Deduction is deductive

The somewhat counter-intuitive fact about both computation and logic is that they actually are destructive processes, although we see them as constructive. When Sherlock Holms deduces that the butler has murdered, he creates the solution, right? Or when we run a program to calculate 100 number of π\pi, we clearly get 100 numbers we have not had a minute ago.

The problem here is that those outputs were not pulled out of nothing, but transformed from input — some casual observations in case of Holmes, and the very π\pi-calculating code. What’s worse, these inputs on their own contained much more information than the outputs retained; Sherlock could probably also established few lovers of the Countess, while the code pull few hundred more digits. As with Michaelangelo sculpting process, deduction only carves solution out of what we already knew, so it could shine with it shameless obviousness.

This can be clearly explained by the determinism of these processes — for a given input, they would always return the same output, so there cannot be more outputs that there are inputs. At most, they would just copy the information, which is not really helpful; otherwise, the transformation is not reversible and thus lossy.

In fact, we don’t even need determinism; the information theory has an important result known as data processing inequality. It states that in a system ABCA\rightarrow B \rightarrow C, so when CC is only influenced by AA through BB, I(A;B)I(A;C).I(A;B)\ge I(A;C). I(X;Y)I(X;Y) here denotes mutual information, that is how much can we say about YY knowing XX (and vice versa). When AA is, say, the state of the world, BB is our cognitive agent and CC its output, we clearly see that regardless of how many petabytes we’ll pull into CC, it won’t have more information about the world that the agent acquired in the first place.

This naturally applies to machine learning — an output of the model has no more useful information than the training set and query, and almost surely much less. And is also contrary to what most people currently expect from contemporary ML, given hopes to replace human creativity with generative models and a fear of model inbreeding. Wikipedia includes a spot-on Charles Babbage quote from his Passages from the Life of a Philosopher:

On two occasions I have been asked, Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out? … I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.

Anyhow, you might be wondering how does human brains work then; a steady stream of novel art and science is pretty self-evident. The key is here though that we interact with our reality, thus pulling more and more information that is still relevant (mutual with the world); this way we won’t ever establish every single truths about our universe, but at least we can be progressively less wrong. Similarly, art neither is inherently artistic — its value lies in how it (and its story) resonates with people; it is also measured in mutual information, not plain entropy.

Previously: Latent supervision.

CC-BY mbq, written 1-9-2023.