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The Milky Way's bulge formed in two episodes: an almost instantaneous early collapse that made ~60% of its stars, then a delayed gas infall around 9 Gyr ago that made the rest.

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2026-08-03 17:47 UTC pith:K7Q2URNZ

load-bearing objection A well-executed, honest parameter search, but the claim that a second infall is 'chemically required' rests on a model contrast the paper never actually runs. the 3 major comments →

arxiv 2512.08090 v2 pith:K7Q2URNZ submitted 2025-12-08 astro-ph.GA astro-ph.SR

The Two-infall Model Revisited: Constraints on Milky Way Bulge Assembly from >30,000 Galactic Chemical Evolution Models and Machine Learning

Niall Miller , Meridith Joyce , Christian I. Johnson , Jamie Tayar , Thomas Trueman , R. Michael Rich This is my paper
classification astro-ph.GA astro-ph.SR
keywords Milky Way bulgetwo-infall modelgalactic chemical evolutionmetallicity distribution functionstar formation efficiencyalpha enhancementage–metallicity relationdegeneracy analysis
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tries to establish that the Milky Way's bulge did not form in a single event: it needs two distinct gas-accretion episodes to explain the observed spread of stellar metallicities and alpha-element abundances. The authors fit a two-infall chemical evolution model to a composite metallicity distribution of bulge stars, searching more than 30,000 model variants with a genetic algorithm plus Markov Chain Monte Carlo refinement. Their best scenario has a nearly instantaneous first collapse (starting ~0.1 Gyr after the universe, lasting ~0.09 Gyr) that forms ~60% of the bulge mass with very high star-formation efficiency, followed by a second infall starting ~5.1 Gyr later that contributes ~40% of the mass with reduced efficiency. If right, the bulge is a hybrid object—classical rapid collapse plus later disk/bar or merger-fed gas—and the chemistry favors a revised, older age scale for bulge stars over previously reported younger ages. The authors are explicit that the exact timing and mass split are degenerate; what they argue is robust is that a non-zero second infall is chemically required.

Core claim

The paper's central claim is that the Milky Way bulge's chemical patterns—a bimodal metallicity distribution with peaks near [Fe/H] ≈ −0.3 and +0.3, and an alpha-to-iron decline at high metallicity—are reproduced only when gas falls into the bulge in two episodes governed by a two-exponential infall law. The maximum-a-posteriori solution places the first infall at t1 ≈ 0.1 Gyr with timescale τ1 ≈ 0.09 Gyr and star-formation efficiency ≈ 2.9 Gyr⁻¹, building about 60% of the mass; the second infall begins at t2 ≈ 5.1 Gyr, lasts τ2 ≈ 1.7 Gyr, carries about 40% of the mass, and runs at roughly 28% lower efficiency. The later episode is presented not as an option but as a chemical necessity: with

What carries the argument

The workhorse is a one-zone galactic chemical evolution model with a time-dependent star-formation efficiency and a two-component exponential infall history. The first mode, a rapid high-efficiency collapse, builds the old alpha-enhanced population; the second mode, a delayed lower-efficiency inflow, dilutes the interstellar medium and lets Type Ia supernovae add iron, producing the metal-rich, low-alpha sequence. The parameter space—infall onset times and timescales, mass ratio between episodes, star-formation efficiencies, IMF upper mass, and supernova Ia normalization, plus categorical yield and IMF choices—is explored with a hybrid genetic algorithm refined by differential-evolution MCMC

Load-bearing premise

The load-bearing premise is that the hand-built composite metallicity distribution—equal-weight averaging of two surveys with different selection effects and a fitted latitude scaling—faithfully represents the bulge's true MDF; if that target is biased, every inferred infall parameter shifts.

