Bayesian photometric redshift estimation is a method that improves upon existing techniques by incorporating prior probabilities and Bayesian marginalization, allowing for the inclusion of valuable information such as redshift distributions and galaxy type mix. This approach quantifies the accuracy of redshift estimation in a way not found in other statistical methods, enabling the selection of galaxy samples with reliable redshifts. The method is tested against 100 HDF spectroscopic redshifts, showing excellent agreement with an rms error of Δz/(1+z_spec)=0.08 up to z<6, with no systematic biases or outliers. The reliability is further tested using only UBVI filters, showing results more accurate than standard techniques even when they include near-IR colors. The Bayesian formalism can be generalized to various problems using photometric redshifts, including estimating galaxy characteristics like metallicity and dust content, studying galaxy evolution, and cosmological parameters from large surveys. It also allows for an integrated statistical method for cluster mass reconstruction using gravitational lensing and photometric redshifts. The paper compares Bayesian probability (BPZ) with traditional methods like maximum likelihood (ML) using the HDF spectroscopic sample and a simulated catalog. BPZ outperforms ML in accuracy, especially at low redshifts, and is more robust in handling color/redshift degeneracies and template incompleteness. The method includes prior calibration using data, allowing for more accurate redshift and morphological priors. Spectroscopic information can also be incorporated to improve the accuracy of redshift estimates. The practical test on the HDF shows that BPZ significantly improves redshift estimation, reducing scatter and eliminating outliers. The results demonstrate that BPZ provides more accurate and reliable redshift estimates compared to traditional methods, making it a valuable tool in photometric redshift estimation.Bayesian photometric redshift estimation is a method that improves upon existing techniques by incorporating prior probabilities and Bayesian marginalization, allowing for the inclusion of valuable information such as redshift distributions and galaxy type mix. This approach quantifies the accuracy of redshift estimation in a way not found in other statistical methods, enabling the selection of galaxy samples with reliable redshifts. The method is tested against 100 HDF spectroscopic redshifts, showing excellent agreement with an rms error of Δz/(1+z_spec)=0.08 up to z<6, with no systematic biases or outliers. The reliability is further tested using only UBVI filters, showing results more accurate than standard techniques even when they include near-IR colors. The Bayesian formalism can be generalized to various problems using photometric redshifts, including estimating galaxy characteristics like metallicity and dust content, studying galaxy evolution, and cosmological parameters from large surveys. It also allows for an integrated statistical method for cluster mass reconstruction using gravitational lensing and photometric redshifts. The paper compares Bayesian probability (BPZ) with traditional methods like maximum likelihood (ML) using the HDF spectroscopic sample and a simulated catalog. BPZ outperforms ML in accuracy, especially at low redshifts, and is more robust in handling color/redshift degeneracies and template incompleteness. The method includes prior calibration using data, allowing for more accurate redshift and morphological priors. Spectroscopic information can also be incorporated to improve the accuracy of redshift estimates. The practical test on the HDF shows that BPZ significantly improves redshift estimation, reducing scatter and eliminating outliers. The results demonstrate that BPZ provides more accurate and reliable redshift estimates compared to traditional methods, making it a valuable tool in photometric redshift estimation.