Bayesian photometric redshift estimation

TL;DR

The paper presents BPZ, a Bayesian framework for photometric redshift estimation that leverages priors and marginalization to overcome degeneracies and template incompleteness inherent in traditional ML/SED methods. By producing full posterior redshift distributions and reliability metrics (e.g., p_Δz and bookmaker odds), BPZ delivers more robust redshift estimates and uncertainty quantification, demonstrated by strong agreement with HDF spectroscopic redshifts ( rms ≈ 0.08 ) and improved performance with limited color information. The approach also enables simultaneous inference of galaxy properties, cluster membership, and cosmological parameters, highlighting its versatile applicability to large multicolor surveys and cluster lensing analyses. Overall, BPZ offers a principled, extensible statistical framework that enhances redshift estimation and supports integrated astrophysical inferences from photometric data.

Abstract

Photometric redshift estimation is becoming an increasingly important technique, although the currently existing methods present several shortcomings which hinder their application. Here it is shown that most of those drawbacks are efficiently eliminated when Bayesian probability is consistently applied to this problem. The use of prior probabilities and Bayesian marginalization allows the inclusion of valuable information, e.g. the redshift distributions or the galaxy type mix, which is often ignored by other methods. In those cases when the a priori information is insufficient, it is shown how to `calibrate' the prior distributions, using even the data under consideration. There is an excellent agreement between the 108 HDF spectroscopic redshifts and the predictions of the method, with a rms error Delta z/(1+z_spec) = 0.08 up to z<6 and no systematic biases nor outliers. The reliability of the method is further tested by restricting the color information to the UBVI filters. The results thus obtained are shown to be more reliable than those of standard techniques even when the latter include near-IR colors. The Bayesian formalism developed here can be generalized to deal with a wide range of problems which make use of photometric redshifts. Several applications are outlined, e.g. the estimation of individual galaxy characteristics as the metallicity, dust content, etc., or the study of galaxy evolution and the cosmological parameters from large multicolor surveys. Finally, using Bayesian probability it is possible to develop an integrated statistical method for cluster mass reconstruction which simultaneously considers the information provided by gravitational lensing and photometric redshifts.

Figures (10)

• Figure 1: a) On the left, VI vs. IK for the templates used in Sec \ref{['test']} in the interval $1<z<5$. The size of the filled squares grows with redshift, from $z=1$ to $z=5$. If these were the only colors used for the redshift estimation every crossing of the lines would correspond to a color/redshift degeneracy. b) To the right, the same color--color relationships 'thickened' by a $0.2$ photometric error. The probability of color/redshift degeneracies highly increases.
• Figure 2: An example of the main probability distributions involved in BPZ for a galaxy at $z=0.28$ with an Irr spectral type and $I\approx 26$ to which random photometric noise is added. From top to bottom: a) The likelihood functions $p(C|z,T)$ for the different templates used in Sec \ref{['test']}. Based on ML, the redshift chosen for this galaxy would be $z_{ML}=2.685$ and its spectral type would correspond to a Spiral. b) The prior probabilities $p(z,T|m_0)$ for each of the spectral types (see text). Note that the probability of finding a Spiral spectral type with $z>2.5$ and a magnitude $I=26$ is almost negligible. c) The probability distributions $p(z,T|C,m_0)\propto p(z,T|m_0)p(C|z,T)$ , that is, the likelihoods in the top plot multiplied by the priors. The high redshift peak due to the Spiral has disappeared, although there is still a little chance of the galaxy being at high redshift if it has a Irr spectrum, but the main concentration of probability is now at low redshift. d) The final Bayesian probability $p(z|C,m_0)=\sum_T p(z,T|C,m_0)$, which has its maximum at $z_B=0.305$. The shaded area corresponds to the value of $p_{\Delta z}$, which estimates the reliability of $z_B$ and yields a value of $\approx 0.91$.
• Figure 3: a)To the left, the photometric redshifts obtained by applying our ML algorithm to the HDF spectroscopic sample using a template library which contains only the four CWW main types, E/SO, Sbc, Scd and Irr. These results are very similar to those of Fernández-Soto, Lanzetta & Yahil, 1998. b) The right plot shows the significant improvement (without using BPZ yet) obtained by just including two of the Kinney et al. 1996 spectra of starburst galaxies, SB2 and SB3, in the template set. One of the outliers disappears, the 'sagging' or systematic offset between $1.5<z<3.5$ is eliminated and the general scatter of the relationship decreases from $\Delta z/(1+z_{spec})=0.13$ to $\Delta z/(1+z_{spec})=0.10$.
• Figure 4: The prior in redshift $p(z|m_0)$ estimated from the HDF data using the prior calibration procedure described in Sec 4., for different values of the magnitude $m_0$ ($I_{814}=21$ to $I_{814}=28$)
• Figure 5: The photometric redshifts obtained with BPZ plotted against the spectroscopic redshifts. The differences with fig. \ref{['comparison']}b are the elimination of 3 galaxies with $p_{\Delta z}<0.99$ (see text). This removes the only outlier present in fig. \ref{['comparison']}b. The rms scatter around the continuous line is $\Delta z_B/(1+z_B)=0.08$.