The true case against the standard deviation is more or less impossible to make without going into the math. For a mathematically advanced Bayesian, the idea i

The true case against the standard deviation is more or less impossible to make without going into the math. For a mathematically advanced Bayesian, the idea is straightforward if you've already been through the lawful derivation of Gaussians from their underlying assumptions: Using standard deviations in your thinking makes sense if you think that an increasingly large error has a decreasing log-probability that goes as the square of the size of that error. The distribution that you get from the sum of independent variables is the classic example. Basically, if you think in standard deviations, then all postulates of the form "The error was that size" will come with probability penalties that you trade off against each other in proportion to the square of the size of that error. Thinking in standard deviations similarly corresponds to models where exceptional events become unlikely in proportion to the square of the size of that exception, only by "unlikely" you have to read log probability rather than probability, and log probability is (always) the right way for Bayesians to think about it anyway for reasons I won't go into. And then the critique of "standard deviation" is that people got taught SD as a statistical tool that you just pull out any time you feel like it, and they don't know what it means in underlying probability-theoretic terms as an assumption about either the world or our own uncertainty, and so it's misused horribly on all sorts of occasions. I'd guess that SDs are appropriate 40-90% of the time depending on which field you work in, but without a lot of Bayesian background with fairly advanced math people will not be able to really appreciate what the other 60-10% of times look like. And the state of education is not anywhere near like that. It's just people being taught to calculate the standard deviation cause, like, that's something you do. They don't know what assumptions it corresponds to even in the cases where SD does apply. Burning SDs to the ground and starting over would not be very much amiss in one of the fields where SDs only make sense 40% of the time, but the practitioners are using them all the time. (Machine learning is one of the fields where SDs make sense maybe 40% of the time, and if I found an ML practitioner who had been taught to think in terms of squared error I'd send them off to learn the underlying probability theory until their entire worldview had been translated into likelihood functions instead.)