Constraints on Cosmic Opacity from Bayesian Machine Learning: The hidden side of the $H_{0}$ tension problem

Bayesian (Probabilistic) Machine Learning is used to probe the opacity of the Universe. It relies on a generative process where the model is the key object to generate the data involving the unknown parameters of the model, our prior beliefs, and allows us to get the posterior results. The constraints on the cosmic opacity are determined for two flat models, $\Lambda$CDM and XCDM (this having $\omega_{de} \neq -1$), for three redshift ranges, $z\in[0,2.5]$, $z \in[0,5]$, and $z\in[0,10]$, in each case. This is to understand how the constraints on the cosmic opacity could change in the very deep Universe, and also to check to what extent there is a redshift-range dependence. The following forms for the opacity, $\tau(z) = 2\epsilon z$ and $\tau(z) = (1+z)^{2\epsilon} -1$, corresponding to an observer at $z=0$ and a source at $z$, are considered. The results of our analysis show that the Universe is not fully transparent, and this may have a significant impact on the $H_{0}$ tension problem. In the generative process, the fact that, owing to cosmic opacity, the flux received by the observer is reduced has been taken into account. In the analysis, the luminosity distance associated with the cosmological model has been employed.