What would settle it

Rebuild the composite MDF target using only one survey at a time, or with a different latitude weighting, and rerun the optimization: if the MAP values for t2, τ2, σ2, and ΔSFE move outside the quoted 68% highest-density intervals, the inferred two-infall history is an artifact of target construction.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • If correct, the old alpha-enhanced bulge population formed within roughly the first 0.2 Gyr of cosmic history—an extreme early starburst that set the chemical baseline.
  • The second infall adds a younger, roughly 40%-mass component around 8–9 Gyr ago; its reduced efficiency is what creates the metal-rich peak and the downturn in alpha/Fe.
  • The bulge's age–metallicity relation implied by reproducing the MDF favors a revised, older age scale for super-solar-metallicity bulge dwarfs over the original younger ages.
  • Because infall timing, mass ratio, and efficiency are strongly covariant, current MDF data constrain only combinations of parameters, not each individually; the existence of a second episode is the robust part.
  • The second infall epoch overlaps both the last major merger and the era of bar formation, so bulge chemistry alone cannot yet distinguish merger-fed from bar-driven late gas supply.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the composite target MDF is an equal 50/50 blend of two surveys with different selection functions; re-running the fit under alternative weightings could shift the MAP values for t2, τ2, and ΔSFE substantially, since no sensitivity analysis is presented.
  • Editorial extension: if reduced second-infall efficiency is the real cause of the low-alpha metal-rich population, spatially resolved multi-zone models should predict a correlation between alpha and vertical metallicity gradients—something the single-zone approach cannot capture.
  • Editorial extension: upcoming large asteroseismic samples in the bulge could test the age-scale choice directly; if the younger ages for metal-rich bulge stars survive, the model's preferred age–metallicity relation would be overturned.
  • Editorial extension: the degeneracy analysis suggests a single observable, such as the height of the super-solar MDF peak or the position of the alpha knee, may be nearly sufficient to certify a second infall; a simple test is to fit single-infall models to each [alpha/Fe] sequence and check whether any can reproduce the downturn.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents a two-infall galactic chemical evolution model of the Milky Way bulge, implemented in an extension of OMEGA+ ("OMEGA++"), and constrains its parameters by fitting a composite MDF assembled from APOGEE DR16 and BDBS data. The optimization uses a hybrid genetic algorithm with DEMC refinement over a 15-dimensional space (10 continuous, 5 categorical), and the resulting weighted ensemble is treated as a pseudo-posterior. The headline results are an early rapid first infall (t1≈0.1 Gyr, τ1≈0.09 Gyr, SFE≈2.9 Gyr−1) that forms ~60% of the bulge mass, followed by a delayed second infall (t2≈5.1 Gyr, τ2≈1.7 Gyr, σ2≈0.69) with reduced SFE, which the authors claim is chemically required to reproduce the metal-rich MDF peak and low-[α/Fe] population. The model's AMR is then compared to observations, favoring the Joyce et al. (2023) age scale over Bensby et al. (2017). Independent MCMC runs for each categorical combination (Appendix A) are used to validate the GA sampler. The manuscript is honest about its pseudo-posterior nature and lists several modeling limitations, but it does not perform a nested single-infall comparison and the composite MDF target is constructed without sensitivity analysis.

Significance. If the central claims hold, the paper would provide a quantitative, observationally constrained picture of bulge assembly with an early rapid collapse and a delayed, sub-dominant second gas infall—relevant to classical versus secular bulge formation debates. The work's main strengths are the unusually wide parameter search, the explicit cross-validation of the GA+DEMC sampler against 288 independent MCMC runs (Appendix A), the transparent treatment of degeneracies via PCA and mutual information, and the out-of-sample use of the AMR rather than fitting it directly. However, the central qualitative claim that a second infall is 'chemically required' is not supported by a nested model test, and the quantitative MAP/HDI values rest entirely on a hand-assembled composite MDF target whose construction is not stress-tested. These issues currently limit the paper's conclusions to conditional on unexamined assumptions.

