The miniJPAS survey: Maximising the photo-z accuracy from multi-survey datasets with probability conﬂation

We present a new method for obtaining photometric redshifts (photo-z ) for sources observed by multiple photometric surveys using a combination (conﬂation) of the redshift probability distributions (PDZs) obtained independently from each survey. The conﬂation of the PDZs has several advantages over the usual method of modelling all the photometry together, including modularity, speed, and accuracy of the results. Using a sample of galaxies with narrow-band photometry in 56 bands from J-PAS and deeper g riz y photometry from the Hyper-SuprimeCam Subaru Strategic program (HSC-SSP), we show that PDZ conﬂation signiﬁcantly improves photo-z accuracy compared to ﬁtting all the photometry or using a weighted average of point estimates. The improvement over J-PAS alone is particularly strong for i & 22 sources, which have low signal-to-noise ratio in the J-PAS bands. For the entire i < 22.5 sample, we obtain a 64% (45%) increase in the number of sources with redshift errors | ∆ z | < 0.003, a factor 3.3 (1.9) decrease in the normalised median absolute deviation of the errors ( σ NMAD ), and a factor 3.2 (1.3) decrease in the outlier rate ( η ) compared to J-PAS (HSC-SSP) alone. The photo-z accuracy gains from combining the PDZs of J-PAS with a deeper broadband survey such as HSC-SSP are equivalent to increasing the depth of J-PAS observations by ∼ 1.2–1.5 magnitudes. These results demonstrate the potential of PDZ conﬂation and highlight the importance of including the full PDZs in photo-z catalogues.


Introduction 1
In this decade, a number of current and upcoming large-area  The Sloan Digital Sky Survey (SDSS; York et al. 2000) 15 showed that five broad bands (ugriz) suffice to obtain relative 16 errors in photo-z, ∆z = (z phot -z spec )/(1+z spec ), as low as σ(∆z) ∼ 17 2% for sources meeting some magnitude and colour cuts (Beck 18 et al. 2016).A similar accuracy level has been reported for much 19 fainter galaxies observed by HSC-SSP (grizy bands) over a wide 20 range of redshifts (HSC Collaboration et al. 2023).However, 21 some applications, such as baryonic acoustic oscillation (BAO) 22 email: ahernan@cefca.esmeasurements, require higher photo-z accuracy, with uncertainties as low as σ(∆z) ∼ 0.3-0.5% (Blake & Bridle 2005;Angulo et al. 2008;Benítez et al. 2009;Chaves-Montero et al. 2018).This level of accuracy is only achievable with narrower pass bands that provide a higher effective spectral resolution.
An important step in determining accurate photo-z over a large area will be taken by the Javalambre-Physics of the Accelerating Universe Astrophysical Survey (J-PAS; Benítez et al. 2009Benítez et al. , 2014)), which will cover one third of the northern hemisphere with a unique set of 56 optical filters (plus i band for detection) and provides, for each 0.48 × 0.48 pixel in the sky, an R∼60 photo-spectrum (J-spectrum) covering the 3800-9100 Å range.Two small surveys were carried out to demonstrate the scientific potential of J-PAS: miniJPAS (Bonoli et al. 2021) and J-NEP (Hernán-Caballero et al. 2023), showing that σ(∆z)=0.3% is attainable with the 56 bands of J-PAS plus u, g, r, and i (Hernán- Caballero et al. 2021Caballero et al. , 2023;;Laur et al. 2022).
Nevertheless, covering the optical range in many narrowband filters is very expensive in terms of telescope time.Thus, the limiting magnitude of J-PAS (m 5σ ∼22.5;Bonoli et al. 2021) is shallower than other large-area (>1000 deg 2 ) broad-band surveys such as KiDS (∼1350 deg 2 , m 5σ ∼25) and HSC-SSP Wide (∼1400 deg 2 , m 5σ ∼26).The results from miniJPAS and J-NEP, as well as previous results from SHARDS and PAUS, indicate that J-PAS photo-z can be improved with the addition of deeper photometry from upcoming broad-band surveys that will cover the J-PAS footprint, such as Euclid (I∼26.2,Y, J, H ∼ 24.5 Euclid Collaboration et al. 2022) and the China Space Station Telescope Optical Survey (CSS-OS; NUV, u, g, r, i, z, y, ∼25 Gong et al. 2019).
Computing accurate photo-z by mixing the photometry of a shallow multi-narrow-band survey and a deep broad-band survey is technically challenging.The two main roadblocks are systematics in the photometry and the choice of the photo-z method.
The broad-and narrow-band observations are obtained with different telescopes or instruments and processed by different pipelines.This implies that the pixel scale and the point spread function (PSF) of the images might differ, as well as the methods for flux calibration and source extraction (including aperture definition), resulting in complex systematic offsets between the broad-and narrow-band fluxes.This problem is exacerbated if the broad-band survey is much deeper because the small nominal photometric uncertainties do not account for the crosscalibration uncertainty.Even if one is to reprocess all the data with the same pipeline, obtaining consistent photometry in all the bands requires convolving all the images to the worst PSF or deblending the emission of sources in wider-PSF images using a sharper one as reference (see Barro et al. 2019).
Another significant limitation comes from the photo-z techniques employed.For observations including only a few broad bands, the best results are obtained with an empirical method (e.g.Beck et al. 2016;HSC Collaboration et al. 2023) that uses a large sample of galaxies with spectroscopic redshifts to calibrate the dependence of the observed colours with redshift.However, the high dimensionality of the colour space that results from having a large number of narrow bands makes the empirical method unfeasible with the small training samples currently available.As a consequence, photo-z measurements from multi-narrow-band photometry rely almost exclusively on SEDfitting with spectral templates (e.g.Wolf et al. 2003;Moles et al. 2008;Pérez-González et al. 2013;Barro et al. 2019;Eriksen et al. 2019;Alarcon et al. 2021;Laur et al. 2022).Doing an SED fit to the combined deep and shallow photometry is tricky because of the cross-calibration issue mentioned above, but also because the optimal number of templates depends on the number of bands with detections (Hernán-Caballero et al. in prep.).
Bright sources with strong detections in the narrow bands benefit from a larger number of templates since that makes it easier to find a good match.However, for faint sources detected only in the broad bands, such diversity of spectral shapes only adds degeneracy to the colour-redshift space, with results improving when just a few templates (representing broad spectral classes) are used (e.g.see Benítez 2000).
An alternative to using the combined photometry from two 110 distinct datasets is to obtain redshift probability distribution 111 functions (PDZs) separately from the broad-and narrow-band 112 surveys and then combine the PDZs.To our knowledge, this ap-113 proach has not been tested so far.Conceptually similar strategies 114 were proposed by Kovač et al. (2010) and Barro et al. (2019) to 115 improve the confidence of spectroscopic redshifts from single-116 line detections using photo-z as priors.

