Simulation-based inference for AGN jet population modelling: Towards more robust comparisons of black hole jet speeds

Clara Lilje; James H. Matthews; Rob Fender

REVIEW 3 major objections 4 minor 287 references

Simulation-based inference for AGN jet population modelling: Towards more robust comparisons of black hole jet speeds

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Simulation-based inference puts AGN jet-speed slope at b = -1.32, matching XRBs.

desk verdict A careful, well-calibrated SBI reanalysis of MOJAVE FSRQs; the b posterior is credible conditionally, but the known maximal-speed bias runs in the direction that makes the 2-sigma XRB agreement look better than it may be. read the letter →

arxiv 2608.08733 v1 pith:PVVUDHRS submitted 2026-08-09 astro-ph.HE

Clara Lilje , James H. Matthews , Rob Fender This is my paper · ORCID

classification astro-ph.HE

keywords activegalacticnucleijetsLorentzfactordistributionsimulation-basedinferencenormalisingflowsneuralposteriorestimationMOJAVEX-raybinaries

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish the parent distribution of jet speeds for radio-loud active galactic nuclei, after correcting for the severe observational bias imposed by a flux limit and relativistic beaming. Using likelihood-free simulation-based inference—neural posterior estimation with a normalising flow—on the 174 FSRQs of the MOJAVE 1.5 Jy sample, it claims the intrinsic Lorentz factor distribution is $N(\Gamma)\propto\Gamma^b$ with $b=-1.32^{+0.20}_{-0.19}$. The authors argue this calibrated posterior is substantially broader and more asymmetric than earlier grid-based estimates, so previous error bars understated uncertainty. If true, the AGN and X-ray binary jet-speed distributions are consistent at the $2\sigma$ level, hinting at shared jet physics across black hole masses. The wider point is that simulation-based inference is a workable route to population inference when multiple observables and selection effects make the likelihood intractable.

What carries the argument

The load-bearing machinery is the combination of a Monte Carlo parent-population simulator and a conditional neural spline flow trained by neural posterior estimation. The simulator generates synthetic MOJAVE-like samples of 174 FSRQs from priors on the Lorentz-factor slope $b$, luminosity-function slope $\gamma$, beaming exponent $p$, and redshift-evolution parameters $k$ and $\eta$, applies the $1.5\,$Jy flux cut, and produces per-object observables (flux, redshift, apparent speed, beamed luminosity, plus a missing-data flag). A permutation-invariant encoder aggregates the per-object features into a context vector, and the normalising flow learns $p(\theta | x)$ directly; calibration is checked with simulation-based calibration, coverage, TARP, an MMD misspecification test, and a local classifier two-sample test. The flow's job is to turn simulator outputs into a full posterior that captures degeneracies and asymmetry without an analytic likelihood.

What would settle it

Rerun the same SBI pipeline on the same MOJAVE data with the simulator's $\Gamma_{\min}$ changed from 1.25 to 2 and with a broken power-law; if the posterior for $b$ shifts by more than the quoted uncertainty, the central claim depends on an unverified cutoff.

Watch

Extended reading notes

Core claim

The central discovery is a new posterior for the underlying jet-speed (Lorentz factor) distribution of the MOJAVE FSRQ population, obtained without writing down a likelihood. The paper's headline result is $b = -1.32^{+0.20}_{-0.19}$ for $N(\Gamma) \propto \Gamma^b$, with $\Gamma$ between 1.25 and 50, which is consistent at $2\sigma$ with the Lorentz-factor exponent inferred for X-ray binary jets. Equally central is the claim that the posterior is non-Gaussian and wider than the grid-based parameter estimates of Lister et al. (2019); the flow-based inference also exposes a strong degeneracy between the two redshift-evolution parameters $k$ and $\eta$, which the Anderson-Darling grid method could not characterise. The authors interpret this as showing that previous comparisons underestimated the statistical uncertainty and that, once properly accounted for, AGN and XRB jet speed populations are statistically compatible.

Load-bearing premise

The load-bearing premise is that the Lister et al. (2019) simulator—power-law Lorentz factors on $[1.25,50]$, isotropic inclination, pure luminosity evolution $(1+z)^k$, and selection purely by the $1.5\,$Jy flux cut—is the true parent population; if any of those forms or cut-offs are wrong, the calibrated posterior can be precise but biased.