major comments (3)
  1. [§5.4.2 / §6] The headline claim that a non-zero second infall is 'chemically required' is not demonstrated by any nested model comparison. Table 1 samples σ2 only over 0.1–10.0, so σ2 = 0 is outside the explored prior. The statement in §5.4.2 that 'models with strictly negligible second infall fail' is unsupported by a figure, loss comparison, or likelihood-ratio test. Moreover, the time-dependent ΔSFE parameter can produce late-time SN Ia enrichment and a declining [α/Fe] track even with zero second infall gas (A2=0 in Eq. 2). Thus the data may require two star-formation phases, but not necessarily two gas infalls. Please add a single-infall (or σ2→0) control model and compare its best ensemble loss and MDF/AMR residuals; if such a model cannot fit, show that explicitly. Otherwise, soften the claim to 'two enrichment phases' rather than 'two infalls.'
  2. [§2, Eq. (1)] The composite MDF target is the only optimization target, yet its construction is not tested for robustness. The latitude scaling N/N0 = 1.029 e^{0.476 b} is fitted to Zoccali et al. (2018) and applied to APOGEE latitude fits, then combined 50/50 by equal weight with the BDBS red-clump MDF. The paper itself notes that BDBS appears less bimodal than APOGEE, so the equal-weight choice is substantive. The MAP/HDI values in Table 2 (σ2≈0.69, t2≈5.15 Gyr, ΔSFE≈0.72) are all derived from this specific target. Please provide a sensitivity analysis: re-run the optimization (or re-weight the existing model ensemble) under alternative weightings (e.g., 30/70, 70/30) or with the BDBS-only and APOGEE-only targets. If t2, σ2, and ΔSFE are stable, report this; if they shift, quantify the shift and adjust the conclusions accordingly.
  3. [§3.5 / §4] The pseudo-posterior weights are computed from an uncalibrated ensemble loss L_ensemble = 0.7 L_WRMSE + 0.2 L_cosine + 0.1 L_Huber, with no noise model for the MDF bins. Consequently, the 68% HDI values in Table 2 are not calibrated posterior intervals in a statistical sense—they depend on arbitrary loss weights and bin choices. Since the paper repeatedly uses these intervals to assert constraints (e.g., 't1 is notably more constrained,' 'σ2 is not well constrained'), the absence of a noise model is load-bearing. Please either (a) formulate a likelihood (e.g., Poisson or Gaussian per bin with the published/estimated uncertainties) and re-derive the posterior, or (b) at minimum, demonstrate that the MAP/HDI conclusions are stable under sensible variations of the loss weights and binning (e.g., 0.6/0.3/0.1, 0.8/0.1/0.1, and varied bin widths). This would also make the 'cannot fit' statemen
minor comments (5)
  1. [Throughout] Notation is inconsistent: the onset of the second infall is called t2 in Table 1 and the text, but tmax,2 in Eq. (2). Please unify. Also, the parameter ΔSFE is written as both 'ΔSFE' and 'δ SFE' in §5.3 and §5.4.4; pick one.
  2. [Abstract / §5.1] The abstract states '>30,000 GCE models', but Table 5 in Appendix A reports 262,144 GA+DEMC model evaluations (and 9,437,184 MCMC evaluations). The count in the abstract appears inconsistent with the total; please clarify what the 30,000 refers to (e.g., unique posterior-weighted models after filtering).
  3. [References] Reference typos: 'Truemam et al. 2025' in §5.2 should be 'Trueman', and the reference list contains both 'Truemam' and 'Trueman' entries; similarly, 'Cˆot´e et al.' appears with inconsistent accents. Also, the software list includes 'ChatGPT, Gemini'—if these were used in manuscript preparation, this is acceptable, but (i) for reproducibility, specify their role (e.g., text editing vs code generation), and (ii) consider whether journal policy requires this disclosure.
  4. [Figure 11 caption] The caption says 'one for each unique choice of categorical model ingredients Table 1)'—a closing parenthesis is missing. Also, the HDI annotations in Figures 3 and 11 are hard to read at the plotted scale; consider enlarging or tabulating the values.
  5. [§5.2 / Fig. 8] The Ti panel shows a systematic underprediction, which the paper attributes to yield uncertainties. Given that Ti is a known problem, it would help to state explicitly whether the model residuals for Ti are included in any quantitative goodness-of-fit metric, or whether the fit is driven entirely by the MDF (as implied by Eq. 7).

