Bayes and Bounds

[This is post 1 in the "Structure and Cognition" series; links to all the posts can be found here]

I wanted to start writing this blog for weeks but was held up by an inescapable feeling that I couldn’t write anything until I had an idea for an introductory post where I explained the blog’s name and laid out the official Purpose of The Blog. 

I intend(ed) part of that purpose to be exploring the structure of problems we face. We often struggle to solve problems not because they are difficult, but because the way they are presented is not well-suited to the mind’s problem-solving abilities. Ideally, we would just reframe the problem to a form that is more useful, but we’re really bad at doing that. 

The irony is that the problem of writing my first post has exactly the structure that the name was supposed to convey, except I didn’t notice, because I wasn’t framing it the right way. 

There is a space of possibilities for topics I might write about on this blog. Some versions of the blog’s future have a single strong theme while others are haphazard collections of ideas or interesting research findings. In all honesty, many of those possibilities involve my writing a post or two and then abandoning the project.

In attempting to write a post explaining the name, I was trying to search through that space of possibilities to find a single best summary of the blog. This is the Bayesian side of the title, attempting to optimize the best possible name, given the early thoughts I had about some posts I wanted to write. 

Much of my intellectual development can be traced to two Bayesian philosophies. The first is my formal education in cognitive science, which (in some corners) views the mind as a Bayesian machine that closely approximates ideal normative performance. The second is the online rationalist community, who prioritize holding true beliefs. Rationalists are Bayesians in a very different sense; they see Bayesian updating of beliefs as aspirational but probably not achievable by the human brain. Instead, we rely on simple heuristic processing that predictably leads to biases and irrational decisions. 

Heuristics are sometimes cast more positively as efficiently producing “good enough” answers, saving time and/or cognitive processing. Some even argue that heuristics are actually quite good most of the time. To the extent that they rarely backfire, this may be because the modern world differs markedly from the environment these heuristics evolved to deal with.  

A less well-known view, often (overenthusiastically, in my opinion) cast in opposition to both Bayesian philosophies, takes a different view of heuristics. We don’t use heuristics to save time because finding the optimal solution is annoying or labor intensive. Instead, for many decision-making problems, optimizing is impossible. This view emphasizes that there are strong bounds on human rationality. Heuristics are necessary when problems are so difficult that the amount of time the heuristic saves you when you skip optimizing may be longer than a human lifespan. 

With those timescales in mind, it’s maybe not surprising that I was 5 weeks out from the first draft of a blog post with no clarity on the best way to characterize the blog. 

Savage (1972) noted that it's inappropriate to rigorously calculate every possible outcome in playing chess or planning a picnic. Given the 35^80 possible moves in a chess game, attempting to optimize for the best move by considering all possible options is computationally intractable. Planning the best picnic also can't be addressed by calculating - it's not a well-formulated problem. There are too many unforeseeable possibilities to “solve” this problem.  

Arguably, finding the best introduction to the blog combines the worst aspects of both problems. There are infinite possible arrangements of ideas that could be candidates for blog posts and I don’t have enough information about what future posts will look like to account for them. 

Exhaustively searching the space of possible blog posts to optimize for the best introduction is impossible. If I was going to write anything at all for an introductory post, it needed to be the product of a much less comprehensive algorithm that might not find the best result but would produce something decent in a reasonable amount of time. 

To the extent that I can predict the theme of this blog, then, it is something like: what kinds of problems are structured in a way that can be solved by optimizing, when do we need to rely on heuristics instead, and how do those heuristics work?