Recovering Gamma-Ray Burst Redshift Completeness Maps via Spherical Generalized Additive Models

Abstract

We present an advanced statistical framework for estimating the relative intensity of astrophysical event distributions (e.g., Gamma-Ray Bursts, GRBs) on the sky tofacilitate population studies and large-scale structure analysis. In contrast to the traditional approach based on the ratio of Kernel Density Estimation (KDE), which is characterized by numerical instability and bandwidth sensitivity, this work applies a logistic regression embedded in a Bayesian framework to directly model selection effects. It reformulates the problem as a logistic regression task within a Generalized Additive Model (GAM) framework, utilizing isotropic Splines on the Sphere (SOS) to map the conditional probability of redshift measurement. The model complexity and smoothness are objectively optimized using Restricted Maximum Likelihood (REML) and the Akaike Information Criterion (AIC), ensuring a data-driven bias-variance trade-off. We benchmark this approach against an Adaptive Kernel Density Estimator (AKDE) using von Mises–Fisher kernels and Abramson’s square root law. The comparative analysis reveals strong statistical evidence in favor of this Preconditioned (Precon) Estimator, yielding a log-likelihood improvement of $\Delta \mathcal{L}\approx 74.3$ (Bayes factor $>{10}^{30}$) over the adaptive method. We show that this Precon Estimator acts as a spectral bandwidth extender, effectively decoupling the wideband exposure map from the narrowband selection efficiency. This provides a tool for cosmologists to recover high-frequency structural features—such as the sharp cutoffs—that are mathematically irresolvable by direct density estimators due to the bandwidth limitation inherent in sparse samples. The methodology ensures that reconstructions of the cosmic web are stable against Poisson noise and consistent with observational constraints.

1. Introduction

In high-energy astrophysics and galactic astronomy, the analysis of the directional distribution of events (e.g., cosmic rays, neutrino sources, or specific stellar populations) requires rigorous non-parametric density estimation [1]. The analysis of cosmological data poses a fundamental epistemological challenge: while theoretical frameworks describe the evolution of continuous physical spaces (e.g., the cosmic matter density space, $\rho \left(\mathbf{x}\right)$), observational campaigns yield discrete point sets (e.g., galaxy coordinates) or pixel intensity maps. Modern cosmological surveys, such as the Sloan Digital Sky Survey, the Dark Energy Survey, and the next-generation missions such as Euclid, provide celestial catalogs of unprecedented volume and depth. These datasets, while rich in information, remain fundamentally discrete-stochastic collections of tracers on the celestial sphere.

1.1. The Cosmological Principle and the Large-Scale Distribution of Gamma-Ray Bursts

The observation of GRBs yields a distribution of events on the whole sky. While the angular coordinates are usually determined immediately upon detection, the event’s cosmological redshift, z, is measured for only a subset, and after the event. These redshift measurements are obtained via two spectroscopic channels: observation of the optical afterglow or the spectroscopic analysis of the host galaxy. A fundamental requirement for any rigorous angular analysis of these sources is the determination of the sky exposure function, which gives the non-uniform detection probability across the sky. Here, we aim to reconstruct this exposure function and we also seek to determine whether separating the GRB population with measured redshift into distinct subsets based on the measurement method (afterglow-derived versus host-derived) yields a statistically superior model compared to a unified analysis. This allows us to assess if the selection effects inherent to these two methods favor a split treatment (Split Analysis) or if the samples are statistically compatible a for unified analysis (Lumped Analysis).

Due to their extraordinary luminosity, gamma-ray bursts serve as a powerful tool for studying distant regions and large-scale structures in the Universe [2,3,4,5]. Since they can be observed across vast cosmological distances, they provide a unique opportunity to study the distribution of matter and cosmic structures, as well as the physical conditions and evolution of galaxies in the early Universe. This includes analyzing the metal content and gas concentration of the interstellar medium (ISM) in their host galaxies [4,6]. Functioning as cosmological beacons, GRBs thus provide critical information about the evolution of the Universe in terms of star formation processes [7] and large-scale environmental effects. However, any cosmological inference drawn from GRB distributions—such as the evolution of the luminosity function or the cosmic event rate—is critically sensitive to observational selection effects. Recent studies have demonstrated that neglecting redshift incompleteness can lead to significant systematic biases in the reconstructed star formation rate and luminosity evolution [8,9,10]. Therefore, reconstructing the selection function (or completeness map) using a mathematically rigorous and numerically stable framework is not just a statistical task but a basic prerequisite for precision cosmology with GRBs.