117
Combining PDZs instead of photometry has some clear ad-118 vantages: i) it eliminates the need to obtain consistent photome-119 try through all the surveys; ii) it allows the use of the most suit-120 able photo-z method for each dataset; iii) it makes it possible to 121 re-use the photo-z already published by other surveys, capitalis-122 ing on the expertise of the teams that produced them and making 123 it easier and faster to add new datasets to the photo-z calculation.124 In this paper, we use a sample of galaxies in the AEGIS field 125 to show that a combination of PDZs from narrow-band (mini-126 JPAS) and broad-band (HSC-SSP) observations results in sig-127 nificantly improved photo-z accuracy compared to the individ-128 ual surveys and also compared to SED-fitting all the photometry 129 together.The structure of the paper is as follows.Section 2 de-130 scribes the probability conflation method for combining PDZs.131 Section 3 presents the sample of galaxies with J-PAS and HSC-132 SSP observations.Section 4 discusses the photo-z obtained with 133 different codes using the two datasets separately.Section 5 de-134 scribes and compares three methods for obtaining photo-z from 135 the two datasets: mixing the photometry, weighted averaging of 136 point estimates, and conflating the PDZs.Finally, Sect.6 com-137 pares the performance of J-PAS, HSC-SSP, and their combined 138 photo-z as a function of the source magnitude and discusses the 139 implications for J-PAS.All magnitudes are expressed in the AB 140 system.141

142
Let X = {X i } and Y = {Y j } be two independent sets of obser-143 vations of the same galaxy, and P X (z) and P Y (z) the probability 144 density distributions for the redshift derived from each of them.145 If we assume P X (z) and P Y (z) to be independent, the combined 146 probability distribution can be expressed as the conflation of the 147 two probability distributions (Hill & Miller 2011): where the integral in the denominator ensures that P C (z) is uni-149 tary.Conflation has multiple advantages over other methods 150 of combining probability distributions, including the following: 151 minimisation of the loss of Shannon information, being a best 152 linear unbiased estimate, and yielding a maximum-likelihood es-153 timator (Hill & Miller 2011). 154 The main disadvantage of the conflation method is that it 155 is sensitive to over-confidence in the individual PDZs.Over-156 confidence (defined as confidence intervals containing the true 157 redshift less often than predicted from the PDZs, implying the 158 PDZs are too narrow) is a common issue in many photo-z codes 159 (see Dahlen et at. 2013 for a review).Over-confidence in one 160 or both PDZs may cause only their wings to overlap or, in ex-161 treme cases, not to overlap at all, resulting in an undefined P C (z). 162 Therefore, successful conflation requires the PDZs to be realistic 163 or under-confident.

