When does it make sense to use a bayesian approach, maybe in context to linear regression? To be more concrete: Assume you measure a certain number of devices and you wanna' check the linear relationship between, e.g., voltage and current. Why bayesian and/or why not? How would it be different?
The Bayesian approach should be used in the case of:
Strong priors - You have preexisting data and / or domain knowledge that you want to incorporate into the analysis.
Distributional estimates - Instead of point estimates, the result of Bayesian analysis will yield distributions. Those distributions will better quantify the uncertainty of predictions.
In the case of voltage and current for devices, Bayesian linear regression will predict a distribution of current for a given voltage. That is useful because many devices are noisy and possibly the degree of noise changes over voltages.