Detection is truncation: studying source populations
with truncated marginal neural ratio estimation

Noemi Anau Montel
GRAPPA Institute
University of Amsterdam, The Netherlands
n.anaumontel@uva.nl

Christoph Weniger
GRAPPA Institute
University of Amsterdam, The Netherlands
c.weniger@uva.nl

Abstract
Statistical inference of population parameters of astrophysical sources is challenging. It requires accounting for selection effects, which stem from the artificial
separation between bright detected and dim undetected sources that is introducedby the analysis pipeline itself. We show that these effects can be modeled selfconsistently in the context of sequential simulation-based inference. Our approach
couples source detection and catalog-based inference in a principled framework thatderives from the truncated marginal neural ratio estimation (TMNRE) algorithm. It
relies on the realization that detection can be interpreted as prior truncation. We
outline the algorithm, and show first promising results.

1

Introduction

Point sources detection is crucial for astronomical surveys, and is the cornerstone for the compilationof source catalogues. Those source catalogues are then typically the basis for the inference of physicalparameters that describe the sources at the population level. Upcoming astronomical facilities, suchas the Square Kilometer Array (SKA) [1] and the Cherenkov Telescope Array (CTA) [2] will deliverlarge and complex datasets. In order to leverage their full potential, it is urgent to develop robust andautomated source detection and source population parameters inference algorithms.
Recent developments in deep learning and more generally automatic differentiation frameworks [3]are increasingly used for tackling difficult astronomical data analysis challenges. The capabilityof deep learning techniques of point sources detection and population characterization has been
demonstrated across different wavelengths surveys, e.g. in γ-ray data [4–7], radio data [8–11] andcosmic microwave background data [12]. In particular simulation-based machine learning approaches
can be highly flexible, allowing to tailor developed pipelines to specific telescopes and science cases.
A range of simulation-based inference (SBI) algorithms have been proposed in the literature (seeCranmer et al. [13] for a review). An appealing feature is that they generally allow to directly estimate
marginal posteriors for parameters of interest [14]. Furthermore, sequential SBI approaches [15–17]
have been shown to be particularly simulation efficient. Among those, truncated marginal neuralratio estimation (TMNRE) [14, 18] is a sequential SBI approach based on neural ratio estimation(NRE) [19], which particularly well composes with marginalization.
Here, we present a strategy for how to use TMNRE 1 to simultaneously perform source detection andpopulation-level parameters inference. This enables to self-consistently combine information fromboth detected and sub-threshold sources, without being affected by detection biases. The key idea isto recast the traditional concept of source detection in terms of prior truncation. This will allow us
to distill information of bright sources directly into the simulation model. Our proposed method ishighly interpretable since it resembles components of traditional survey analysis workflows.
1

We use swyft TMNRE implementation that can be found at https://github.com/undark-lab/swyft.

Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

2

Methodology

Background TMNRE (and NRE in general) performs posterior estimation by solving a binaryclassification problem. Given a model p(x, z) = p(x|z)p(z), where x is data, and z a set ofparameters of interest, one trains a network to distinguish joined samples x, z ∼ p(x, z) frommarginal samples x, z ∼ p(x)p(z). The networks learns to estimate the likelihood-to-evidence
ratio r(z; x) = p(x|z)/p(x) 2 , which we can use to obtain weighted samples from the posteriorp(z|x) = r(z; x)p(z). In order to improve the network’s learning and maximise the simulator
efficiency, TMNRE concentrates in stages the regions in parameter space from which training
examples are drawn based on a target observation. Hence, this truncation scheme restricts the priordistribution’s support without modifying its shape, as opposed to other sequential methods that employ
a posterior estimate as proposal distribution for generating simulations for the next round [15].
Simulation model We consider here a simple Bayesian hierarchical source model,p(x, ⃗s, ϑ) = p(x|⃗s)p(ϑ)