164
In a Bayesian framework, the probability P X (z) can be writ-165 ten as where L(X | z, σ, θ) is the likelihood of the observations X given 167 the uncertainties σ = {σ i } and the parameters θ = {θ k } that de-168 fine the model1 .P(z, θ) is the prior on z and θ, which repre-169 sent our knowledge (obtained elsewhere) about the distribution 170 of redshift and spectral properties for galaxies of a given magni-171 tude (N(z,θ|m), hereafter N(z)).
Since the prior does not depend on the observations X or 173 Y, P X (z) and P Y (z) are not entirely independent.This results in PDZ.The width δz of the peaks in L(z) is proportional to the width δλ of the spectral features that can be resolved by the photometry: Therefore, everything else being equal, a factor of two reduction in the FWHM of the filters roughly translates into a factor of two decrease in photo-z errors.On the other hand, the S/N of the observations impacts the number and contrast of the peaks in L(z).
At high S/N, L(z) is usually unimodal since the observed colours are only compatible with a reduced range of intrinsic colours and redshifts 2 .As the S/N per band decreases, the larger photometric uncertainties can make the observations compatible with combinations of intrinsic colours and redshift that are very different from the actual ones, resulting in spurious peaks in L(z) located far from the true redshift of the galaxy.At S/N 1, the number of peaks in L(z) increases dramatically, while the contrast between peaks and valleys decreases, resulting in a roughly flat distribution with minor fluctuations.At this point, L(z) contains very little information, and the PDZ becomes dominated by the prior.
Figure 1 shows representative examples of PDZs obtained from miniJPAS observations (using only the narrow bands) and from the much deeper broad-band observations of HSC-SSP, with the S/N per band ∼25-30 times higher (see Sect. 3).For relatively bright sources (i 21) with median S/N 3 in the narrow bands, the J-PAS PDZ usually has a single peak concentrating most of the probability density.This peak is narrower than the PDZ from HSC-SSP despite the much higher S/N because its spectral resolution, not sensitivity, limits the latter.For fainter sources (i 22), the PDZ from HSC-SSP is still unimodal and has the same width, while the PDZ of J-PAS is now much broader since it is composed of multiple overlapping peaks whose intensity is modulated by the prior.This implies that point estimates, such as the mode of the PDZ, can be highly inaccurate (an outlier) if photometric errors or the prior favour the wrong peak.In fact, the broad and multi-modal PDZs of faint J-PAS sources result in a high rate of outliers (see Sects.4.2 and 6).
Throughout this paper, we use several quantities related to photometric redshifts and their accuracy.While some are relatively standard, others are not.Here, we define all of them for convenience: z phot : A generic term for any point estimate of the photometric redshift.It can be obtained from the PDZ in several ways, such as taking the mode, the mean, the median, or a randomly sampled value.In this paper, z phot represents the mode of the PDZ unless otherwise indicated.
-∆z : The relative error in z phot , defined as odds : An indicator of the confidence in z phot .It represents the probability that |∆z| is smaller than a given threshold δ.
In this work, we used δ = 0.03.The odds value is computed by integration of the PDZ as where the index i runs over the sources in the sample.2020) surveys using a 1 search radius.The spectroscopic coverage from DEEP spans a narrow strip that includes ∼50% of the miniJPAS footprint (Fig. 2).Outside this strip, only a few hundred SDSS spectra are available.In total, there are 7123 sources in miniJPAS with spectroscopic redshifts and i<22.5.
From this sample, we removed the sources for which the spectroscopic redshift is unreliable (zWarning > 0 in SDSS; ZQUALITY < 3 in DEEP), or with spectral classification other than galaxy (stars and quasars), or with unreliable miniJPAS photometry (SExtractor flags > 0 or mask_flags > 0; see Bonoli et al. 2021 for details).Finally, we removed four galaxies with z spec >1.5, which are beyond the search range that we used to compute photo-z for J-PAS galaxies (0<z<1.5).After these cuts, we were left with 4448 sources.
