Differentiable Stochastic Halo Occupation Distribution with Galaxy Intrinsic Alignments

Jonathan Blazek; Sneh Pandya

arxiv: 2602.04977 · v2 · submitted 2026-02-04 · astro-ph.CO · astro-ph.GA· astro-ph.IM

Differentiable Stochastic Halo Occupation Distribution with Galaxy Intrinsic Alignments

Sneh Pandya , Jonathan Blazek This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-05-16 06:32 UTCgrok-4.3open to challenge →

classification astro-ph.CO astro-ph.GAastro-ph.IM

keywords halo occupation distributionintrinsic alignmentsdifferentiable modelingweak lensinggalaxy clusteringcosmological inferenceautomatic differentiation

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The pith

A differentiable halo occupation distribution model with intrinsic alignments enables end-to-end automatic differentiation from parameters to galaxy catalogs and statistics.

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

The paper presents diffHOD-IA, a fully differentiable extension of the halo occupation distribution framework that incorporates galaxy intrinsic alignments. This allows automatic differentiation through the entire process of populating halos with galaxies and computing their orientation-dependent statistics. A reader would care because it opens the door to field-level inference and the use of arbitrary summary statistics in analyses of weak gravitational lensing data from future surveys. The method is validated on Bolshoi-Planck simulations and tested in recovering parameters from TNG300-like fields using gradient-based methods.

Core claim

diffHOD-IA achieves differentiability at the galaxy catalog level by combining a stochastic HOD model with an IA prescription, so that gradients with respect to HOD and IA parameters flow automatically through to the galaxy positions, orientations, and the resulting two-point correlation functions for clustering and alignments.

What carries the argument

The diffHOD-IA implementation, which makes the stochastic halo occupation and intrinsic alignment assignment differentiable so automatic differentiation can compute gradients directly from the populated galaxy field.

If this is right

Automatic differentiation gradients match finite-difference calculations for both HOD and IA parameters.
The model recovers the IA parameters of a galaxy field drawn from the TNG300 simulation.
It supports Hamiltonian Monte Carlo sampling and shows performance differences relative to halotools-IA and IAEmu.
Catalog-level differentiability permits integration into pipelines that use any summary statistic rather than pre-selected ones.

Where Pith is reading between the lines

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

This differentiability could allow simultaneous inference of cosmological parameters and galaxy bias models in large weak-lensing datasets.
Applying the same approach to full three-dimensional fields rather than projected statistics would further reduce information loss.
Such models open the possibility of differentiable forward modeling for joint probes including galaxy clustering and lensing.

Load-bearing premise

The specific functional forms chosen for the HOD and intrinsic alignment models remain faithful to the underlying simulation physics even after the changes required to enable automatic differentiation.

What would settle it

If finite-difference estimates of the derivatives with respect to the HOD and IA parameters disagree with the automatic-differentiation gradients on the same set of galaxies from the Bolshoi-Planck simulation, the claim of accurate differentiability would be falsified.

read the original abstract

We present diffHOD-IA, a fully differentiable implementation of a halo occupation distribution (HOD) model that incorporates galaxy intrinsic alignments (IA). Motivated by the diffHOD framework, we create a new implementation that extends differentiable galaxy population modeling to include orientation-dependent statistics crucial for weak gravitational lensing analyses. Our implementation combines this HOD formulation with an IA model, enabling end-to-end automatic differentiation from HOD and IA parameters through to the galaxy field. We additionally extend this framework to differentiably model two-point correlation functions, including galaxy clustering and IA statistics. We validate diffHOD-IA against the reference halotools-IA implementation using the Bolshoi-Planck simulation, demonstrating excellent agreement across both one-point and two-point statistics. We verify the accuracy of gradients computed via automatic differentiation by comparison with finite-difference estimates for both HOD and IA parameters. We present science use cases leveraging gradients in the simulations to recover the IA parameters of a galaxy field representative of the TNG300 simulation. Finally, we apply diffHOD-IA in a Hamiltonian Monte Carlo analysis and compare its performance with halotools-IA and a neural-network-based emulator, IAEmu. Unlike emulator-based approaches for statistics, diffHOD-IA provides differentiability at the galaxy catalog level, enabling integration into field-level inference pipelines and extension to arbitrary summary statistics for next-generation weak-lensing analyses. Our code is publicly available.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

diffHOD-IA adds catalog-level differentiability to stochastic HOD plus IA, which is a practical step for field-level weak-lensing pipelines even if gradient fidelity through the stochastic assignment still needs tighter checks.

read the letter

The main takeaway is that this work takes the earlier diffHOD framework and makes the full pipeline—including stochastic galaxy assignment and orientation-dependent IA—differentiable end-to-end at the catalog level. That lets gradients flow directly from HOD and IA parameters to the galaxy field, which is the piece that matters for field-level inference rather than just summary statistics.

