Bayesian Photometric Redshift Estimation

Photometric redshifts are quickly becoming an essential tool of observational cosmology, although their utilization is somewhat hindered by certain shortcomings of the existing methods, e.g., the unreliability of maximum-likelihood techniques or the limited application range of the "training-set" approach. The application of Bayesian inference to the problem of photometric redshift estimation effectively overcomes most of these problems. The use of prior probabilities and Bayesian marginalization facilitates the inclusion of relevant knowledge, such as the expected shape of the redshift distributions and the galaxy type fractions, which can be readily obtained from existing surveys but are often ignored by other methods. If this previous information is lacking or insufficient—for instance, because of the unprecedented depth of the observations—the corresponding prior distributions can be calibrated using even the data sample for which the photometric redshifts are being obtained. An important advantage of Bayesian statistics is that the accuracy of the redshift estimation can be characterized in a way that has no equivalents in other statistical approaches, enabling the selection of galaxy samples with extremely reliable photometric redshifts. In this way, it is possible to determine the properties of individual galaxies more accurately, and simultaneously estimate the statistical properties of a sample in an optimal fashion. Moreover, the Bayesian formalism described here can be easily generalized to deal with a wide range of problems that make use of photometric redshifts. There is excellent agreement between the ≈130 Hubble Deep Field North (HDF-N) spectroscopic redshifts and the predictions of the method, with a rms error of Δ z ≈ 0.06(1 + z spec ) up to z < 6 and no outliers nor systematic biases. It should be remarked that since these results have not been reached following a training-set procedure, the above value of Δ z should be a fair estimate of the expected accuracy for any similar sample. The method is further tested by estimating redshifts in the HDF-N but restricting the color information to the UBVI filters; the results are shown to be significantly more reliable than those obtained with maximum-likelihood techniques.