The miniJPAS footprint is partially covered by deep imaging (∼26 mag at 5σ) in the grizy bands from the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP; Aihara et al. 2018).The roughly circular footprint of HSC-SSP is larger than miniJ-PAS but overlaps only ∼80% of its area (Fig. 2).Out of the 4448 miniJPAS sources selected above, 3712 are inside the HSC-SSP footprint.To cross-match our sample with the HSC-SSP catalog, we retrieved all the sources with cmodel magnitude i<23.5 from the HSC-SSP database. 4 5Inside the overlap region, all but two miniJPAS sources with spectroscopy have an HSC-SSP counterpart within 1 .Our final sample includes the 3710 miniJPAS sources with i<22.5, no photometry flags, reliable z spec <1.5, and HSP-SSP photometry.
Figure 3 compares the depth of the J-PAS and HSC-SSP observations using the median S/N of all the bands for each of the sources.The median S/N correlates with the i-band magnitude of the sources, with some dispersion caused by differences in their spectral type and redshift.On average, the S/N of a galaxy is ∼25-30 times higher in the HSC-SSP bands compared to the J-PAS bands.
We note that the 5-σ depth in a 3 aperture of the miniJPAS images ranges between ∼21.5 and ∼24 magnitudes, depending on the band and pointing, with an average of ∼22.5 (see Fig.  2018), for the codes DNNz and mizuki we chose the value z best that minimises the probability of the inferred redshift being an outlier as the default point estimate (z phot ) (see Tanaka et al. 2018 for details).For DEmP, we instead used the mode of the PDZ, which performs slightly better than z best for this particular code.
In Fig. 4, we compare the accuracy of z phot obtained by the three codes for the galaxies in our sample.The statistics f 03 , σ NMAD , and η are shown at the bottom right corner of each panel.DEmP clearly outperforms the other two codes in these three scores (we note that, for f 03 , higher is better).DEmP is also the least affected by systematic issues at z<0.3 (see HSC Collaboration et al. 2023 for a discussion).
While evaluating the accuracy of PDZs for individual sources is not possible, we can perform statistical tests to determine if, on average, z spec falls within a given confidence interval of the PDZ with the expected frequency.Figure 5 shows the distribution of odds and the relation between odds and η for the three codes.If the PDZs are realistic, the expected value of η is related to the mean odds by This theoretical relation is represented by the dotted diagonal line in Fig. 5.The outlier rate for DNNz is systematically below the expected relation (i.e. the odds are under-confident).In contrast, mizuki produces odds that are under-confident in the intermediate range 0.5 odds 0.7) but over-confident in the high end (odds 0.9).Finally, the odds from DEmP are realistic in the entire range except for the few sources with odds 0.4.Additionally, DEmP is the only code that obtains high odds for a large fraction of the sample.Therefore, in the following analysis, we only focus on DEmP results (see Appendix A for the results of conflation of PDZs from each of the three codes with the J-PAS likelihood).
The distribution of ∆z for DEmP (top panel in Fig. 6) shows that z phot is slightly biased towards z phot >z spec .Assuming that the bias scales with (1+z), we obtain δz ∼ 0.003(1+z).After correcting for this bias, the f 03 score increases from 0.196 to 0.267, while the impact on σ NMAD and η is negligible.images using SExtractor in dual mode (the extraction aperture defined in the detection band is applied to all bands).We selected the PSF-corrected aperture (PSFCOR) photometry, which we corrected for Galactic extinction and for systematic offsets in the colour indices between J-PAS bands (see Sect. 3 in HC21).Then, we scaled the fluxes to match the flux in the larger AUTO aperture for the i band.This maximizes the S/N and accuracy of the observed colour indices and prevents underestimation of the luminosity of the galaxies.To estimate the photo-z, LePhare computes the likelihood L(z i ) ∝ exp[-χ 2 min (z i )/2] for a discrete set of redshifts, z i , where χ 2 min corresponds to the χ 2 value of the best fitting template at redshift z i .The templates are a set of 50 synthetic galaxy spectra generated with cigale (Boquien et al. 2019).The physical parameters that define the synthetic spectra (such as star formation history, metallicity, or extinction) are chosen by finding the values that best reproduce the observed photometry of individual galaxies in a small training sample.
To compute the probability distribution for the redshift, P(z), LePhare modulates the raw likelihood with a redshift prior N(z).The default prior for LePhare is obtained from galaxy counts in the Vimos VLT Deep Survey (Le Fèvre et al. 2005).The N(z) prior helps break degeneracies in the colour-redshift space and improves the photo-z accuracy for most galaxies, but it can also bias z phot values for populations at the faint end of the luminosity distribution (such as dwarf elliptical galaxies; see Fig 18