Circularity Check

0 steps flagged

No circular reduction; the central inference is an independent parameter fit with an out-of-sample AMR by-product.

full rationale

The paper's quantitative core is a fit of a two-infall GCE model to an observational composite MDF using an explicit ensemble loss (Eq. 7). The MAP/HDI values for t1, tau1, t2, tau2, sigma2, SFE, DeltaSFE, etc. are outputs of that fit, not inputs to it. The AMR comparison is explicitly stated to be a post-optimization by-product and is not used as an optimization target, so the preference for the Joyce et al. (2023) ages is an out-of-sample model output rather than a fitted quantity. The Johnson et al. (2022) MDF and Joyce et al. (2023) AMR are externally anchored observational/age data sets even though they share authors; self-citation overlap of this kind is bias evidence, not circularity. The main scientific caveat is that the claim that a second infall is 'chemically required' is not supported by a fitted single-infall baseline: Eq. 2 imposes a two-infall form and Table 1 samples sigma2 only over 0.1-10.0, so sigma2=0 is outside the explored prior, and no nested model comparison is shown. That is a model-comparison and inference-calibration gap, not a circular reduction of the kind where a prediction equals its input by construction. The paper also transparently acknowledges that its one-zone model cannot uniquely identify the number of infall episodes. Because no derivation step reduces to its own input or to an unverified self-citation, no circularity is established.

Axiom & Free-Parameter Ledger

11 free parameters · 8 axioms · 0 invented entities

No new physical entities (particles, forces, dimensions) are introduced; OMEGA++ is a software extension, not a physical postulate. The load-bearing ledger is otherwise substantial: 10 continuous parameters plus loss weights are fitted to one constructed histogram (the composite MDF), and the claimed two-infall history is conditioned on the imposed functional form (Eq. 2), the one-zone inflow-only geometry, the pre-enriched STELLAB gas initialization, and the hand-assembled MDF target. The yield, IMF, and SN Ia grids are honest inputs from prior literature, with the Ti tension indicating residual input error.

free parameters (11)
  • t1 (first infall onset) = 0.098 Gyr
    Fitted to the composite MDF; drives the early starburst timing.
  • tau1 (first infall timescale) = 0.093 Gyr
    Fitted to the MDF; sets rapidity of early enrichment.
  • t2 (second infall onset) = 5.145 Gyr (HDI 3.25–8.45)
    Fitted; only weakly constrained, anchors the GSE/bar inflow discussion.
  • tau2 (second infall timescale) = 1.74 Gyr (HDI 0.5–3.7)
    Fitted to the MDF; controls late-time dilution and the metal-rich tail.
  • sigma2 (second/first infall mass ratio) = 0.69 (HDI 0.11–3.1)
    Fitted; the paper concedes this is weakly constrained, with almost order-of-magnitude HDI.
  • SFE (first-phase star formation efficiency) = 2.93 Gyr^-1 (bimodal; secondary peak ~25)
    Fitted to the MDF; posterior is bimodal and the HDI is misleadingly broad.
  • dSFE (multiplicative SFE drop at t2) = 0.72 (HDI 0.39–0.85)
    Fitted; controls low-alpha late population.
  • Mmax (IMF upper mass cutoff) = 108.4 Msun
    Fitted to the MDF; weakly constrained with substantial mutual information with sigma2.
  • MBulge (final stellar mass normalization) = 1.01e10 Msun
    Fitted; acts as a normalization on integrated gas inflow.
  • NIa/Msun (SN Ia normalization) = 5.8e-4
    Fitted to the MDF; controls the [alpha/Fe] knee and late iron enrichment.
  • Loss weight coefficients (0.7/0.2/0.1) = 0.7, 0.2, 0.1
    Hand-chosen in Eq. 7; they define the pseudo-posterior and hence all MAP/HDI numbers.
axioms (8)
  • domain assumption Bulge is a single well-mixed gas reservoir with instantaneous mixing
    Section 3.1; precludes spatial gradients and migration; authors note it likely underestimates the metal-poor tail by 10–20% and overestimates tau2 by ~0.5–1 Gyr (Section 4.1).
  • domain assumption Inflow-only evolution, no outflows
    Section 3.1; justified by the bulge's deep potential, but excludes wind and feedback scenarios included in some previous GCE studies.
  • ad hoc to paper Gas infall rate has the two-exponential form of Eq. 2
    Section 3.1; the paper acknowledges that any additional minor or closely spaced inflows are absorbed into the effective second episode (Section 4.1), so the form is imposed, not derived.
  • domain assumption Infalling and initial gas is pre-enriched according to the STELLAB library
    Section 3.1; non-zero initial metallicity for gas and inflows shapes the entire early enrichment path and the metal-poor tail, independent of the fitted parameters.
  • ad hoc to paper Composite MDF target defined by Eq. 1 latitude scaling and 50/50 equal weighting of APOGEE and BDBS
    Section 2; both the fitted scaling (to Zoccali et al. 2018) and the equal-weight choice are hand-set, and no sensitivity analysis is presented.
  • domain assumption Nucleosynthetic yield grids (LC18, Karakas, Nomoto, Shen, Gronow) and SN Ia delay-time distributions are correct
    Section 3.1; treated as external categorical inputs; the systematic Ti underproduction (Section 5.2) shows residual yield-model error.
  • ad hoc to paper Final-mass window 5e9 < Mfinal < 3e10 Msun used as a model filter
    Section 3.5; the 'physical plausibility constraint' truncates the ensemble and therefore shapes the pseudo-posterior.
  • domain assumption IMF family spans the true bulge IMF
    Section 3.2, Table 1; posterior fractions stay broad (Chabrier 0.58, Kroupa 0.40, Salpeter 0.02), so the data do not validate the IMF choice.