NYp(si |ϑ) ,

(1)

i=1

where x is the observed sky map, si ≡ (Fi , Ωi ) denotes the flux Fi and position Ωi of point sourcei, and ϑ ≡ {N, Σ, h} collects source population parameters, namely the number of sources N , andthe parameters Σ and h that control the flux and spatial distributions respectively. Finally, we add
instrumental effects to simulated maps. We give more details on the model in appendix A. We show
examples of our simulated maps in the top row of the right panel of fig. 2.
Source detection For source detection we consider the following likelihood-to-evidence ratio
r1 (Ω, Fth ; x) ≡p(Ix (F ≥ Fth ) = 1, Ω|x).
p(Ix (F ≥ Fth ) = 1, Ω)(2)Here, the denominator corresponds to the prior probability of having a source at position Ω with aflux F ≥ Fth that exceeds some threshold flux Fth . The numerator is the corresponding posterior.
We model the source detection ratio estimator in eq. (2) as an image-to-image neural network thatsolves a binary classification problem in each image pixel. For simulated data, we call a simulated
source si ‘detected’ when there is a corresponding compact region as function of Ω where thedetection significance is above threshold, r1 (Ω, Fth ; x) > 5. This effectively leads to a split between
sources that are clearly identifiable and ‘sub-threshold’ sources that are difficult to detect a individual
instances. Below, we assign the detection label di = 1 (di = 0) to detected (undetected) sources.
In order to characterize the split between detected and sub-threshold sources, we introduce a sourcesensitivity function, S(F, Ω), which provides the probability that a source with flux F and at positionΩ would be detected by the ratio estimator in eq. (2). This function can be estimated by training theratio estimatorp(d|F, Ω, x)(3)
r2 (d; F, Ω, x) ≡p(d)which is marginalised over all other sources and source parameters. By omitting the dependence on
the map, x, which is then effectively marginalized, the ratio estimator can then simply be modeled asa R3 → R multi-layer perceptron. The source-sensitivity function can then be estimated as 
p(d = 1|F, Ω)S(F, Ω) = σ log,(4)p(d = 0|F, Ω)
where we have introduced the sigmoid function σ(y) ≡ 1/(1 + e−y ).
We can make the concept of source detection part of our model as follows. In a random realization,
each source i will be either detected, di = 1 (with probability S(Fi , Ωi )), or not detected, di = 0(with probability 1 − S(Fi , Ωi )). To keep notation simple, we omit di and instead group detectedp(a,b|c)In the next section we will use the notation r(a; b|c) = p(a|c)p(b|c), where with ‘|’ we refer to conditioning
on specific variables for all factors in the ratio definition. If necessary, multiple variables are comma separated,p(a,b,c|d)for example r(a, b; c|d) = p(a,b|d)p(c|d). Training conditional ratios is a straight-forward extension of NRE.
22

Figure 1: Illustration of our inference framework (see section 2 for details). Left panel: A source
detection network r1 and the corresponding sensitivity network r2 are trained based on the full
simulation model. Right panel: Bright sources are constrained in the truncated simulation model,
while sub-threshold sources vary freely. Two inference networks are trained to capture informationfrom sub-threshold sources (r3 ) and detected sources (r4 ).

and non-detected (sub-treshold) sources together in vectors with corresponding subscripts, and write⃗s ≡ (⃗sdet , ⃗ssub ). Taking into consideration this split, our model becomesp(x, ⃗sdet , ⃗ssub , ϑ) = p(x|⃗sdet , ⃗ssub )p(⃗sdet |ϑ)p(⃗ssub |ϑ)p(ϑ) .

(5)Importantly, although p(⃗sdet/sub |ϑ) depends on the sensitivity function S(Fi , Ωi ), the distributionof all sources p(⃗s|ϑ) is the same as in eq. (1).
Detection as truncation Truncating source priors in eq. (1) is difficult due to the the label switchingproblem (x is invariant under relabeling sources). However, once specific sources are detected, they
can be labeled and ordered arbitrarily. Let us assume that Ndet sources were detected by the ratio
estimator in eq. (2) in the regions Ri with i = 1, . . . , Ndet for a given observation of reference xo .
Our ansatz for the indicator function, which selects a specific prior region, is
Ixo (⃗sdet ) =

N
det
Y

Ixo (Ωi ∈ Ri )Ixo (Fi ≥ Fth ) .

