This is an expanded version of my talk. I assume a high degree of familiarity with Bayesian probability theory.

Toward a New Technical Explanation of Technical Explanation -- an attempt to convey the practical implications of logical induction -- was one of my most-appreciated posts, but I don't really get the feeling that very many people have received the update. Granted, that post was speculative, sketching what a new technical explanation of technical explanation might look like. I think I can do a bit better now.

If the implied project of that post had really been completed, I would expect new practical probabilistic reasoning tools, explicitly violating Bayes' law. For example, we might expect:

A new version of information theory.
- An update to the "prediction=compression" maxim, either repairing it to incorporate the new cases, or explicitly denying it and providing a good intuitive account of why it was wrong.
- A new account of concepts such as mutual information, allowing for the fact that variables have behavior over thinking time; for example, variables may initially be very correlated, but lose correlation as our picture of each variable becomes more detailed.
New ways of thinking about epistemology.
- One thing that my post did manage to do was to spell out the importance of "making advanced predictions", a facet of epistemology which Bayesian thinking does not do justice to.
- However, I left aspects of the problem of old evidence open, rather than giving a complete way to think about it.
New probabilistic structures.
- Bayesian Networks are one really nice way to capture the structure of probability distributions, making them much easier to reason about. Is there anything similar for the new, wider space of probabilistic reasoning which has been opened up?

Unfortunately, I still don't have any of those things to offer. The aim of this post is more humble. I think what I originally wrote was too ambitious for didactic purposes. Where the previous post aimed to communicate the insights of logical induction by sketching broad implications, I here aim to communicate the insights in themselves, focusing on the detailed differences between classical Bayesian reasoning and the new space of ways to reason.

Rather than talking about logical induction directly, I'm mainly going to explain things in terms of a very similar philosophy which Richard Jeffrey invented -- appare...

Epistemology