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Information theory

math

In progress

Nothing written up yet. This is a placeholder for notes I am actively taking.

I keep using the vocabulary without having derived any of it. Things to work out properly:

  • entropy as expected surprise, and why the log is not an arbitrary choice
  • cross-entropy and KL divergence from the coding perspective — bits wasted using the wrong code — rather than as formulas that happen to share gradients
  • why KL is not a metric, and when the asymmetry actually matters in practice
  • mutual information, and what contrastive objectives like InfoNCE are really bounding
  • the data processing inequality, and what it says about representation learning

This overlaps heavily with probability and likelihood, which already covers the DKL(pq)=H(p,q)H(p)D_{KL}(p \,\|\, q) = H(p, q) - H(p) identity and where KL blows up. The goal here is the coding intuition underneath it.