pith-pipeline@v1.3.0-alltime-deepseek · 35570 in / 17695 out tokens · 161616 ms · 2026-08-03T17:47:25.455549+00:00 · methodology

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Cite this review

Pith. "Pith review of The Two-infall Model Revisited: Constraints on Milky Way Bulge Assembly from >30,000 Galactic Chemical Evolution Models and Machine Learning." pith.science (2026). https://pith.science/paper/K7Q2URNZ

@misc{pith2026251208090,
  author       = {Pith},
  title        = {Pith review of: The Two-infall Model Revisited: Constraints on Milky Way Bulge Assembly from >30,000 Galactic Chemical Evolution Models and Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K7Q2URNZ}},
  note         = {Machine review of arXiv:2512.08090}
}
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read the original abstract

We constrain the formation history of the Milky Way bulge using a two-infall galactic chemical evolution (GCE) algorithm implemented in the N'OMEGA+ code. We recover a best-fit scenario in which the bulge forms through an early, rapid starburst ($t_1 \sim 0.1$ Gyr, $\tau_1 \sim 0.09$ Gyr, and star formation efficiency (SFE) $\sim 3~\mathrm{Gyr}^{-1}$), followed by a delayed, lower-mass second infall ($t_2 \sim 5.1$ Gyr, $\tau_2 \sim 1.7$ Gyr, and $\sigma_2 \sim 0.69$). Our model adopts mass- and metallicity-dependent nucleosynthetic yields from modern stellar grids and explores a wide GCE parameter space in infall timing, SFE, mass partitioning, initial mass function upper mass, and type Ia supernova normalization, optimized via a hybrid genetic algorithm with Markov Chain Monte Carlo refinement. The later infall features a reduced SFE ($\Delta\mathrm{SFE} \sim 0.72$), reproducing the metal-rich peak of the bulge metallicity distribution function (MDF) and the decline in [$\alpha$/Fe] at high [Fe/H]. Our model naturally favors the M. Joyce et al. age--metallicity relation over the ages in T. Bensby et al. Degeneracy and principal component analyses show that the infall history, SFE, and mass partitioning are strongly covariant---the bulge's observed MDF, abundance trends, and age distribution constrain only their combinations, not each parameter independently. The results support a composite bulge origin---an early, rapid collapse builds the majority of the mass, while a younger component is required to match the late-stage enrichment.

Figures

Figures reproduced from arXiv: 2512.08090 by Christian I. Johnson, Jamie Tayar, Meridith Joyce, Niall Miller, R. Michael Rich, Thomas Trueman.