The Cosmological Principle states that the Universe is homogeneous and isotropic on a large scale. This principle theoretically limits the largest size within which structures could have formed without violating the age of the Universe. This limit is generally set at around 260 h⁻¹ Mpc [11]. However, precision observations in recent years have identified an increasing number of structures arising from the clustering of bright astronomical objects that exceed this theoretical limit. These challenging structures include large-scale features traced by gamma-ray bursts, such as the Giant GRB Ring [12,13] and the Hercules-Corona Borealis Great Wall (HerCrbGW) [14,15]. As the largest known structure, with an estimated diameter of 2.5–3 Gpc, the HerCrbGW raises serious questions about its compatibility with the Cosmological Principle, underscoring the need for a closer connection between large-scale cosmological models and local observations. The HerCrbGW was initially discovered in a sample of 283 known redshifted GRBs in the range $1.6\le z\le 2.1$ and was later confirmed and extended [16,17]. However, since GRBs are transient events and their detection rate is strongly influenced by selection effects, the true size and shape of these large-scale structures remain uncertain [18]. Overcoming these uncertainties is critical and requires robust, non-parametric methodologies for density estimation, such as those presented in this article. Previous attempts to characterize these distributions have relied on density-ratio methods or geometric tessellations, which often introduce artifacts especially in the low-exposure regions. Our goal is to create a framework that preserves structural fidelity without the strict bandwidth limitations imposed by sparse datasets.

1.2. The Problem of Sky Coverage Completeness for Redshifted GRBs

The observed angular distribution of GRBs is not uniform, but is modulated by a complex, varying ‘sky exposure function’ [19]. This function represents a combined probability density that depends on the celestial mechanics and technical specifications of the detecting satellite (e.g., detector sensitivity, field of view, and pointing history). Accurately reconstructing this modulator is essential for distinguishing between instrumental non-uniformity and clustering of matter on cosmological scales. Furthermore, the probability of detecting optical afterglow, a prerequisite for determining redshift and detailed characterization, is governed by logistical factors related to the ground-based tracking network. These factors include telescope availability, local meteorological conditions, lunar phase, and human-induced variables such as academic schedules or maintenance periods. Consequently, the effective selection function is a complex surface probability density derived from the convolution of satellite dynamics and ground constraints.

Let ${\mathcal{D}}_{total}$ be the complete catalog of detected bursts (e.g., from Swift BAT), and ${\mathcal{D}}_z\subset {\mathcal{D}}_{total}$ be the subset with spectroscopically confirmed redshifts [20]. Our goal is to identify the regions exhibiting a statistically significant relative excess or deficiency of redshift measurements on the sky.

Adaptive geometric methods like Voronoi Diagram Field Estimator (VDFE) [21] and Delaunay Tessellation Field Estimator (DTFE) [22] are standard for cosmic web analysis in surveys such as DESI [23] and Euclid [24]; the probabilistic reconstruction of the relative exposure function presented here offers a superior framework for handling complex selection functions in the GRB population.

While the VDFE and DTFE are standards, they were excluded from the primary comparison in favor of Kernel Density Estimation (KDE) for GRB sky exposure functions [25]. Simple but optimized Gaussian density estimators demonstrated that both VDFE and DTFE are prone to generating spurious local maxima, or ‘bogus hot spots’, particularly in regions like the North Galactic Pole. To recover a physically realistic detection probability function, these geometric estimators require an additional stage of kernel smoothing to suppress these artifacts. In contrast, it was shown that KDE yields a self-consistent reconstruction of the density field without introducing significant artificial features or requiring secondary post-processing. Therefore, the KDE method was selected as the more robust density-based baseline for this study too.

Cosmologists and researchers working on large-scale studies are the target audience for this work. By implementing a Bayesian-driven preconditioning technique this method recovers some high-frequency structural features in the selection function that were previously lost, exceeding the bandwidth constraints of conventional estimators. It can be a useful tool for the catalogs of the new high-redshift missions (e.g., SVOM [26], Einstein Probe [27] and Theseus [28]), where it is critical to differentiate between physical clustering and observational bias.

2. Data Description

Here, we use a comprehensive database of the Swift GRBs with measured positions and spectroscopic redshifts. The Swift GRB Table1 is a public catalog of all Swift’s GRB observations. It contains the most $\gamma$ parameters, such as the positional information, the different durations, fluxes and fluences of a GRB event [29]. Several X-ray and UV-optical parameters and comments from all three Swift instruments are also included. Additionally, the Swift-XRT GRB Catalogue2 from the UK Swift Science Data Centre (UKSSDC) contains all published XRT light curves and spectra with XRT positions. The data was accessed on 17 November 2025, with GRB 251112A beeing the last event included.

The Swift mission employs a hierarchical positioning strategy initiated by the Burst Alert Telescope (BAT), a wide-field coded-aperture instrument operating in the 15–150 keV range [30]. Upon triggering, the BAT reconstructs sky images to provide an initial localization with an accuracy of 1–4 arcminutes (90% confidence) within seconds [31]. Once the spacecraft slews to the BAT target, the XRT position repository, maintained by the UK Swift Science Data Centre, generates the most accurate astrometric localizations. The primary data product, known as the ‘Enhanced Position’, utilizes a sophisticated cross-correlation technique to minimize systematic errors inherent in the spacecraft’s pointing [32]. It significantly reduces the radial systematic error to approximately 1.4 arcseconds (90% confidence) [33]. Here, we use this enhanced positional information, if available for a given event; otherwise, we use the Swift BAT position. On the figures, like Figure 1, black points show the actual sky positions of the GRB dataset.

