The library

Everything I’m
reading & writing.

hover a spine to ease it out · click a book to open it · click a scroll on the top shelf to read an essay

It’s so imperative to discern the signal over the noise. You need to continually ask yourself, “what is the most important thing” — for a particular system, for whatever you’re doing, for anything at all. You need to cut away the noise and focus on the most significant parts.

For example, asking “what are the limitations of this machine learning model” is such a great way to gauge the places where we can actually improve. It immediately collapses a fog of possibilities down to the few that matter.

It’s great, of course, to be excited by something — to come up with new ideas and to become a visionary. But you need to remain close to the truth to make real advancements. Excitement points you at a direction; signal tells you whether the direction is real.

Many people have said that Bayesian updating is a good framework for understanding how beliefs change over time. And it is. But humans can also significantly update their beliefs by reaching a set of logical conclusions — reasoning itself is a major driver of belief change.

I think this is a big reason humans can reach such high reasoning capabilities with so little data: we reason as a form of synthetic data generation. We use a small amount of data to build a fundamental set of axioms. Then, by applying logical reasoning to those axioms, we generate a vastly larger and better space of inferences than the data alone could justify.

There’s enormous value in boiling knowledge down into more fundamental principles. This is, famously, how Elon Musk describes reasoning: rather than taking textbook knowledge as truth, he reduces it to something like Newton’s laws, or the Lagrangian and Hamiltonian formalisms. Then you return to the advanced, textbook-level knowledge with that axiomatic footing — and now you have a solid base from which to absorb genuinely new ideas.

The huge idea

What if an LLM continually boiled its entire corpus of generated knowledge down into a vector database of axiomatic truths, likely truths, and false statements — and then somehow folded that structure back into the model itself?

We essentially want to build circuits in a neural network grounded in truths — a subgraph of truths the rest of the model can lean on. A “truth-sensing detector,” built in.

What I find very interesting is that I arrived at this conclusion without seeing it as a direct consequence of my initial Bayesian thinking — which, in a way, proves the point. A single talk by François Chollet fundamentally updated my beliefs. That’s not gradual Bayesian updating on a stream of evidence; that’s reasoning restructuring the prior.

Say we have a simple neural network. How do we know which input neurons mattered most for the final prediction?

One family of answers is perturbation: poke each input, see how much the output moves. That works, but it’s expensive and a little blunt. Layerwise Relevance Propagation (LRP) takes a different route.

The core idea

A neural network is, generally speaking, a stack of nonlinear functions. The trick behind LRP is to approximate each of those nonlinear functions as a first-order Taylor expansion — a local linearization — and then use that linear structure to push “relevance” backward, layer by layer, from the output down to the inputs. Each neuron hands its relevance to the neurons that fed it, in proportion to how much each one actually contributed.

Run that all the way to the input layer and you get a relevance score per input neuron: a map of what the network leaned on.

Conservation and positivity

LRP comes with some nice properties. There’s a conservation property — relevance is neither created nor destroyed as it flows down through the layers — and a positivity flavor in the classic rules. Both are satisfying, and they make the attributions easier to trust.

It’s worth noting, though, that these constraints are not strictly required, and a lot of more recent work relaxes or ignores them in pursuit of sharper attributions.

Where it’s going

The interesting frontier is attention. AttnLRP extends these ideas to transformer attention, and the line of work around Chefer et al. shows how to get clean relevance maps through attention layers specifically. That’s where I want to take this note next — into how relevance propagates through attention, and what it tells us about what these models are really doing.

This is an evolving note; I’ll expand the derivation and the attention case over time.

So you want to learn computational chemistry but haven’t taken quantum mechanics, chemistry, physical chemistry, and numerical algorithms? No problem. In this series we’ll build up the approximations used in computational chemistry with no prerequisites other than precalculus, high-school calculus, and an enthusiastic reader.

Motivation

Chemistry is an expensive and dangerous field. Doubtless you’ve heard stories of chemists blowing up their labs, inhaling toxic fumes, or needing very expensive equipment to run experiments. A more convenient way to do chemistry is to sit at home and simulate experiments with our wonderful computers.

Our goal today is to simulate the behavior of atoms with programming.

How can we do that? What even is an atom?

Starting

Many of these questions are better reserved for an expert in quantum mechanics — but here I’ll present a few key assumptions and build our approximations up from there.

