How do people handle uncertainty in simulation/experimental data for a model?

Let’s say you have inputs and outputs given by:


  1. Matrix/Array of predictors (X X)
  2. Matrix/Array of predictor uncertainties (Xsd \sigma_X)
  3. Vector of response values (y y)
  4. Vector of response uncertainties (ysd \sigma)
  5. Matrix/Array of new predictors (X2 X_2)
  6. (Optional) Matrix/Array of new predictor uncertainties (Xsd2 \sigma_{X2})


  1. Vector of new responses (ypred y_{pred})
  2. Vector of new response uncertainties (ysd2 \sigma_2)
  3. (Optional) predictor covariance matrix (cov \Sigma_2)

How do you get something like this (or even something along the way to this)?

(related SE post)