Using mean squared error corresponds to using the negative log-likelihood of the data under a normal distribution with fixed standard deviation.

“Even if you look at something like regression or uh yeah uh let's say you do like image uh detection, like uh let's say you do depth estimation, something like that, you s uh with like your mean square error, you're still doing inference, right? It's the mean square error is just a log uh negative log likelihood of the uh of your data under a normal distribution with a fixed standard deviation. And so you are still”