Editorial extensions

If this is right

Previous grid-based error bars for AGN parent-population parameters are too small; reanalyses using full posteriors will widen confidence intervals in population comparisons.
The AGN and XRB jet Lorentz-factor distributions are consistent at $2\sigma$, so differences between them cannot be claimed as significant without stronger data.
The method can be applied to other flux-limited, multi-observable astrophysical populations where the likelihood is intractable but forward simulation is possible.
Calibration diagnostics (SBC, TARP, MMD, L-C2ST) should accompany neural posterior estimates to guard against over-confidence.
The $k$–$\eta$ degeneracy shows that redshift evolution is not independently constrained by this sample; external redshift-evolution measurements would be needed to break it.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the posterior is as broad as reported, then earlier claims of a significant AGN–XRB speed difference may have been driven by underestimated uncertainties; re-testing with the full posterior would quantify this.
The low value of $\Gamma_{\min}=1.25$ is degenerate with the slope $b$; applying the same flow to simulated samples with $\Gamma_{\min}=2$ or 5 would test how much of the result depends on this choice.
The consistency of $b$ across radio-, X-ray- and $\gamma$-ray-selected samples hints that the Lorentz-factor distribution is set by jet launching physics rather than by the band used to select sources, but confirming this requires samples selected at other wavelengths with the same missing-data treatment.
The missing-data handling via a $-1$ flag and indicator channel could be reused for other surveys with partial measurements, and the permutation-invariant encoder is a template for population-level SBI beyond jets.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The thing to know: this is a methodologically serious reanalysis of the MOJAVE FSRQ parent population, and the headline b = -1.32 (+0.20, -0.19) is probably the most carefully calibrated estimate of that slope to date. The XRB comparison is suggestive, not established, because the simulator-level validation cannot fix a known model-data mismatch that pushes b in the direction of agreement.

What is genuinely new: previous grid-based errors from Lister et al. (2019) were essentially grid spacings; the SBI posteriors here are broader, asymmetric, and explicitly calibrated. The validation is unusually thorough for this literature—SBC, TARP, empirical coverage, L-C2ST, and an MMD misspecification test are all reported, and the authors check that fixing p = 2.0 leaves the results stable. Those tests do what they claim: they show the flow recovers the posterior of the simulator. They do not, and cannot, show the simulator is the right description of the sky.

That limitation matters more than the paper lets on. The MMD p-value of 0.994 is read as "strongly disfavours misspecification," but a non-rejection on 174 objects is weak evidence for the model class. More telling is the acknowledged mismatch in Section 4.2: observed apparent speeds are maximal values over multiple jet components, while the simulator draws one beta_app per source. The paper concedes this can bias b toward flatter power laws—the same direction that brings AGN closer to the XRB posterior. So the central numerical result is well-calibrated conditional on Lister 2019's generative model, but the point estimate could be precise-and-biased if that model or the maximal-speed treatment is wrong.

The XRB comparison is a same-group continuation and also depends on the posterior from Lilje et al. (2025). That is not circular, but it is another reason to want independent confirmation before taking the 2-sigma consistency as evidence for scale-invariant jet physics.

I would send this to peer review and require, as a condition, release of the simulator, training scripts, seeds, or pretrained flow artifacts, plus a robustness appendix varying Gamma_min, Gamma_max, and the single-component versus maximal-beta_app choice. The methodology and the correction to previous error bars are valuable; the cross-mass consistency claim needs one more layer of scrutiny.

Referee Report

3 major / 4 minor

Summary. The paper applies simulation-based inference (SBI) with neural posterior estimation to model the parent population of AGN jets using the MOJAVE 1.5 Jy FSRQ sample. The authors adopt the generative model of Lister et al. (2019): a power-law Lorentz factor distribution N(Γ) ∝ Γ^b on [1.25, 50], isotropic inclinations, a power-law luminosity function with pure luminosity evolution, and selection by a 1.5 Jy flux limit. They train a neural spline flow with per-object and permutation-invariant embeddings, and validate it with SBC, TARP, empirical coverage, an MMD misspecification test, and L-C2ST. The main quantitative claim is b = -1.32 (+0.20, -0.19), with posteriors that are broader and more asymmetric than previous grid-based estimates. They further claim that this AGN Lorentz factor distribution is consistent at the 2σ level with the XRB jet population posterior taken from a companion paper (Lilje et al. 2025). The paper concludes that SBI is well suited to population studies of jetted AGN with multiple observables.