Hernán-Caballero et al. 2023). To evaluate the impact of the N(z)
409 prior on the results, we also obtain photo-z using a flat prior.

410
We take the mode of P(z) as the best point estimate, z phot .We note that since the contrast correction does not change the mode of the PDZ, it does not affect in the value of z phot .

Combined photometric redshifts
In this section, we describe and compare three methods for obtaining photo-z with the combined HSC-SSP and J-PAS datasets: SED-fitting using the mixed photometry (broad bands and narrow bands fitted together with the same model), a weighted mean of point estimates from the two surveys, and the conflation of the two PDZs.

SED-fitting the mixed photometry
One of the main challenges of photo-z estimation is ensuring that colour indices are accurate.While errors in the absolute flux calibration can introduce a systematic bias, the primary source of uncertainty is often the amount of flux lost outside the extraction aperture, which depends on the aperture size, the PSF of the image, and the light profile of the source.In point sources, aperture losses are easy to quantify and correct, but in extended sources, the uncertainty can be very considerable, especially if a small aperture is used to maximise the S/N of the photometry.
To decrease this uncertainty, photo-z are often computed using aperture photometry extracted from images that are convolved to the same PSF in all the bands or using equivalent methods to compensate for PSF variation, such as PSFCOR.This approach can yield accurate colours if the differences in PSF FWHM are small and the light profile of the source does not change significantly from band to band.More sophisticated methods that model both the light profile of the source and the PSF of the image, such as The Tractor (Lang et al. 2016a,b), The Farmer (Weaver et al. 2023), or SourceXtractor++ (Bertin et al. 2020;Kümmel et al. 2020) can overcome PSF variation without convolving the images.
Given that both miniJPAS and HSC-SSP already provided PSF-corrected photometry and that they have three bands in common (g, r, i), we chose a more straightforward strategy.We found a magnitude offset (different for each source) that when applied to the HSC-SSP bands minimises the average of the magnitude difference between miniJPAS and HSC-SSP in g, r, and i.From miniJPAS, we took the photometry described in Sect.4.2.From HSC-SSP, we took the photometry from the columns (g|r|i|z|y)_convolvedflux_2_kron_flux in the table pdr3_wide.forced4.This photometry is obtained using a Kron aperture (Kron 1980) on images convolved to 1.1 FWHM (similarly to the typical FWHM of miniJPAS images) with the 'afterburner' method (see Aihara et al. 2018 for details).This ensures that the PSF FWHM and the aperture sizes are as close as possible to those in miniJPAS (the PSFCOR photometry of miniJ-PAS is also based on the Kron aperture; see Sect.2.2 in HC21 for details).Tanaka et al. (2018) confirmed that photometry from convolved images provides the most accurate colours and photoz for both crowded and isolated objects in HSC-SSP.The most accurate photo-z from HSC-SSP (obtained with the DEmP code; see Sect.4.1) also use the Kron aperture on convolved images.
We corrected the HSC-SSP photometry for Galactic extinction using the coefficients a x provided in the table pdr3_wide.forced.Then, for each source, we computed the magnitude offset that we applied to the HSC-SSP bands as the average colour difference: where the suffixes H and J correspond to HSC-SSP and miniJ-PAS, respectively.
After applying the offsets, we find a 1-σ dispersion between HSC-SSP and miniJPAS magnitudes measured in the same band of 0.12, 0.08, and 0.09 mag for g, r, and i, respectively.This dispersion may be in part a consequence of differences in the transmission profiles of the filters.
We computed photo-z for the 61-band dataset (56 narrowband filters of J-PAS plus g,r,i,z,y from HSC-SSP) with LePhare using the same configuration described in Sect.4.2 for J-PAS alone (including the N(z) prior).We added a quantity ∆m to the nominal errors of the HSC-SSP photometry to account for the systematic uncertainty between the J-PAS and HSC-SSP photometry.We tested the values ∆m = 0, 0.05, 0.1, and 0.2 mag.
We obtain the best photo-z accuracy for ∆m = 0.1 mag.The results are described in Sect.5.4.

Conflation of the PDZs
The PDZs of J-PAS and HSC-SSP are represented by 32-bit floating point arrays containing the probabilities P(z i ) sampled at a discrete set of redshifts {z i }.Each survey uses a different {z i }: from z = 0 to z = 1.5 in steps of 0.002 in J-PAS; from z = 0 to z = 6 in steps of 0.01 in HSC-SSP.
To compute the conflation of the two PDZs, we linearly interpolate the PDZ from HSC-SSP, P H (z), at the {z i } of the J-PAS PDZ, P J (z).For this, we apply the systematic shifts δz=0.003(1+z) for HSC-SSP and δz=0.0015(1+z) for J-PAS found in Sect. 4. Since, by construction P J (z>1.5) = 0, we discarded the z>1.5 range in P H (z).Then, Eq. 1 becomes where zmax = 1.5 and P − J (z) is the version of the J-PAS PDZ obtained with a flat prior.The limited numerical precision implies that very small probabilities are rounded to zero.If the intervals where P − J (z) > 0 and P H (z) > 0 do not overlap, then their product is always zero and P C (z) becomes undefined.In practice, zero overlap can happen only if one of the PDZs is unrealistically over-confident (P(z spec ) = 0) or if the source association between the two surveys is wrong.None of the 3710 sources in our sam-533 ple has undefined P C (z).For an arbitrary PDZ, the expected value of the redshift, E(z) = 536 µ, and its variance, σ 2 , are given by If the PDZ is Gaussian, it is entirely determined by the values 538 of µ and σ 2 .Furthermore, the conflation of two Gaussian PDZs, 539 P X (z) and P Y (z), is also Gaussian with the same mean and vari-540 ance as the weighted mean of µ X and µ Y (Hill & Miller 2011): Therefore, assuming Gaussian PDZs, a weighted average of 543 point estimates from miniJPAS and HSC-SSP would result in the 544 same point estimates that we obtained from the conflated PDZs, 545 with no need to use the PDZs.Unfortunately, the PDZs are often 546 far from Gaussian (especially at a low S/N) and the information 547 on the position and relative strength of the multiple peaks is lost 548 in the Gaussian approximation.As a consequence, using Eq. 13 549 results in much lower photo-z accuracy compared to PDZ con-550 flation, as we show in the next section.J-PAS datasets.Table 1 summarises the accuracy statistics for all 555 the datasets and methods.

