  The need to analyze the available large synoptic multi-band surveys drives
the development of new data-analysis methods. Photometric redshift estimation
is one field of application where such new methods improved the results,
substantially. Up to now, the vast majority of applied redshift estimation
methods have utilized photometric features. We aim to develop a method to
derive probabilistic photometric redshift directly from multi-band imaging
data, rendering pre-classification of objects and feature extraction obsolete.
A modified version of a deep convolutional network was combined with a mixture
density network. The estimates are expressed as Gaussian mixture models
representing the probability density functions (PDFs) in the redshift space. In
addition to the traditional scores, the continuous ranked probability score
(CRPS) and the probability integral transform (PIT) were applied as performance
criteria. We have adopted a feature based random forest and a plain mixture
density network to compare performances on experiments with data from SDSS
(DR9). We show that the proposed method is able to predict redshift PDFs
independently from the type of source, for example galaxies, quasars or stars.
Thereby the prediction performance is better than both presented reference
methods and is comparable to results from the literature. The presented method
is extremely general and allows us to solve of any kind of probabilistic
regression problems based on imaging data, for example estimating metallicity
or star formation rate of galaxies. This kind of methodology is tremendously
important for the next generation of surveys.