The redshift dataset was constructed using GCN data and was cross-checked using Jochen Greiner’s GRB Big Table3, publications and the Gamma-Ray Burst Online Index (GRBOX)4. We used here the same spectroscopic redshift dataset used earlier by [16,34,35], with redshift observations corrected and updated up until 31 August 2022 [17]. The list of the events was limited to agree with the Swift observations.

Figure 1. Adaptive KDE estimation of the entire population according to Abramson [36]. This map serves as the basis for the structural constraints of the Preconditioned model. The black points show the actual GRBs’ sky positions. They form the dataset for the AKDE producing the density map. The intensities are non-normalized relative values.

Figure 1. Adaptive KDE estimation of the entire population according to Abramson [36]. This map serves as the basis for the structural constraints of the Preconditioned model. The black points show the actual GRBs’ sky positions. They form the dataset for the AKDE producing the density map. The intensities are non-normalized relative values.

We explicitly excluded photometric redshifts and estimates based on limits (e.g., Lyα limits), as these data tend to introduce significant uncertainties in radial distance. Since our final goal is the investigation of the full spatial distribution, we restricted our analysis to precise spectroscopic redshifts only.

The selection results in a total of 1665 GRBs with Swift positional information (BAT positions and, where available, XRT enhanced positions). Of this sample, 427 GRBs possess precise spectroscopic redshifts, with 300 derived from afterglow observations and 127 from host galaxy measurements.

3. The Kernel Density Estimation

The Kernel Density Estimation is a non-parametric alternative for density reconstruction [37]. In the classical approach, two independent intensity functions, ${\lambda }_{total}\left(\mathbf{x}\right)$ and ${\lambda }_z\left(\mathbf{x}\right)$, are estimated using KDE. The angular relative probability (or redshift completeness), denoted as $R\left(\mathbf{x}\right)$, is then derived by calculating their ratio:

$$R\left(\mathbf{x}\right)={\displaystyle \frac{{\widehat{\lambda }}_z\left(\mathbf{x}\right)}{{\widehat{\lambda }}_{total}\left(\mathbf{x}\right)}}$$

(1)

In this naive approach, we determine relative probability by dividing the density estimate of the redshifted subset (${\mathcal{D}}_z$) by the density estimate of the total observed population (${\mathcal{D}}_{\mathrm{total}}$), applying standard KDE to both datasets independently.

Applying the KDE method poses significant statistical and numerical challenges [38,39]. Unlike parametric models, which assume a specific underlying shape (e.g., a single cluster), non-parametric methods allow the data to determine the complexity of the angular structure, but standard KDE on the sphere often suffers from the classic bias-variance trade-off. A fixed bandwidth tends to oversmooth dense clusters (erasing high-frequency angular information) or undersmooth sparse background regions (introducing spurious noise). Insufficient bandwidth ($h\to 0$) results in an undersmoothed, high-variance estimate that is sensitive to the stochastic placement of individual points (overfitting). In contrast, excessive bandwidth ($h\to \infty$) produces an oversmoothed, high-bias estimate in which significant structural details are obscured. This is a classic example of the bias-variance trade-off known in statistical learning [40]. Although standard KDE approaches can be used, they introduce the aforementioned bandwidth dilemma [41]: an inadequately optimized bandwidth can result in the deletion of true large-scale anisotropies (oversmoothing) or the generation of false, noise-driven peaks (undersmoothing).

The most critical problem with this ratio-based approach is the singular behavior: in areas where the background density of ${\lambda }_{total}\left(\mathbf{x}\right)$ approaches zero (e.g., zones obscured by the Galactic plane), the ratio tends to diverge asymptotically to infinity, causing extreme variance and meaningless artifacts (spikes) in the resulting map. Furthermore, the error propagation is significant, as the biases and variances in two independent estimators accumulate during the division process, often amplifying Poisson noise. Finally, the bandwidth selection dilemma complicates the analysis, as the two density functions may require different optimal smoothing parameters, the coordination of which poses a non-trivial optimization problem for ratio formation.

To solve this problem, we apply the adaptive method of Abramson [36], adapted to spherical geometry [42], and introduce a robust Bayesian logistic regression framework (GAM/SOS), which is detailed below.

Adaptive Spherical KDE Pilot Estimation and Bandwidth Selection

One way to solve density estimation problems is to use adaptive bandwidth. This method varies the kernel width in inverse proportion to the square root of the local density, which reduces the asymptotic bias of the estimation function from $\mathcal{O}\left(h^2\right)$ to $\mathcal{O}\left(h^4\right)$.

