Yudhowsky gets a lot wrong even in a few sentences:

Previously, the most popular philosophy of science was probably Karl Popper's falsificationism - this is the old philosophy that the Bayesian revolution is currently dethroning.

First, Popper's philosophy cannot be accurately described as falsificationism - that is just a component of it, and not the most important component. Popperian philosophy consists of many inter-related ideas and arguments. Yudhowsky makes an error that Popperian newbies make. One suspects from this that Yudhowsky is making himself out to be more familiar with Popper than he actually is. His claim to be dethroning Popper would then be dishonest as he does not have detailed knowledge of the rival position. Also he is wrong that Popper is popular: he isn't. Furthermore, Popper is familiar with Bayesian epistemology and actually discusses it in his books. So calling Popper's philosophy old and making out that Bayesian epistemology is new is wrong also.

Karl Popper's idea that theories can be definitely falsified, but never definitely confirmed, is yet another special case of the Bayesian rules;

Popper never said theories can be definitely falsified. He was a thoroughgoing fallibilist and viewed falsifications as fallible conjectures. Also he said that theories can never be confirmed at all, not that they can be partially or probabilistically confirmed, which the above sentence suggests he said. Saying falsification is a special case of the Bayesian rules also doesn't make sense: falsification is anti-induction whereas Bayesian epistemology is pro-induction.