Significance. If the central claim holds, the paper makes a useful methodological contribution by showing that SBI can produce calibrated, non-Gaussian posteriors for AGN jet parent-population parameters where grid-based Anderson-Darling fits cannot. The extensive calibration suite (SBC, TARP, coverage, MMD, L-C2ST) is a genuine strength, and the authors are appropriately cautious in several places about flow pathologies. The claim that AGN and XRB jet speed distributions are compatible at 2σ, if robust, would strengthen the case for common jet physics across black hole mass scales. However, the headline value of b is a fitted quantity whose accuracy depends on the simulator being an adequate model of the true parent population, and the paper's own discussion acknowledges a known mismatch in how apparent speeds enter the analysis. Because the 2σ agreement is the paper's main astrophysical conclusion, the robustness of b to this and other model-form choices is load-bearing.

major comments (3)

[Section 2.2.2 and Section 3] The MMD test is reported as yielding p = 0.994 and is described as 'strongly disfavours misspecification'. This interpretation is not supported: a high p-value only means that the 174-object observed sample is not unusual under the simulated distribution; it does not establish that the simulator is a good proxy for the true parent population, and the test has limited power against many structured misspecifications. Please replace this wording with a more neutral statement, and ideally provide a power analysis or a demonstration that the MMD test can reject alternative simulators (e.g., with different Γmin, broken power laws, or different missingness mechanisms).
[Section 4.2] The paper acknowledges that observed apparent speeds are maximal values over multiple jet components, while the simulator draws a single β_app per source from one Γ and θ, and states that this may bias b toward flatter power laws. Since the central scientific result is the value of b and its 2σ agreement with the XRB posterior, this is a load-bearing caveat, not a peripheral one. The direction of the suspected bias is exactly the direction that would ease agreement with the XRB sample. Please quantify the impact: for example, modify the simulator to draw multiple components per source and take the maximum, or explicitly demonstrate that the inferred b shifts by less than the reported uncertainty under such a change. Without this, the central claim can be precise but biased.
[Table 1 and Section 2.1] The posterior for b is only meaningful if the assumed generative model is approximately correct: N(Γ) ∝ Γ^b on [1.25, 50], isotropic inclinations, pure luminosity evolution, and selection solely by the 1.5 Jy flux limit. The calibration tests validate the flow against its own simulator and cannot detect misspecification of this model class. The paper's MMD test is too weak to resolve this. I recommend adding sensitivity analyses that vary the functional form (e.g., a broken power law or a high-Γ cutoff) and the fixed boundaries (Γmin, Γmax), and showing how the inferred b changes. The current manuscript gives no quantitative indication of the robustness of b to these choices.

minor comments (4)

[Section 2.2.2] The name 'Kalmogorov-Smirnov' should be 'Kolmogorov-Smirnov'.
[Section 3] The statement that a L-C2ST p-value of 0.82 allows one to 'say with 95% confidence that the estimator is valid' is statistically imprecise; a p-value above 0.05 only means the test does not reject validity, not that validity has 95% confidence.
[Section 2.2.1] The missingness mechanism is described as randomly choosing 10%–50% of apparent speed values to be missing, but the actual missingness in the MOJAVE sample may not be missing at random. A brief justification or a sensitivity test of the missingness fraction and pattern would strengthen the preprocessing description.
[Section 3] The comparison to Lister et al. (2019) uses the spacing between adjacent grid points as the literature uncertainty; this is not a statistical uncertainty and should be labelled as such, since the SBI posteriors are on a different footing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the AGN b-posterior is an empirical fit to MOJAVE data, and the XRB comparison uses an independent kinematic sample.

full rationale

The paper's central result, b = -1.32(+0.20/-0.19), is obtained by training a normalising flow on simulations generated from a parent-population model and then applying it to the observed MOJAVE FSRQ data. This is a standard likelihood-free inference, not a fit-then-predict loop: the observed data enter as the evaluation input, not as a fitted calibration target. The functional form N(Gamma) proportional to Gamma^b is an assumption taken from Lister et al. (2019), but estimating b within that family from real data is exactly what the paper claims, so it is not self-definitional. The calibration tests (SBC, TARP, coverage, L-C2ST) are self-consistency checks on the flow; the paper explicitly acknowledges this in Section 3: 'these tests are exclusively simulation-based calibration, so they can only tell us how well the flow performs on simulations generated with the same framework it is trained on.' The additional MMD test compares simulated and observed data, and the reported p=0.994 is a goodness-of-fit statement, not a circular prediction. The XRB comparison in Section 4.2 relies on the posterior from Lilje et al. (2025), a companion paper by overlapping authors, but that posterior is derived from XRB jet kinematics, a distinct data set. The agreement at 2-sigma is an external comparison; the current paper does not define the XRB posterior in terms of its own AGN fit. The explicit caveat about maximal apparent speeds biasing toward flatter power laws is an honest limitation and further indicates that the authors do not treat the comparison as forced. No equation or fitted parameter is renamed as a prediction, and no load-bearing argument reduces to a self-citation chain. Therefore the derivation is self-contained and no circularity is present.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests almost entirely on the parent-population simulator inherited from Lister et al. (2019). The free parameters are all fitted via the normalizing flow; none is independently measured. No new physical entities are introduced. The main systematic risk is model-form misspecification, which the MMD test cannot exclude.