For a unit vector $\mathbf{x}$ and a mean vector $\mathit{\mu }$, the spherical counterpart of the Gaussian distribution is the Von Mises–Fisher distribution [43], whose probability density function is given by:

$$f_{vMF}(\mathbf{x};\mathit{\mu },\kappa )=C\left(\kappa \right)exp\left(\kappa {\mathit{\mu }}^T\mathbf{x}\right)$$

(2)

where $\kappa \ge 0$ is the concentration parameter (which is analogous to the inverse variance, $\kappa \sim 1/{\sigma }^2$). The normalization constant $C\left(\kappa \right)$ is critical for strict likelihood estimation [44] and is defined as follows:

$$C\left(\kappa \right)={\displaystyle \frac{\kappa }{4\pi sinh\left(\kappa \right)}}$$

(3)

In the first step, a pilot density estimate, $\tilde{f}\left(\mathbf{x}\right)$, is created using a fixed global concentration parameter ${\kappa }_{pilot}$. To determine the ${\kappa }_{pilot}$, we use Likelihood Cross-Validation, also known as Maximum Likelihood Cross-Validation. This procedure, which was established by Stone [45] in statistical prediction and standardized by Silverman [37] for density estimation, ensures data-driven bandwidth selection. The specific extension of the method to spherical data and directional distributions, as well as the optimization of the concentration parameter ($\kappa$), is discussed in detail in the work of Hall et al. [42]. We define the Leave-One-Out estimator for the density of data point ${\mathbf{X}}_i$ using all other points $j\ne i$:

$${\widehat{f}}_{-i}({\mathbf{X}}_i;\kappa )={\displaystyle \frac{1}{N-1}}\sum _{j\ne i}C\left(\kappa \right)exp\left(\kappa {\mathbf{X}}_j^T{\mathbf{X}}_i\right)$$

(4)

The optimal pilot parameter is obtained by maximizing the log-likelihood of the data:

$${\kappa }_{pilot}=arg\underset{\kappa }{max}\sum _{i=1}^{N}log{\widehat{f}}_{-i}({\mathbf{X}}_i;\kappa )$$

(5)

This optimization is performed numerically, without constraints, allowing the data to objectively dictate the underlying smoothness.

Using the optimized $\tilde{f}\left(\mathbf{x}\right)$ pilot estimate, we calculate the local bandwidth modifiers (${\lambda }_i$) for each data point according to Abramson’s square root law [36]:

$${\lambda }_i={\left({\displaystyle \frac{\tilde{f}\left({\mathbf{X}}_i\right)}{g}}\right)}^{-1/2}$$

(6)

where g is the geometric mean of the pilot densities, which serves as a scaling factor. In areas where the pilot density $\tilde{f}\left({\mathbf{X}}_i\right)$ is high (clustering), ${\lambda }_i<1$, which prescribes a narrower kernel function in order to preserve detail. In areas of low density, ${\lambda }_i>1$, which prescribes a wider kernel function to smooth the variance. Since the concentration of vMF $\kappa$ is inversely proportional to the square of the bandwidth ($h^2\propto 1/\kappa$), the derivation of the local concentration parameter ${\kappa }_i$ associated with the i-th event is as follows:

$${\kappa }_i={\displaystyle \frac{{\kappa }_{pilot}}{{\lambda }_i^2}}$$

(7)

The final adaptive $\widehat{f}\left(\mathbf{x}\right)$ probability density estimate is the normalized sum of the variable kernel functions at any arbitrary point $\mathbf{x}$ on the sphere:

$$\widehat{f}\left(\mathbf{x}\right)={\displaystyle \frac{1}{N}}\sum _{i=1}^{N}C\left({\kappa }_i\right)exp\left({\kappa }_i{\mathbf{X}}_i^T\mathbf{x}\right)$$

(8)

Here, $C\left({\kappa }_i\right)$ ensures that the integral of each kernel function on the sphere is unity, guaranteeing that the total estimate $\widehat{f}\left(\mathbf{x}\right)$ is a valid PDF:

$${\int }_{S^2}\widehat{f}\left(\mathbf{x}\right)d\mathsf{\Omega }=1$$

(9)

4. Methodology: Bayesian Preconditioning via Spherical GAMs

To address the inherent instabilities of the traditional density ratio approach, we reformulate the completeness problem as a conditional probability estimation task. Instead of independently estimating the generative densities for the total and redshift samples—which leads to divergence and bandwidth mismatch—we utilize a statistical framework. See Appendix B for a detailed mathematical derivation of the Restricted Maximum Likelihood (REML) optimization and the Splines on the Sphere (SOS) basis functions.

4.1. The Bayesian Approach: High-Statistics Priors

Using the entire Swift BAT GRB sample ($N_{total}=1665$) as an informative spatial prior is our main innovation. For every detection, we establish a binary indicator $y_i\in \{0,1\}$ that indicates if a measured redshift is present. We aim to recover the latent probability surface $p\left(\mathbf{x}\right)=P(z>0|\mathbf{x})$. By embedding this in a Generalized Additive Model (GAM) with a logit link function, we can guarantee that the resulting efficiency map is physically consistent and rigorously constrained within $[0,1]$. This method prevents the phantom leaks and spikes characteristic of Adaptive KDE ratios by forcing the model to satisfy the geometric restrictions of the survey (e.g., low probability areas) inherent in the entire sample.

4.2. Structural Optimization

To provide an isotropic, pole-singularity-free depiction of the celestial sphere, we use Splines on the Sphere. SOS enables continuous interpolation and sub-pixel resolution, in contrast to grid-based techniques. The Akaike Information Criterion (AIC) and Restricted Maximum Likelihood objectively control the model’s smoothness and complexity, providing a data-driven bias-variance trade-off.

