  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.