556
Out of the three combination methods, the mixed photome-557 try is the worst performing, with scores only slightly better than 558 those obtained from J-PAS alone and worse than HSC-SSP with 559 DEmP in σ NMAD and η (but not in f 03 ).This suggests that adding 560 the HSC-SSP photometry does little to improve the photo-z from 561 the 56 bands of J-PAS, possibly due to the uncertainty in the 562

565
The weighted mean method results in roughly the same f 03 566 as the mixed photometry method or using J-PAS alone.How-567 ever, both σ NMAD and η are much better.This suggests that the 568 weighted mean is improving the photo-z of faint sources, where 569 the much larger uncertainty in the J-PAS z phot results in a mean 570 value is closer to the HSC-SSP value (which is much less likely 571 to be an outlier).For the bright sources, the uncertainty in the J-572 PAS z phot is small compared to HSC-SSP, therefore the solution 573 is closer to the J-PAS value, which is usually the most accurate.

574
Finally, the PDZ conflation performs much better than the 575 other two combination methods in f 03 and σ NMAD (but only 576 slightly better than the weighted mean method in η).PDZ con-577 flation is also much better than either J-PAS or HSC-SSP sep-578 arately in every score.We find the fact that PDZ conflation is In the previous section, we established that point estimates with the conflation method are the most accurate.However, many applications also require confidence intervals derived from the PDZs be realistic.Figure 10 shows the relation between η and odds and the PIT test for the PDZs from HSC-SSP with DEmP (corrected for systematic offset), the narrow-band photometry of J-PAS (LePhare code with flat prior, also corrected for systematic offset), and the conflation of the two PDZs.
The odds values for J-PAS are strongly underconfident, while those for HSC-SSP correctly predict the outlier rate, except for the lowest odds bin (0.2<odds<0.4),where they are also underconfident.On the other hand, the conflated PDZs are slightly overconfident for odds>0.4(but consistent with the η = 1odds relation within their 1-σ uncertainties).The PIT test (bottom panel in Fig. 10) shows a roughly flat distribution for the conflated PDZs, which suggests that the PDZs are, on average, realistic.However, since the source counts increase steeply with the i-band magnitude, results from the PIT test are dominated by the faintest sources.
Another powerful test for the PDZs is the fraction of galaxies F(c) in which z spec falls within a given confidence interval c of the PDZ (e.g.Fernández-Soto et al. 2002;Dahlen et at. 2013;Schmidt & Thorman 2013).For realistic PDZs, we expect F(c) = c.Out of the many possible definitions of a confidence interval, the most useful is the highest probability density confidence interval (HPDCI), as proposed initially by Fernández-Soto et al. (2002) and illustrated by Wittman et al. (2016).The HPDCI is the shortest interval (or union of disjoint intervals) that contains a given fraction of the total area under the PDZ distribution.Therefore, it always contains the mode of the PDZ.HC21 obtained F(c) versus c relations for different magnitude bins, finding that, before applying a contrast correction, miniJPAS PDZs are slightly over-confident at bright magnitudes but increasingly under-confident at fainter ones.
In Fig. 11, we show the same test on the PDZs from J-PAS, HSC-SSP, and their conflation.J-PAS PDZs present the same magnitude dependence found by HC21, while HSC-SSP PDZs show the opposite trend: they are more under-confident at brighter magnitudes.Also, in the case of HSC-SSP, the arcs defined by the F(c) versus c relation are not symmetric with respect to the diagonal line, F(c) = c, suggesting that the underconfidence is stronger for small values of c.Since the PDZs from HSC-SSP are usually unimodal, this implies that the probability density around the peak is underestimated at all magnitudes, but even more so for the brightest ones.This might be a consequence of the relatively coarse sampling (constant steps of 0.01 in z) of the PDZs published by the HSC Collaboration, which would be insufficient to resolve the actual shape of the PDZ near the peak.Interestingly, the combination of the J-PAS and HSC-SSP PDZs results in F(c) versus c relations that are roughly symmetric with respect to F(c) = c, with much less underconfidence compared to J-PAS alone, and with no clear magnitude dependence (except maybe for the brightest sources, where there is a hint of overconfidence that might be the result of small number statistics).This is probably a consequence of the opposite magnitude dependencies in J-PAS and HSC-SSP cancelling each other out.
These tests indicate that conflating the PDZs from J-PAS and HSC-SSP results in realistic PDZs irrespective of the odds and magnitude of the sources.Therefore, we consider that there is no need for an ad hoc contrast-correction of the J-PAS likelihood in this case.When conflating other datasets, a contrast correction of Fraction of galaxies F(c) for which z spec is inside the highest probability density confidence interval (HPDCI) as a function of the confidence level c, computed separately for the colour-coded magnitude bins i = {17.5-18.5, 18.5-19.5, 19.5-20.5, 20.5-21, 21-21.5, 21.5-22, 22-22  664 Our three main photo-z accuracy statistics, f 03 , σ NMAD , and η, are all better for HSC-SSP compared to J-PAS 7 .This might seem surprising given the factor ∼ 8.5 increase in spectral resolution of J-PAS over HSC-SSP.However, this is entirely a consequence of pushing the magnitude limit of our sample selection down to i=22.5, which, given the red colours of most faint galaxies, implies that many sources have a very low S/N in most of the J-PAS narrow bands.The low S/N results in unreliable z phot estimates at faint magnitudes, as evidenced by the large fraction of sources with low J-PAS odds (Fig. 13).
The statistics f 03 , σ NMAD , and η all show a much stronger dependence on the limiting magnitude for J-PAS compared to HSC-SSP (left panel in Fig. 14).J-PAS obtains higher f 03 than HSC-SSP for i<22.1, lower σ NMAD for i<21.9, and even lower η for i<21.At bright magnitudes (i 20), both surveys saturate in   J-PAS overtakes HSC-SSP in f 03 when the fraction of the sample selected is f <0.8 and in σ NMAD for f <0.63.However, even very strong odds cuts do not achieve a lower η than HSC-SSP at the same f (both converge towards η = 0 at odds = 1, as expected).
These trends reflect the different nature of the uncertainties in the broad-band and narrow-band photo-z, which are a consequence of the shape of their PDZs (see Fig. 1); broad-band photo-z provide highly confident (low η) estimates thanks to the high S/N of the photometry, but with little accuracy due to the limited spectral resolution (therefore, the PDZs are broad and unimodal).On the other hand, narrow-band photo-z provide higher accuracy due to higher spectral resolution but with lower confidence given the low S/N (the PDZs contain multiple narrow peaks).