For the full Lumped dataset, the AIC selects an optimal base dimension of $k=44$ (Figure 2). This parameter sets the number of basis function nodes in the SOS framework, allowing the model to accurately capture the distribution without overfitting.

5. Statistical Analysis

To determine whether the proposed Bayesian methodology provides a more accurate representation of physical reality, we perform a comparative analysis between the two approaches with respect to the signal population (events with redshift). One of the models examined is $M_{KDE}$, a traditional Adaptive Kernel Density Estimator (AKDE), which is derived exclusively from the distribution of signal events ($N_z$) (Figure 1). This contrasts with the $M_{Precon}$ model, a complex Preconditioned Estimator, defined as the product of the background density ${\rho }_{total}$ and the conditional probability $P\left(\mathrm{Redshift}\right|x)$ derived using Bayesian Generalized Additive Models (Figure 3):

$${\widehat{f}}_{Precon}\left(x\right)\propto P\left(\mathrm{Redshift}\right|x)·{\rho }_{total}\left(x\right)$$

(10)

We use two different statistical frameworks for the objective evaluation of the realism and performance of the models: Likelihood and Discrimination Analysis (ROC/AUC).

5.1. Likelihood Analysis

The likelihood principle provides a rigorous and objective measure for comparing probability density functions [37,45]. To scientifically prove that the Precon distribution is superior to the standard AKDE, we must demonstrate that it minimizes information loss, i.e., it maximizes the predictive likelihood of the true distribution of the signal.

We compare the log-likelihood ($\mathcal{L}$) of the observed GRB events for both density maps. We define the log-likelihood of the $\widehat{f}$ model for the signal data $X_z=\{{\mathbf{x}}_1,\dots ,{\mathbf{x}}_n\}$ as follows:

$$\mathcal{L}\left(\widehat{f}\right|X_z)=\sum _{i=1}^n\ln \left(\widehat{f}\left({\mathbf{x}}_i\right)\right)$$

(11)

A higher $\mathcal{L}$ value indicates that the model has correctly concentrated a significant portion of the probability mass on the areas where the events actually occur. This is a measure of calibration and precision. To compare the two models, we calculate the likelihood ratio (or Bayes factor proxy):

$$\Delta \mathcal{L}={\mathcal{L}}_{Precon}-{\mathcal{L}}_{KDE}$$

(12)

We interpret this difference based on the scale of Kass and Raftery [46] ($2\Delta \mathcal{L}$), where the range $0\dots 2$ is negligible, $2\dots 6$ is positive, $6\dots 10$ is strong, and $2Delta\mathcal{L}>10$ represents very strong (decisive) evidence in favor of the model.

5.2. Discrimination Analysis (ROC/AUC)

Since the goal is to distinguish the signal (GRBs with redshift) from the rest of the distribution, in the absence of true non-events, we applied the Presence-Background ROC analysis methodology [47]. To evaluate the discriminative power of the density estimators, we generated $N=$ 12,288 HEALPix grid points uniformly distributed on the sphere, which served as pseudo-negative class (background).

The Area Under the Curve, which quantifies performance, represents the probability that a randomly selected signal event has a higher estimated density than a randomly selected background point:

$$AUC=P(\widehat{f}\left(x_{signal}\right)>\widehat{f}\left(x_{noise}\right))$$

(13)

It is important to note that the AUC is a rank-order metric; so, it is invariant with respect to the absolute values of density [48]. If a model physically underestimates probabilities but preserves the order, it can still achieve a high AUC value [1].

6. Results and Discussion

6.1. Results of Statistical Comparative Analysis

We evaluated the Information Criteria (Log-Likelihood) and Discriminative Power (ROC/AUC) of the proposed Precon model against the standard AKDE. During the empirical evaluation of the datasets, the goodness-of-fit analysis (Likelihood) revealed overwhelming evidence in favor of the Precon Estimator (Figure 4). The $\Delta \mathcal{L}\approx 74.3$ Log-Likelihood difference (${\mathcal{L}}_{Precon}\approx -519.3$ vs. ${\mathcal{L}}_{KDE}\approx -593.6$), which corresponds to a standard evidence score of $2\Delta \mathcal{L}\approx 148.6$, represents a Bayes factor of $K>{10}^{30}$. This indicates that the Precon distribution provides a significantly more accurate description of the event generation process, minimizing information loss.

In contrast, in terms of discrimination analysis (ROC), i.e., signal-to-noise separation, both models performed similarly (Figure 5). AKDE achieved a marginally higher Area Under the Curve ($AU{C}_{KDE}\approx 0.682$) compared to the Precon model ($AU{C}_{Precon}\approx 0.674$). This difference of less than $1.2\%$ is negligible and is likely due to the sensitivity of ranking metrics to local noise spikes.