Figure 1
Figure 1. Figure 1: Schematic illustration of the two–infall framework used in this work. Left: Face-on (looking down the Galaxys rotation axis) view of the early bulge depicting the first infall (Shown by the blue arrows), represented as a rapid, centrally directed collapse that builds the classical bulge (Orange circle) in the absence of a bar or extended disk. Right: Diagonal view of the later galactic bulge during the sec… view at source ↗
Figure 2
Figure 2. Figure 2: The figure shows the [Fe/H] distribution for the Milky Way bulge. Dashed lines indicate individual MDF fits from APOGEE DR16 across various Galactic latitude bands ‘|b|’. The APOGEE Composite (thick black line) is the lat￾itude-weighted average of the APOGEE fits. The BDBS MDF (blue line) is derived from red clump stars Johnson et al. (2022). The composite MDF (thick red line) represents the final, equally… view at source ↗
Figure 3
Figure 3. Figure 3: Corner plot of posterior showing 1D histograms (diagonals) and 2D density (off–diagonals) continuous parameter groups. The black cross indicates the maximum a posteriori (MAP). The red square highlights the Highest Density Interval (HDI) preserve a viable MDF. A similar negative correlation is seen between the onset time of the second infall (t2) and SFE (ρw ≃ −0.20), while τ2 is negatively correlated with… view at source ↗
Figure 4
Figure 4. Figure 4: Parameter dependency analysis. Left: Fitness-weighted Pearson correlation matrix. Right: Mutual information matrix which highlights both linear and nonlinear dependencies. Parameter pairs with high mutual information but low corre￾lation indicate strong nonlinear coupling [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Principal component analysis of the top-performing models. Left: Parameter loadings on the first six principal components, showing which parameters contribute most strongly to each mode of variation. Right: The variance explained by each principal component. The first six PCs capture approximately 77.5% of the total variance, indicating substantial degeneracy in the parameter space. tional form. The poster… view at source ↗
Figure 6
Figure 6. Figure 6: Posterior corner plot for the continuous model parameters. Diagonal panels show 1D marginalized distributions with MAP (solid line) and 68% highest–density intervals (dashed lines). Lower–triangle panels show the MDF–weighted model ensemble as a background point cloud with overlaid smoothed posterior density. The red crosses mark the MAP location in each 2D plane. Coloured stars and guide lines indicate pa… view at source ↗
Figure 7
Figure 7. Figure 7: Metallicity distribution function (MDF) of the bulge for the best-fit two-infall model compared to the observed MDF. In the upper panel, the blue shading shows the posterior predictive distribution of model MDFs as a function of [Fe/H], the red curve marks the single best-fit model realization, and the black crosses show the empirically measured, normalized MDF. The lower panel displays the residuals (mode… view at source ↗
Figure 8
Figure 8. Figure 8: Posterior [α/Fe]–[Fe/H] relations for individual bulge α-elements Mg, Si, Ca, and Ti for the best-fit two-infall model. In each panel, blue shading shows the posterior predictive distribution of the model abundance ratios, the red curve traces the median (best-fit) model track, and black points indicate the observed stellar abundances. The top and right insets give the corresponding one-dimensional [Fe/H] … view at source ↗
Figure 9
Figure 9. Figure 9: Posterior age–metallicity relation (AMR) for the bulge. The blue density field shows the weighted posterior ensemble of chemically acceptable models; the solid red line traces the MAP model. Individual stellar measurements are overplotted for comparison: Joyce et al. (2023) ages as red stars and Bensby et al. (2017) ages as blue triangles. The lower panel shows residuals in [Fe/H] (model minus data) as a f… view at source ↗
Figure 10
Figure 10. Figure 10: Posterior age–abundance relations obtained by mapping each model’s [α/Fe]–[Fe/H] track through the median posterior age–metallicity relation. Blue shading indicates the weighted posterior density, and the solid red curve shows the best model. Observed bulge abundances are transformed into age using the same AMR mapping and shown as black points. The panels show the intermediate-age, metal-rich population … view at source ↗
Figure 11
Figure 11. Figure 11: Corner plot of the joint posterior distribution in the continuous two–infall parameters obtained by combining 288 independent MCMC runs, one for each unique choice of categorical model ingredients [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Convergence of the GA posterior. Solid lines show the evolution of the 68% HPD ellipse size for several parameter pairs as a function of generation for a single GA run. Dotted lines show the corresponding HPD sizes measured from the combined pseudo–posterior used in the main corner plot. The rapid early decrease and subsequent plateau, together with the close agreement between solid and dotted curves, ind… view at source ↗

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