Photo-z accuracy from the conflated PDZs
We show z phot computed as the mode of the conflated PDZ versus z spec in the right panel of Fig. 12. Visually, the distribution looks similar to that of HSC-SSP, albeit with less dispersion, particularly at low z.The improvement in the scores is substantial: ∼50% more sources with |∆z|<0.3%, a factor ∼2 decrease in σ NMAD compared to HSC-SSP (factor ∼3 compared to J-PAS), and an ∼25% decrease in the outlier rate compared to HSC-SSP (∼70% compared to J-PAS).
Figure 14 shows that the conflation of the HSC-SSP and J-PAS PDZs improves all the scores, not only for the whole i<22.5 sample, but also for any other magnitude cut or selected fraction using an odds cut.At bright magnitudes, the conflation results converge to those of J-PAS alone.For fainter sources, results are expected to eventually converge towards the HSC-SSP values as the S/N in the J-PAS bands approaches zero.This is confirmed by Figure 15, which shows a steep increase with magnitude in the correlation between errors in z phot from HSC-SSP and the conflated PDZs.However, even in the faintest magnitude bin (22.0<i<22.5),we obtain ρ<1, which is indicative of some contribution from the J-PAS PDZ to the z phot with conflation.On the other hand, the correlation between errors in z phot values for HSC-SSP and J-PAS is consistent with ρ=0 at all magnitudes,

2
imaging surveys such as the Rubin Observatory Legacy Survey 3 of Space and Time (LSST; LSST Science Collaboration et al. 4 2009), the Dark Energy Survey (DES; Dark Energy Survey Col-5 laboration et al. 2016), the Hyper Suprime-Cam Subaru Strategic 6 Program (HSC-SSP; Aihara et al. 2018), and the Kilo-Degree 7 Survey (KiDS; Hildebrandt et al. 2021) will cover thousands of 8 square degrees in the sky to depths that previously were only 9 possible for small surveys of a few square degrees at most.These 10 surveys use sets of broad-band filters that are carefully designed 11 to maximise their performance for photometric redshifts (photo-12 z), allowing for new applications in galaxy evolution and cos-13 mology (e.g.see Newman & Gruen 2022 and references therein). 14

Fig. 1 .
Fig.1.Comparison of redshift probability distributions obtained from narrow-band J-PAS photometry (blue solid line) and deep broad-band photometry from HSC-SSP (red dashed line) in four galaxies from the AEGIS field sample.The spectroscopic redshift is marked with a green vertical line.Annotations in the top left corner of each panel indicate (from top to bottom) the source ID in the miniJPAS public data release catalogue, its i-band magnitude, and the median S/N per band in the J-PAS photometry and the HSC-SSP photometry.