(6)

i=1

Our truncation strategy is now to focus on the parameter space where Ixo (⃗sdet ) = 1 for our data ofinterest xo . We can then write our truncated model asp(x, ⃗sdet , ⃗ssub , ϑ, Ixo (⃗sdet )) = p(x|⃗sdet , ⃗ssub )p(⃗sdet |ϑ, Ixo (⃗sdet ))p(⃗ssub |ϑ)p(ϑ)p(Ixo (⃗sdet )|ϑ). (7)Population parameters inference with truncation Given some observation xo , we want toestimate the posterior p(ϑ|xo ). Since the truncation affects parameters ⃗sdet whose prior depends on
population parameters ϑ, the truncation volume is not a constant factor, and the procedure requires
extra care. We can estimate the posterior p(ϑ|x) by considering the ratior(ϑ; x) =p(ϑ|x) p(ϑ|x, Ixo (⃗sdet ) = 1) p(ϑ|x, Ixo (⃗sdet ) = 1) p(ϑ|Ixo (⃗sdet ) = 1)
≃
=
. (8)
p(ϑ)
p(ϑ)
p(ϑ|Ixo (⃗sdet ) = 1)
p(ϑ)
|
{z
}|
{z
}
≡r3 (ϑ;x|Ixo (⃗sdet )=1)≡r(ϑ;Ixo (⃗sdet )=1)

The second step in eq. (8) corresponds to the truncation approximation. It is exact (in the senseof leaving p(ϑ|xo ) unaffected) in the limit where p(xo |Ixo (⃗sdet ) = 0) → 0. Training that ratio
estimator directly with TMNRE would be challenging since Ixo (⃗sdet ) = 1 has very small support in
the training data. Instead, we split the ratio into two computationally feasible ratios (this is in spirit
similar to the telescoping ratio estimation approach presented in Rhodes et al. [20]).
3

Figure 2: Left: Observation of reference xo . Right: In the first row we show samples from our full
simulation model. In the second row we show targeted samples from the truncated model. Targeted
data are visually more close to xo , the main dissimilarities are due to different sub-threshold sources
and instrumental effects realizations.

The first ratio, r3 (ϑ; x|Ixo (⃗sdet ) = 1), can be estimated by training a peak-count network [21–23]
on targeted data, that is truncated to Ixo (⃗sdet ) = 1. The second ratio can be estimated asp(Ixo (⃗sdet ) = 1|ϑ)r(ϑ; Ixo (⃗sdet ) = 1) =p(Ixo (⃗sdet ) = 1)ZZp(⃗sdet |ϑ) p(⃗sdet )Ixo (⃗sdet )p(⃗sdet |ϑ)= d⃗sdet= d⃗sdetp(⃗sdet |Ixo (⃗sdet ) = 1) , (9)p(⃗sdet ) p(Ixo (⃗sdet ) = 1)p(⃗sdet )| {z }≡r4 (⃗
sdet ;ϑ)so it can be estimated by training a network on detected sources lists on un-truncated training data.
In practice, we generate weighted samples from the full posterior p(ϑ|x) by sampling ϑ, ⃗sdet ∼p(ϑ)p(⃗sdet |Ixo (⃗sdet ) = 1) with weights w = r3 (ϑ; x|Ixo (⃗sdet ) = 1) · r(ϑ; Ixo (⃗sdet ) = 1).
In the process, we trained four ratio estimation networks that are directly connected with traditionalsource analysis pipeline components: r1 (Ω, Fth ; x) performs source detection; r2 (d; F, Ω, x) is thesource sensitivity function; r3 (ϑ; x|Ixo (⃗sdet ) = 1) constraints ϑ based on sub-threshold sources(because detected sources are assumed to be fixed in the parameter space where Ixo (⃗sdet ) = 1); andr4 (⃗sdet ; ϑ) constraints ϑ using the detected sources catalog. We show a schematic overview of theinference framework used in this work in fig. 1.

3

Results

We apply the proposed methodology to the target observation xo shown in fig. 2. We first train thesource detection ratio estimator in eq. (2) on data simulated from the full model shown in eq. (1),and then apply it to xo to obtain a detection map. From the detection map we derive a catalog ofdetected sources ⃗sdet , and define a truncated parameter space of interest, where Ixo (⃗sdet ) = 1. Inorder to make source detection part of our model, we train the sensitivity ratio estimator in eq. (3) toestimate the sensitivity function S(F, Ω). We then generate targeted training data from our truncatedsimulation model in eq. (7). We show samples from the full model and the truncated one in fig. 2.
Finally, we train two inference networks to capture information regarding population parameters fromsub-threshold and detected sources, as explained in section 2. We show the constraints on population
parameters ϑ from sub-threshold sources, detected ones, and their combination in fig. 3. The
different posteriors are consistent with each other, indicating that the proposed inference framework
automatically accounts for detection biases. We see that different constraints are dominated bydifferent neural networks, e.g. the number N of point source is better constrained by sub-thresholdones, whereas the spatial distribution parameter h by detected ones. The weaker constraint on theflux parameter Σ inferred from detected sources is due to the fact that in eq. (9) we average overdifferent detected sources realisations, always re-sampling the parameter Σ (see appendix A for thehierarchical model details).
The four neural networks were trained on a NVIDIA GeForce RTX 3080 Ti GPU, the total computational time cost to obtain the results shown in fig. 3 is ∼ 2 hours. We intend on describing in details
the networks used for each task and choice of hyperparameters in future work.
44Conclusions