The Duality of Matching and Ranking

Based on the above, we can observe an apparent contradiction: the Precon model is significantly better in terms of likelihood ($\Delta \mathcal{L}\gg 10$), but marginally weaker or equivalent in terms of discrimination ($\Delta AUC\approx 0$). This phenomenon can be explained by the fundamental difference between fitting and ranking.

AKDE, because it comes from a small sample ($N_z$), tends to overfit local variance, often creating narrow ‘spikes’ (high variance) around observed redshift events [49]. On the one hand, in the case of AUC, KDE places the spikes exactly on the training points, thus ranking these points higher than the background, which artificially inflates the ranking-based metrics. On the other hand, in the case of Likelihood, although KDE fits well to specific points, it often fails to model the distribution shape between points. It may assign a probability close to zero to valid regions in the immediate vicinity of events (undersmoothing). In Likelihood calculations, an estimated probability of nearly zero for a real event results in a huge mathematical penalty ($\ln \left(0\right)\to -\infty$), which causes a significant disadvantage in the fit test compared to the Precon model.

By conditioning the signal probability $P\left(\mathrm{Redshift}\right|x)$ on the high-statistic total density ${\rho }_{total}$, the Precon method incorporates structural information from the background that is missing when analyzing isolated subsets with small sample sizes [1]. This effectively acts as a Bayesian prior, stabilizing the density estimation and better handling the bias-variance trade-off.

6.2. Subgroup Analysis and Validation of the Split Model

As a supplement to the lumped model analysis, we separately examined two specific subgroups of the signal population (afterglow and host galaxy based redshifts) and evaluated the effectiveness of this model separation compared to the AKDE approach.

In the dataset with afterglow based redshifts (optimal $k=41$), the Precon model retained its statistical superiority (top of Figure 6). The likelihood analysis resulted in a value of $\Delta \mathcal{L}\approx 16.11$ in favor of Precon (Precon: $-361.54$, KDE: $-377.65$), which is very strong evidence on the Kass–Raftery scale (Section 5.1). In terms of discriminator power, the two models performed similarly ($AU{C}_{Precon}\approx 0.65$, $AU{C}_{KDE}\approx 0.67$).

For host-galaxy-based redshifts ($k=23$, Figure 7), which comprise a smaller sample size, the likelihood difference decreases significantly ($\Delta \mathcal{L}\approx 0.29$, Figure 6, center). Consequently, this model imposes a more stringent penalty on complexity to avoid misinterpreting stochastic Poisson noise as genuine physical clustering.

Here, the advantage of the Precon model is negligible, while the AUC values showed the superiority of KDE ($AU{C}_{Precon}\approx 0.63$, $AU{C}_{KDE}\approx 0.70$). This suggests that for data with lower information content, the structural constraints of Precon are less effective.

Finally, to verify model complexity, we performed a Split vs. Lumped hypothesis test. We compared the lumped model with the split model, where the density was the weighted sum of the two sub-models (separate datasets for the afterglow and host galaxy based redshifts). The result (bottom of Figure 6) of the likelihood analysis shows that the lumped model is statistically superior (${\mathcal{L}}_{Lumped}\approx -1599.44$, ${\mathcal{L}}_{Split}\approx -1615.57$, $\Delta {\mathcal{L}}_{diff}\approx -16.13$). The afterglow-host galaxy partitioning added redundant complexity, resulting in overfitting without improving the model’s accuracy in describing the real distribution.

Figure 6. Precon density estimate for the different groups. The black dots marks the GRB events’ angular positions. The intensities are non-normalized relative values, but the maximums are the same. (Top) afterglow determined redshift subgroup: based on the likelihood analysis, this model statistically significantly ($\Delta \mathcal{L}\approx 16.1$) outperforms the KDE. (Center) host-galaxy-determined redshift subgroup: The likelihood ratio ($\Delta \mathcal{L}\approx 0.29$) here indicates only a marginal advantage for Precon, reflecting the weaker relationship between the model and the data in this subgroup. (Bottom) superposition of the two density estimates of the two independent submodels (afterglow and host galaxy) forms the Split Density Map. The statistical comparison shows that the Lumped model outperforms the Split model ($\Delta {\mathcal{L}}_{Split}\approx -16.13$); thus, it is suboptimal (Figure 8). A comparison of these figures reveals that the afterglow-derived map shows a high-density region in the lower left region, a feature absent in the host-derived map. Furthermore, the under-density void at $(l\approx {60}^{\circ },b\approx 0^{\circ })$ in the host-galaxy map is less pronounced in the (combined) Split Density map. Due to the disparity in sample sizes ($N_{\mathrm{afterglow}}=300$ versus $N_{\mathrm{host}}=127$), the result map is primarily governed by the afterglow distribution.