Fig. 2 .
Fig. 2. Footprint of multi-band photometry observations in the AEGIS field.Orange dots represent i<23.5 sources in the SUBARU HSC-SSP PDR3 catalogue with photometry in the grizy bands.Blue dots represent i<22.5 sources in the miniJPAS PDR201912 catalogue.Black dots mark the miniJPAS sources with spectroscopic redshift.
∆z) : The standard deviation of ∆z in a sample.253254 -σ NMAD : A robust statistic equivalent to σ(∆z), but less sen-255 sitive to outliers.It takes the same value of σ(∆z) if the dis-256 tribution of ∆z is Gaussian.It is defined as

258-
f K : The fraction of sources in a sample with relative errors 259 in z phot smaller than a threshold K.We mainly used f 03 to 260 represent the fraction of sources with |∆z|<0.003(0.3%), a 261 threshold relevant to BAO studies.We also provide f 1 (frac-262 tion with relative error <1%), which is more relevant to the is comprised of all 21235 sources with mag-266 nitude i<22.5 in the AUTO aperture from the public data release 267 of miniJPAS 3 .We cross-matched this sample with a spectro-268 scopic redshift catalogue that combines redshifts from the DEEP 269 DR4 (Newman et al. 2013) and SDSS DR16 (Ahumada et al.

Fig. 3 .
Fig. 3. Comparison of median S/N per band in the narrow-band photometry of miniJPAS (blue dots) and the broad-band photometry of HSC-SSP (red dots) as a function of the i-band magnitude of the galaxies.Dashed lines show the best-fitting log-linear regression model.

Fig. 4 .Fig. 5 .
Fig. 4. Comparison of photometric versus spectroscopic redshift for the individual galaxies in our sample.Each panel shows the results from a different photo-z code using the photometry from HSC-SSP in the grizy bands.Symbols are colour-coded for the odds parameter.

Fig. 8 .
Fig. 8. Close-up of [-0.03,0.03]interval of the distribution of ∆z (top) and the distribution of PIT values (bottom) for photo-z measurements with LePhare using either a N(z) prior (left) or a flat prior (right).Solid and open histograms represent the distributions obtained before and after correcting for a systematic offset δz=0.0015(1+z),respectively. 411

Figure 7
Figure 7 shows z phot versus z spec for the two prior options.Using412 440

Fig. 9 .
Fig. 9. Comparison of dispersion of z phot estimates obtained from the combined HSC-SSP + J-PAS dataset using SED-fitting of the mixed photometry (left panel), a weighted average of z phot values obtained independently in the two datasets (middle panel), and conflation of the probability distributions (right panel).The three panels contain the same sources.Symbols are colour-coded for the odds parameter.

551 5 . 4 .Figure 9
Figure9compares the dispersion of the z phot versus z spec relation 553 obtained by the three methods for combining the HSC-SSP and 554 J-PAS datasets.Table1summarises the accuracy statistics for all 555 the datasets and methods.

Fig. 10 .
Fig. 10.Outlier rate (η) as a function of the odds parameter (top panel) and the PIT test (bottom panel) using PDZs from HSC-SSP alone, J-PAS alone, and a combination of both with probability conflation.Error bars are as in Fig. 5.

579
the only combination method that can significantly increase the 580 fraction of galaxies with very small photo-z errors with respect 581 to what can be achieved with HSC-SSP or J-PAS alone particu-582 larly interesting.This establishes PDZ conflation as a simple yet 583 powerful method for increasing the photo-z accuracy for highly 584 demanding tasks such as BAO measurements and highlights the 585 importance of including the full PDZs in published photo-z cat- Fig. 11.Fraction of galaxies F(c) for which z spec is inside the highest probability density confidence interval (HPDCI) as a function of the confidence level c, computed separately for the colour-coded magnitude bins i = {17.5-18.5, 18.5-19.5,19.5-20.5, 20.5-21, 21-21.5, 21.5-22,  22-22.5}.The left, middle, and right panels show results for the PDZs from HSC-SSP, J-PAS, and their conflation, respectively.Values above (below) the diagonal line indicate under-(over-)confidence in the PDZs.
figure since most HSC-SSP photo-z have high odds.

Fig. 13 .
Fig.13.Comparison of distribution of odds values for z phot from HSC-SSP (red dotted lines), J-PAS (blue dashed lines), and the conflated PDZs (black solid lines).Histograms indicate galaxy counts in bins of odds, while curved lines represent the fraction of the sample with odds higher than a given threshold value (right axis).

Fig. 14 .
Fig.14.Dependence of statistics f 03 , σ NMAD , and η with the cut in magnitude applied (left panels) or with the fraction of the total sample selected using a cut in odds (right panels).

689Fig. 15 .
Fig. 15.Pearson correlation coefficient between ∆z values for z phot measured on HSC-SSP and conflated PDZs (black squares) and between HSC-SSP and J-PAS (blue circles) as a function of the i-band magnitude.Error bars represent 1-σ confidence intervals obtained with bootstrap resampling.