We have introduced a novel method to selfconsistently perform point sources detection andsource population parameters inference using
TMNRE. With this approach, we can exploit
information of detected as well as sub-thresholdsources for population-level parameter inference.
Detection biases are automatically accounted
for in our approach. Exemplary results of ourapproach are shown in fig. 3, where we show
inference results on source population parameters from both detected point sources and subthreshold sources separately, as well as their
combination. Since the proposed method is essentially a specific implementation of TMNRE,
we expect that it inherits its positive properties
in terms of simulation-efficiency and scalability [18]. A possible shortcoming of this approach is that multiple neural networks need to
be trained self-consistently, which on the other
hand have a clear interpretation in terms of traditional analysis pipeline components. A potential
application beyond those directly intended isto detectable and sub-threshold substructures instrong gravitational lenses.

Figure 3: Marginal posteriors inferred by TMNRE
on target observation xo for population parameters ϑ. We show constraints from sub-thresholdsources (violet), detected sources (orange), andcombined results (yellow). In the 2D marginal
posterior we indicate the 68% and 95% credible
regions with solid and dashed lines respectively.
For more details see section 3.

Broader Impact This work is focusing on the analysis of astronomical surveys that contain point
sources population via TMNRE. Variants of the presented approach could find application in otherareas of the physical sciences. We do not expect any negative societal impact of the presented
methods. However, we recommend the usual caution in inferring scientific conclusions based on acomplex methodology.

Acknowledgments and Disclosure of Funding
This work is part of a project that has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (Grant agreement No.
864035 – Undark).
We acknowledge the use of the python [24] modules, matplotlib [25], numpy [26], scipy [27],
PyTorch [28], tqdm [29], and jupyter [30].

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Table 1: Point source simulation model parameters and priors.
Parameter
Prior

A

Population parameters
number of point sources
flux distribution parameterspatial distribution parameterNΣhU(10, 500)
U(1, 3)
U(1, 20)

Point source parameters
flux
positionFΩ ≡ (l, b)

log N (1, Σ)(N (0, 20), N (0, h))Appendix: Simulation model

We describe in more detail the Bayesian hierarchical point source model adopted in this work. To
generate an observation, first, we sample point sources population parameters ϑ ≡ {N, Σ, h} fromtheir priors, given in table 1. Then, for each point source, we draw its flux F and position on themap Ω ≡ (l, b) from the priors given in table 1. We then generate a 128 × 128 pixels map. To model
instrumental effects, we add a point-spread function (PSF) with Gaussian kernel standard deviation
ϵ = 1.5 and Poisson noise, obtaining the final simulated map x.

8

Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
contributions and scope? [Yes] As stated in the abstract and contribution, the main scopeof this work is to propose the methodology described in section 2, whose exemplary
results we show in section 3(b) Did you describe the limitations of your work? [Yes] We have presented work-inprogress results and briefly mentioned possible shortcomings of this method in section 4.
(c) Did you discuss any potential negative societal impacts of your work? [N/A] The
proposed methodology applies to astronomical data, we do not anticipate any negative
societal impacts from it. However, we have recommended caution when using acomplex analysis machinery in section 4.
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were run using publicly available code, while our simple simulation model is describedin details in appendix A. We plan on describing the networks used for each task infuture works.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
were chosen)? [No](c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No](d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)? [Yes]4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes](b) Did you mention the license of the assets? [No](c) Did you include any new assets either in the supplemental material or as a URL? [No](d) Did you discuss whether and how consent was obtained from people whose data you’reusing/curating? [N/A] We only use open-source code and data simulated from ourmodel.
(e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content? [N/A] The data is generated by a physical models
which abstractly relate parameters to astronomical data.
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
applicable? [N/A](b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable? [N/A](c) Did you include the estimated hourly wage paid to participants and the total amountspent on participant compensation? [N/A]9