parameters as tightly as TT or EE.","marker":"Galli et al. 2014"},{"why":"Supports the equivalence of maximum-likelihood and Bayesian posterior covariances under the stated assumptions.","marker":"Raveri & Hu (2019)"},{"why":"Justifies interpreting the maximum-likelihood parameter covariance as the posterior covariance.","marker":"Gelman et al. (2013)"},{"why":"Provides ACTPol TE and EE data used to confirm that TE-EE correlations weaken with added noise as expected.","marker":"Louis et al. 2017"}],"fun_headline_variants":["SPTpol TE/EE agree when covariances included: PTE 0.53","Analytic covariances resolve SPTpol TE-EE tension as fluctuation","Parameter correlations: 9% H0, 25% As, 32% ns; TE-EE consistent","No SPTpol TE-EE inconsistency: gap is statistical fluctuation","New formula: SPTpol TE/EE differences consistent with zero"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For the SPTpol application, the load-bearing premise is that $\\Lambda$CDM is the true model and that the published SPTpol bandpower covariance matrix correctly describes the scatter in the TE and EE spectra; if the covariance is misestimated, the analytic cross-covariances, the PTE of 0.53, and the claimed independence of the two tests would all shift.","fun_headline_variants_meta":{"raw":{"variants":["SPTpol TE/EE agree when covariances included: PTE 0.53","Analytic covariances resolve SPTpol TE-EE tension as fluctuation","Parameter correlations: 9% H0, 25% As, 32% ns; TE-EE consistent","No SPTpol TE-EE inconsistency: gap is statistical fluctuation","New formula: SPTpol TE/EE differences consistent with zero"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00038,"raw_usage":{"total_tokens":2131,"prompt_tokens":1172,"completion_tokens":959,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":788,"completion_tokens_details":{"reasoning_tokens":850}},"tokens_in":788,"tokens_out":959,"duration_ms":8556,"temperature":1.0,"reasoning_tokens":850,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:08:05.047585+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the TE-only and EE-only parameter-difference covariance using a bandpower covariance matrix estimated from end-to-end simulations that include non-Gaussian lensing and a deliberately perturbed TE-EE cross-block (for example, inflated by 50%); if the resulting PTE for the parameter differences moves substantially away from 0.53, or if the PTE distribution from repeated simulations is no longer uniform, the analytic claim fails.","supporting_citations":[{"cited_title":"E., & Bennett, C","cited_arxiv_id":null,"evidence_quote":"Provides the Gaussian-likelihood and linear-response assumptions that the derivation adopts as its starting point."},{"cited_title":"2002, PhRvD, 66, 103511","cited_arxiv_id":null,"evidence_quote":"Supplies the MCMC sampler used for the maximum-likelihood simulations that validate the analytic covariance."},{"cited_title":"2014, PhRvD, 90, 063504","cited_arxiv_id":null,"evidence_quote":"Motivates the comparison by showing TE can constrain several parameters as tightly as TT or EE."},{"cited_title":"2019, PhRvD, 99, 043506","cited_arxiv_id":null,"evidence_quote":"Supports the equivalence of maximum-likelihood and Bayesian posterior covariances under the stated assumptions."},{"cited_title":"2017, JCAP, 6, 031","cited_arxiv_id":null,"evidence_quote":"Provides ACTPol TE and EE data used to confirm that TE-EE correlations weaken with added noise as expected."}],"review_version":1}