Figure 6. Precon density estimate for the different groups. The black dots marks the GRB events’ angular positions. The intensities are non-normalized relative values, but the maximums are the same. (Top) afterglow determined redshift subgroup: based on the likelihood analysis, this model statistically significantly ($\Delta \mathcal{L}\approx 16.1$) outperforms the KDE. (Center) host-galaxy-determined redshift subgroup: The likelihood ratio ($\Delta \mathcal{L}\approx 0.29$) here indicates only a marginal advantage for Precon, reflecting the weaker relationship between the model and the data in this subgroup. (Bottom) superposition of the two density estimates of the two independent submodels (afterglow and host galaxy) forms the Split Density Map. The statistical comparison shows that the Lumped model outperforms the Split model ($\Delta {\mathcal{L}}_{Split}\approx -16.13$); thus, it is suboptimal (Figure 8). A comparison of these figures reveals that the afterglow-derived map shows a high-density region in the lower left region, a feature absent in the host-derived map. Furthermore, the under-density void at $(l\approx {60}^{\circ },b\approx 0^{\circ })$ in the host-galaxy map is less pronounced in the (combined) Split Density map. Due to the disparity in sample sizes ($N_{\mathrm{afterglow}}=300$ versus $N_{\mathrm{host}}=127$), the result map is primarily governed by the afterglow distribution.

Figure 7. Model selection for the host galaxy determined redshift subgroup. The minimum of the AIC curve is found at a lower dimension, $k=23$. This more economical model suggests that this dataset contains less data points and less detailed spatial information.

Figure 7. Model selection for the host galaxy determined redshift subgroup. The minimum of the AIC curve is found at a lower dimension, $k=23$. This more economical model suggests that this dataset contains less data points and less detailed spatial information.

Figure 8. Density estimate using Precon map and Lumped model. This map is proved to be statistically better to the Split model.

Figure 8. Density estimate using Precon map and Lumped model. This map is proved to be statistically better to the Split model.

6.3. Pixel-to-Pixel Correlation Analysis

To check the concordance between the two estimators and to identify their behaviors across distinct signal regimes, a pixel-by-pixel correlation analysis of the probability density functions was performed. Both the Precon and AKDE intensity fields were normalized such that their total probability mass equaled unity, thereby ensuring a comparison independent of arbitrary scaling factors.

The resulting correlation density plot (Figure 9) reveals three distinct regimes of model behavior driven by the conservation of probability mass.

In the Low-Density Regime ($\lesssim 2.0\times {10}^{-5}$), a significant asymmetry is visible where the distribution deviates above the identity line. This indicates that the AKDE estimator consistently assigns higher probability to the low-probability (void) regions than the Precon model. This is a characteristic signature of boundary bias: whereas the Precon model successfully fits the sharp cutoff of the Galactic Disc, the AKDE is spatially bounded by its kernel width, causing probability mass to ‘spill’ into unobserved regions.

In the Intermediate Regime ($2.0-4.0\times {10}^{-5}$), to compensate for the excess mass assigned to the voids and peaks by the AKDE, the Precon model concentrates slightly more mass in the intermediate signal domain. The grid points cluster tightly along the diagonal (Pearson’s $r\approx 0.974$), demonstrating that Precon preserves the general signal structure of the standard estimators.

In the latex High-Intensity Regime (>$4.0\times {10}^{-5}$), a slight scatter and upward deviation is observable. This reflects the different regularization strategies: the Abramson adaptive bandwidth ($h\propto {\widehat{f}}^{-1/2}$) allows AKDE to fit narrow, high-amplitude spikes (potentially overfitting Poisson noise), whereas the Precon model is governed by a global smoothness penalty (REML), resulting in slightly more conservative peak estimates.

6.4. Spatial Artifact Analysis: Spurious Spikes and Signal Leakage

We conducted a point-by-point difference analysis of the probability density fields in order to quantify the structural limitations of the Adaptive Kernel Density Estimator in comparison to the Precon model. Three classes can be seen in the resulting difference map (Figure 10). The map is punctuated by sharp, high-amplitude areas of excess density, where ${\widehat{f}}_{AKDE}\gg {\widehat{f}}_{Precon}$. These correspond to the spurious spikes where the AKDE adaptive bandwidth, scaling as $h\propto {\widehat{f}}^{-1/2}$, narrows excessively in high-density regions. The Galactic central region appears to be covered by a wide, diffuse region of excess probability. The Signal Leakage artifact is demonstrated by this. The AKDE kernel width grows unduly in data-sparse zones to make up for the absence of local support. As a result, probability mass from high-latitude sources is artificially dispersed throughout the Galactic Disk. Surrounding these features, large void regions appear in Cyan/Blue, indicating where the AKDE density is lower than the Precon estimate (${\widehat{f}}_{AKDE}<{\widehat{f}}_{Precon}$). This shows the overall cost of the unphysical spikes. The estimator must decrease the background density elsewhere in order to meet the normalization constraint because these spikes take up such a significant portion of the overall probability.

6.5. Limitations and Physical Biases

It is important to note that the current analysis is based on the Swift BAT catalog, which is inherently biased towards harder X-ray energies and dominated by long gamma-ray bursts. Short GRBs, typically originating from neutron star mergers, are probably under-represented in redshift catalogs due to their fainter optical afterglows and different host galaxy environments. Furthermore, missions operating in different energy bands, such as the new SVOM or Einstein Probe, will probe sligthly different populations of the GRB.