free parameters (5)

b (Lorentz factor power-law index) = -1.32 (+0.20, -0.19)
Central parameter of the claimed result; inferred via NPE and reported as 68% credible interval (abstract, Figure 1).
gamma (luminosity function power-law index) = -2.86 (+0.20, -0.26)
Inferred by the same flow; quoted in Section 4.1 when comparing with the Fermi-LAT sample.
eta (redshift evolution normalization exponent) = not quoted; prior [-0.65, -0.25]
Inferred parameter entering e(z) = (1+z)^k e(z/eta); shows a strong degeneracy with k (Figure 1).
k (redshift evolution power index) = not quoted; prior [4.5, 9.5]
Inferred parameter, degenerate with eta, which limits the physical interpretation of the redshift evolution.
p (beaming exponent, p = 2 - alpha) = not quoted; posterior width at prior grid resolution
Inferred beaming exponent; the flow finds it very constrained, and fixing p = 2 leaves the other posteriors consistent.

assumptions (5)

domain assumption Parent population functional forms from Lister et al. (2019) are correct: N(Gamma) prop Gamma^b, luminosity power law with pure luminosity evolution, isotropic inclination.
Table 1 and Section 2.1. If these forms are misspecified, the b posterior is systematically biased.
domain assumption Selection is fully described by the 1.5 Jy radio flux limit.
Section 2.1. No other incompleteness or wavelength-dependent selection is modelled.
domain assumption Apparent-speed missingness is random and well approximated by masking 10% to 50% of simulated values.
Section 2.2.1. If missingness correlates with jet orientation or speed, the posteriors can shift.
domain assumption The trained normalizing flow is a faithful posterior approximator, and uniform priors are uninformative.
Section 2.2 and Appendix. Supported by SBC, TARP and L-C2ST, but these are mostly internal consistency checks.
domain assumption The XRB posterior from Lilje et al. (2025) is a valid external benchmark.
Section 4.2. The 2 sigma consistency claim inherits the assumptions of that companion paper.

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Cite this review

Pith. "Pith review of Simulation-based inference for AGN jet population modelling: Towards more robust comparisons of black hole jet speeds." pith.science (2026). https://pith.science/paper/PVVUDHRS

@misc{pith2026260808733,
  author       = {Pith},
  title        = {Pith review of: Simulation-based inference for AGN jet population modelling: Towards more robust comparisons of black hole jet speeds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PVVUDHRS}},
  note         = {Machine review of arXiv:2608.08733}
}

abstract

We present the most complete modelling of the MOJAVE 1.5 Jansky Quarter Century active galactic nuclei (AGN) jet population, using likelihood-free simulation-based inference. Due to the complex impact of a flux-limit on observed AGN data sets, careful modelling of the parent population is required. In particular, when observing and fitting to multiple data distributions likelihoods become non-intuitive. Parameter degeneracies further complicate the problem and make it suitable for likelihood-free, simulation-based inference. This method relies on a normalising flow learning the likelihood surface or posteriors directly. We extensively validate the flow to show that previous parameter estimates for the AGN jet speed distributions underestimated parameter errors significantly and do not capture the non-gaussianity of the parameter posteriors. The new results enable a better statistical comparison to other AGN population studies, but also a more accurate comparison of supermassive black hole jets with their lower mass counterparts, X-ray binaries (XRB). We find that the AGN follow a Lorentz factor distribution of the shape $N(\Gamma) \propto \Gamma^b$ with $b= -1.32_{-0.19}^{+0.20}$. This slope is consistent with the XRB Lorentz factor distribution at $2\sigma$. Simulation-based inference as a method is generally well-suited to many astrophysical problems, and this paper shows the convenient applicability of this methodology to parent population studies of jetted AGN with multiple observables specifically.

Figures

Figures reproduced from arXiv: 2608.08733 by the authors.

**Figure 1.** The parameter posteriors from the simulation-based inference model for the FSRQ sub-sample of the MOJAVE 1.5 Jy Quarter century AGN parent population. The red stars indicate the best fit values from Lister et al. (2019). The uncertainties quoted for each parameter are at the 1𝜎 level. 0.82 we can once again say with 95% confidence that the estimator is valid for our observation. When investigating the correlation of… view at source ↗

**Figure 2.** Comparison of posterior for the Lorentz factor exponent 𝑏, if 𝑁 (Γ) ∝ Γ 𝑏. We show both the posteriors for the XRB large-scale jet population as taken from Lilje et al. (2025) and the posterior for the FSRQ sub-sample of the MOJAVE AGN sample obtained by SBI in this work. A vertical line marks the mean of the population and the arrow markers indicate the 2𝜎 intervals of both posteriors. As a reference we also show t… view at source ↗

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