Choosing the right likelihood for your data

I am trying to set up a problem where my observed data (y) has a most likely value and comes with upper & lower bounds (the spread around each point may not necessarily be Gaussian for all points). What is an appropriate likelihood function to be used for this kind of data?

Choosing a likelihood is a very flexible process with many options that may or may not make sense given the context you are applying it to. Can you share more details about your data?

I have a dataset that contains measurements for a physical process(continuous set of values). However, these values are not measured in reality but are rather best estimates based on interpretations. So they come with a high side & low side for each measurement /interpretation. I am trying to do a probabilistic calibration of a parametric model that predicts the data. The goal here is to learn the parameters of this model (joint distribution) that accounts for the uncertainty in the calibration data.

Sounds like censored regression (not to be confused with truncated regression!). There’s been quite a bit of discussion about these models here and now there is a tutorial in the docs.

Or if you are just interested in the overall shape of the distribution of y, not the effects of predictors on its values, you could rescale the data between [0, 1] and model it as a beta distribution.