However, the distinction between these GRB populations is not only a matter of intrinsic physics but is also significantly influenced by instrumental effects. Comparative studies between Swift BAT and Fermi GBM have shown that for the exact same set of jointly detected GRBs, the measured durations ($T_{90}$) systematically differ [50]: Fermi GBM tends to measure longer durations for short GRBs, while Swift BAT measures longer durations for long ones. This instrumental dependency is further highlighted by the fact that the $T_{90}$ distribution of the same GRB sample requires three lognormal components when observed by Swift, but only two when observed by Fermi. These discrepancies suggest that the traditional classification into short and long events is instrument-specific and does not uniquely represent the underlying physical mechanisms. Consequently, while certain types like neutron star mergers might be under-represented in our current redshift-complete sample, the lack of a clear selection effect in the redshift measurement process itself—independent of these fuzzy classification boundaries—suggests that our completeness maps is a useful tool for characterizing the observational biases of the respective satellite missions.

The Bayesian GAM framework, however, is intended to be mission-neutral. Once sufficient statistics are available, the factorized estimation approach (${\widehat{\lambda }}_z={\widehat{\lambda }}_{total}·ϵ$) can be directly applied to any future catalog or particular sub-population (e.g., short GRBs only). Our approach objectively recovers the selection function for any given observational setup, providing the mathematical framework to account for these intrinsic biases. Recovering high-frequency features like sharp low probability areas is critical to prevent observational artifacts from being misidentified as genuine cosmological anisotropies. Accurate recovery minimizes signal leakage into zero-visibility regions, enabling the use of physically realistic exposure maps instead of those degraded by sparse sample constraints. These high-resolution maps are essential for rigorous large-scale isotropy investigations.

7. Summary and Conclusions

We concluded that the Precon Estimator constitutes a statistically improved framework for recovering the underlying angular distribution of GRBs with determined redshift compared to traditional adaptive kernel methods. The empirical analysis demonstrates that the Precon model yields a log-likelihood improvement of $\Delta \mathcal{L}\approx 74.3$ over the standard AKDE, which corresponds to a standard evidence score of $2\Delta \mathcal{L}\approx 148.6$, implying a Bayes factor exceeding ${10}^{30}$. This illustrates that the Precon model offers a significantly more accurate probabilistic description of the event selection process. While the discrimination analysis suggests comparable performance ($\Delta \mathrm{AUC}<1.2\%$), this parity is attributable to the rank-order invariance in the AUC metric, which rewards the tendency of AKDE to overfit local noise spikes in sparse datasets without penalizing poor calibration. Conversely, the Precon Estimator minimizes information loss, providing a robust reconstruction of the physical intensity field that is stable against Poisson noise.

The theoretical efficacy of the Precon Estimator is derived from its ability to circumvent the fundamental bandwidth limitation that limits direct adaptive estimators. The Asymptotic Mean Squared Error analysis shows that the angular resolution of any direct estimator is asymptotically bound by the sample size of the signal population (n), scaling as $h\propto n^{-1/6}$. For the sparse redshifted GRB sample ($n\ll N$), this constraint forces the smoothing kernel to function as a low-pass filter with a soft ceiling on resolution, rendering it mathematically impossible to resolve high-frequency angular features. By decomposing the problem into a wideband exposure map and a narrowband selection efficiency, the method recovers structural information beyond the Nyquist–Shannon sampling limit of the sparse redshift subset. This ensures that the resulting cosmological maps remain physically consistent with global observational constraints.

The resulting high-fidelity redshift completeness maps have direct implications for both GRB population synthesis and observational cosmology. In population studies, the ability to assign a coordinate-dependent completeness weight to individual events allows for a more precise derivation of the intrinsic rate density and luminosity function. This approach effectively corrects for spatial spatial variance induced by observational constraints and satellite mechanics [19]. Furthermore, the model serves as a noise-filtered background for Large-Scale Structure studies and tests of the Cosmological Principle.

This methodology is directly applicable to data from current and upcoming missions, including SVOM, THESEUS, and the Einstein Probe. Implementing this Bayesian framework on early mission catalogs will facilitate the rapid identification of sky-dependent efficiencies, ensuring a more informed interpretation of the high-redshift Universe. While the empirical validation presented here uses the Swift-BAT catalog, the Precon framework is fundamentally mission-agnostic. Due to its generalizable nature, the technique can be applied to any observational setup in which the recovery of a sparse subset can be preconditioned using a high-statistics background.

From an information-theoretic perspective, the Precon approach represents a minimization of the generalized Kullback–Leibler divergence between the estimated and true intensity fields. By conditioning the signal probability on the high-statistics background density, the method injects essential structural constraints—effectively acting as informative priors—that prevent the ‘probability leakage’ into unobserved regions (such as the Zone of Avoidance) that characterizes standard KDE. While the direct AKDE maximizes entropy solely under the weak constraints of the sparse redshift sample, the Precon model incorporates side-channel information provided by the non-redshifted population. This yields a Maximum Entropy solution that is consistent with global observational constraints, validating the hypothesis that a factorized, physics-aware statistical framework is essential for the mapping of cosmological objects subject to complex selection functions.