# Reasoning about modelling uncertainty

I am a beginner and am trying to build up my reasoning about uncertainty modelling and ways of modelling it.

What I am trying to essentially get at is how changes in input variables results in different posteriors(given e.g multicollinearity, constant inputs, high variance, few/many datapoints etc).

Lets take an model into account:
e^(monthdummiex1_t) * inputx1 ^ exponentmonthdummiex1_t + e^(monthdummiex2_t) * inputx2 ^ exponentmonthdummiex2_t

where as the monthdummies are modeled by lets say some normal distribution with 0 mean and sd=2.

The exponentdummies are modelled by lets say some beta distribution thus having the domain [0, 1]-

Lets say I generate data from a generator that simulates the above function + some noise. Given an constant input x1 and a non constant input x2 we will probably have equally wide posteriors… I would really like to model this type of uncertainty w.r.t inputrange in some sort of way such that the corresponding x1 parameters are much wider than x2 due to x1 being constant, is there a way to model this with gaussian processes given an timedependent model operating in a highly stochastic environment?

Or is my assumption wrong? Will the constant input in fact yield wider posteriors – according to my simulations it will not.

Given x1 and x2’s generated from some normal distribution thus none of the being constant:

W.r.t multicollinearity I assume the posteriors will in fact be wider since the model will have a hard time deducing from where the actual response came thus resulting in RV’s(parameters) with wider tails.

W.r.t datapoints I assume that as long as the datapoints point in the direction of the model-setup the posterios will get narrower thus more datapoints → narrower posteriors. However if they e.g start to change in the last lets say 10-20 datapoints(in a dataset of 50 datapoints) such that they point in another direction I would expect the posterior to get wider.

Can someone elaborate on my reasoning.

I highly appreciate any elaboration on my thought.

I am specifically interested in the input case(constant vs high variance inputs)