Bayesian Neural net with non probibalistic Data? Is it possible to construct a Bayesian Neural Network without Probability Distributions as dependent Variable for purpose of predictive modeling?

I mean, if id like to Infer on a Specific Value, like y(e.g. y=5), with a vector of explanatory Variables X(e.g. X=[3,5,1.3,(.....)])
The Bayesian Neural Network infers on a distribution of ymean with standard deviation sigma(e.g. ymean =5, sigma =0.5).

Does it even make sense? Is the loss function of the Neural net able to work by comparing y with ymean, without taking simgma into account?

partial answer:
I think sigma is the result of the distributions in the weight matrices of the Neural net, and it should work. But I want to be sure and understand.

PS: I work in ecology, so getting a probability distribution as result would serve my goals.
It is possible to predict a single value from a Bayesian Neural Network. Given a set of input data, conduct the forward pass to generate the resulting probability distribution. Then convert that probability distribution to a single specific value in one of the following common ways:


Sample - Take a random sample. That random sample will automatically be weighted by the posterior probability distribution. This type of sampling is similar to Thompson sampling.
Use a measure of central tendency. Given that posterior probability distribution, calculate the most useful measure of central tendency (e.g., mean, median, or mode).