## DELVE DR2 MDN Photometric Redshifts

We derive a sample of photometric redshift estimates for objects in the DELVE DR2 catalog using a Mixture Density Network (MDN) trained on *griz* magnitudes and colors.
This network outputs the weights, means, and standard deviations of 20 Gaussian distributions, which are then combined into a single probability density function (PDF) from which samples of the photo-*z* can be drawn.
The resulting photo-*z* information can be accessed using the interfaces described on the Data Access page, and the delve_dr2.photoz table can be browsed using the Data Lab table browser.

More information about the construction and quality of this value added catalog can be found below.

### Training Data

The MDN photo-*z* estimator was trained on a sample of objects with measured spectroscopic redshifts. This spec-*z* catalog was assembled by cross-matching sources from DELVE DR2 with several spectroscopic catalogs: SDSS, 2DF, 6DF, VIPERS, GAMA, 2dFLens, DES_AAOMEGA, DES_IMACS, WIGGLEZ, DEEP2, 3D-HST, VVDS, CLASH-VLT, ACES, N17B331, SAGA, SPT_GMOS, UDS, C3R2, ATLAS,
, VANDELS, SPARCS, GLASS, CDB, ELG FIGS, VUDS, ZFIRE, and MOSFIRE.

- SNR > 5 in the
*g*band and SNR > 3 in the*riz*bands (when each band was available). *g*< 22.5- 0.01 <
*z*-spec < 2 - We removed objects with 178 < RA < 182 and Dec < 5 due to issues in a preliminary version of the DELVE catalog.

- Objects with
*g*,*r*,*i*and*z*measurements available were requited to have:*(g-r)*< 4, -1 <*(r-i)*< 4, and -1 <*(i-z)*< 4 - Objects with just
*g*,*r*and*i*measurements available were required to have:*(g-r)*< 4 and -1 <*(r-i)*< 4

- 13.478 <
*g*< 22.500 - 12.634 <
*r*< 23.297 - 12.270 <
*i*< 22.751 - 12.016 <
*z*< 22.859

*griz*photometry measured by DELVE. Below we show the magnitude distributions of the objects used to train and evaluate the MDN model Below we show the spec-

*z*distribution (bin size = 0.01) of the training and testing samples. The training sample was selected to be as uniform as possible in redshift to avoid biases. In practice, this resulted in a flat distribution up to

*z*= 0.6.

### Performance Metrics

We used point estimates (taking the most probable value from the PDF) to build our metrics (“point-like metrics”). We define the following metrics:- The photo-
*z**bias*is defined as Δ*z*=*z*_{phot}-*z*_{spec}. - The
*median bias*, median(Δ*z*), is useful for identifying possible systematic effects in the redshift estimation - The
*outlier fraction*is defined as |Δ*z*|/( 1 +*z*_{spec}) > 0.15 within a given redshift bin. - The
*normalized median absolute deviation*measures the dispersion of the bias and is defined as*σ*_{NMAD}= 1.48 × median(| (Δ*z*- median(Δ*z*)) / (1 +*z*_{spec}) |) - The
*relative errors*are defined as mean( Δ*z*/ (1 +*z*) ) within photo-*z*intervals of 0.05.

*z*interval of 0.05. Below we show the normalized median absolute deviation, median bias, outlier fraction, and relative error for the MDN photo-

*z*sample evaluated on our test sample of objects.