Are there any linear regression models that account for uncertainty in $X$ and $Y$? Bayesian Linear Regression

Bayesian Linear Regression

 Yes, Bayesian Linear Regression Accounts for Uncertainty in X and Y  Bayesian linear regression (BLR) is a classical statistical modeling approach used to account for errors and uncertainties in linear regression models. Bayesian probability theory is used to explain this approach. It states that the data required for prediction are uncertain, and should be understood probabilistically. (Tui, 2021). BLR can be used in many fields, such as finance, health, and science, like environmental science. (Endo (2015) The uncertainty variable in this model is represented as a probability distribution reflecting prior beliefs. BLR can capture uncertainty by computing the posterior distributions of regression parameters using a computational approach. This results in an accurate prediction of the response variable. BLR is used to identify causal relationships and test hypotheses (Kuhn 2002).   BLR is a method to determine the estimate level of an error term. This allows for better understanding of correlations and strength of linear associations between variables. BLR provides better prediction accuracy than OLS (ordinary least squares) and a greater fit efficiency (Chavent. 2021). BLR can also be used to model data that has outliers or missing data (Gelman and. BLR is an appealing method for modelling complex data with heteroscedasticity and non-normality.  References  Chavent, M. (2021). Bayesian linear regression fundamentals. Retrieved from https://mathalino.com/bayesian-linear-regression-fundamentals  Endo, T., Goto, A., Ohara, K., Daido, Y., Abe, M., & Hotta, R. (2015). Bayesian linear regression analysis for quantitative finance. Mathematical methods in Economics and Finance 2, 91–106.  Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B. (2015). Bayesian data analysis. Boca Raton, FL: Chapman & Hall/CRC.  Kuhn, M. (2021). Bayesian linear regression. Retrieved from https://towardsdatascience.com/bayesian-linear-regression-e66e60791ea7  Tui, M., Karim, M., Yusuf, S., & Ismail, S. (2021). Bayesian-linear regression in scientific research: An overview. Pakistan Journal of Statistics 37(1): 10-17.