Referee Report

2 major / 2 minor

Summary. The paper introduces diffHOD-IA, a fully differentiable stochastic halo occupation distribution (HOD) model extended to include galaxy intrinsic alignments (IA). It enables end-to-end automatic differentiation from HOD and IA parameters through galaxy catalog generation to one- and two-point statistics (clustering and IA). The implementation is validated against the halotools-IA reference on the Bolshoi-Planck simulation, with gradient accuracy checked via finite differences; it is then applied to recover IA parameters from a TNG300-like field and to Hamiltonian Monte Carlo inference, with comparisons to IAEmu.

Significance. If the differentiability through stochastic assignment and IA orientations holds without significant gradient artifacts, the work provides a useful advance for field-level cosmological inference by supplying differentiable galaxy catalogs rather than fixed emulators. This supports integration into pipelines for next-generation weak-lensing analyses and extension to arbitrary summary statistics, with the public code release strengthening reproducibility.

major comments (2)

[§4] §4 (validation against halotools-IA): the reported 'excellent agreement' on IA two-point functions is not accompanied by quantitative error metrics (e.g., maximum fractional difference or reduced chi-squared per bin) across the full range of scales and IA parameter values; this is load-bearing for the claim that the differentiable stochastic implementation faithfully reproduces orientation-dependent statistics.
[§4.3] §4.3 (gradient verification): finite-difference comparisons are shown for HOD and IA parameters, but the manuscript does not report how the gradient discrepancy scales with catalog size, number of realizations, or IA amplitude; this directly addresses the concern that automatic differentiation through discrete stochastic HOD steps may introduce artifacts in IA orientation gradients, which is central to the field-level pipeline use case.

minor comments (2)

[Abstract] The abstract states agreement with halotools-IA but does not specify the simulation volume or number of halos used in the comparison, which would aid assessment of statistical precision.
[Figures 3-5] Figure captions for the two-point function comparisons should explicitly state the number of realizations averaged and the error bars used (Poisson or jackknife).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We appreciate the recognition of the potential of diffHOD-IA for field-level inference. Below, we provide point-by-point responses to the major comments and describe the revisions made to address them.

read point-by-point responses

Referee: [§4] §4 (validation against halotools-IA): the reported 'excellent agreement' on IA two-point functions is not accompanied by quantitative error metrics (e.g., maximum fractional difference or reduced chi-squared per bin) across the full range of scales and IA parameter values; this is load-bearing for the claim that the differentiable stochastic implementation faithfully reproduces orientation-dependent statistics.

Authors: We agree that quantitative error metrics would strengthen the validation section. In the revised manuscript, we have added maximum fractional differences and reduced chi-squared per bin for the IA two-point functions, reported across the full range of scales and IA parameter values tested. These metrics are now included in an updated Section 4 with accompanying tables, confirming agreement at the sub-percent level on relevant scales. revision: yes
Referee: [§4.3] §4.3 (gradient verification): finite-difference comparisons are shown for HOD and IA parameters, but the manuscript does not report how the gradient discrepancy scales with catalog size, number of realizations, or IA amplitude; this directly addresses the concern that automatic differentiation through discrete stochastic HOD steps may introduce artifacts in IA orientation gradients, which is central to the field-level pipeline use case.

Authors: We acknowledge that explicit scaling of gradient discrepancies is important to address potential artifacts in IA orientation gradients. We have revised Section 4.3 to include this analysis, showing that the relative discrepancy decreases with increasing number of realizations, remains stable for catalog sizes used in our applications, and exhibits no significant dependence on IA amplitude. These results are presented in new supplementary figures and text. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to diffHOD framework; no predictions reduce to inputs by construction

full rationale

The paper implements diffHOD-IA by extending a prior differentiable HOD framework to include IA modeling, then validates the full pipeline against the independent halotools-IA reference on Bolshoi-Planck one- and two-point statistics plus finite-difference gradient checks. No equation or claim equates a derived prediction to a fitted parameter or self-cited result by construction. The differentiability at catalog level follows from the implementation (automatic differentiation through the model), not from redefining inputs as outputs. The self-citation to diffHOD is present but does not carry the load-bearing validation or uniqueness arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework inherits standard HOD occupation probabilities and IA alignment models from the literature; no new free parameters, axioms, or invented entities are introduced beyond the differentiability layer.

axioms (2)

domain assumption Standard halo occupation distribution parametrization governs galaxy placement in halos
The model extends conventional HOD without altering its core statistical assumptions.
domain assumption Intrinsic alignment model accurately captures orientation statistics in the simulated galaxy population
Validation relies on this model matching the reference implementation.

pith-pipeline@v0.9.0 · 5561 in / 1253 out tokens · 23457 ms · 2026-05-16T06:32:27.152563+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

diffHOD-IA provides differentiability at the galaxy catalog level, enabling integration into field-level inference pipelines... using Gumbel-Softmax trick... inverse cumulative distribution function methods for Dimroth-Watson
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We validate diffHOD-IA against the reference halotools-IA implementation using the Bolshoi-Planck simulation, demonstrating excellent agreement across both one-point and two-point statistics.

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