We can represent the full state of an atom with a symbol, $\psi$ . By “state of an atom” I mean this: $\psi$ will tell us everything we could ever want to know about the atom.
Applying functions¹ to $\psi$ gives us information about the atom. We’ll soon define operations on $\psi$ that extract physical quantities.
Every atom has an intrinsic “energy.” You can think of this as how much it can move now plus how much movement it has stored up for the future. The energy of a car, for instance, is related to how fast it’s moving and how much it will be able to move later.

Now for a little history. At the turn of the 20th century, physicists like Niels Bohr, Albert Einstein, and Max Planck discovered something striking about the atoms we’ve been describing: every atom can only take on certain energies. We can’t have a continuous range of energies for an atom — the allowed energies are discrete. It’s a bit like how you can have one plate or two plates, but half a plate doesn’t quite make sense.

This single fact — quantization — is where everything interesting begins. In the next part, we’ll turn these assumptions into something we can actually compute.

This is part one of an evolving series — more soon.

Technically, we apply operators to $\psi$ — functions that take in a function and return another function (often scaled by a number we care about). ↩

Books

Deep Learning — Goodfellow, Bengio & Courville. The book that taught a generation the math under the magic. Dense, but the chapters on optimization and representation learning are worth returning to whenever a model misbehaves. The intuition it builds for why depth helps has aged better than most of its architectures.
Molecular Biology of the Cell — Bruce Alberts et al.. The cell, rendered in full. What strikes me most is how much of it reads like systems engineering — feedback, error correction, modularity — written in a language we're only beginning to model computationally. A reference I keep open more than I read cover-to-cover.
The Machinery of Life — David S. Goodsell. Goodsell paints the molecular world to scale — proteins, membranes, and machinery crowded together the way they actually are inside a cell. It permanently changed how I picture structure. If you want to feel why protein shape is function, start here.
Reinforcement Learning: An Introduction — Sutton & Barto. The clearest exposition of how an agent learns from consequence. The early chapters on value, policy, and the Bellman equations are some of the best-written technical prose in ML. I keep coming back to it while thinking about RL for synthesis planning.
Organic Chemistry — Clayden, Greeves & Warren. The organic chemistry text that actually explains why reactions go the way they do, mechanism first. It rewired how I think about electrons pushing around a molecule — and made the bench at Caltech make sense. Mandatory for anyone who wants to reason, not memorize.
Gödel, Escher, Bach — Douglas Hofstadter. A book about how meaning and minds can emerge from formal rules — told through math, art, music, and a lot of mischief. It's the closest thing I have to a personal manifesto for why intelligence (artificial or otherwise) is worth chasing. Slow, strange, and unforgettable.
What Is Life? — Erwin Schrödinger. A physicist asks how life cheats entropy — and half-predicts the genetic code decades early. Proof that the most interesting questions live in the gaps between fields. Short enough to read in an afternoon, big enough to think about for years.
Surely You're Joking, Mr. Feynman! — Richard P. Feynman. Less physics, more a manual for curiosity. Feynman picking locks, learning to draw, and refusing to take anything on authority is the disposition I try to bring to everything. The funniest serious scientist who ever lived.
Pattern Recognition and Machine Learning — Christopher Bishop. The Bayesian counterweight to the deep-learning canon. Bishop treats learning as probabilistic inference, and the discipline of that view has saved me from a lot of hand-waving. The graphical-models chapters reward every hour you give them.
The Selfish Gene — Richard Dawkins. The idea that the gene, not the organism, is the unit of selection — and that so much complexity follows from that one shift in frame. A masterclass in reasoning from the right level of abstraction, which turns out to be a transferable skill.
The Emperor of All Maladies — Siddhartha Mukherjee. A biography of cancer — the disease, the science, and the people who spent lifetimes fighting it. On the shelf because drug discovery is, in the end, about this. Next up.
The Structure of Scientific Revolutions — Thomas S. Kuhn. Where "paradigm shift" comes from. Kuhn's argument that science advances in ruptures, not smooth increments, feels especially live in a field being remade by machine learning right now. Queued, and overdue.
The Logic of Chemical Synthesis — E. J. Corey & Xue-Min Cheng. Retrosynthesis, formalized by the man who won the Nobel for it. Learning to take a target apart along strategic bonds — to think backward from the molecule you want — is the single most transferable idea I picked up at the bench. The worked syntheses still read like chess.
Structure and Interpretation of Computer Programs — Abelson & Sussman. The wizard book. It treats programming as a way of thinking about thinking — building up abstraction, then tearing it down to interpreters and evaluators. Taking the Berkeley course built on it (61A) rewired how I see computation. Few books are this fundamental.
The Art of Electronics — Horowitz & Hill. The bible of practical circuit design — opinionated, funny, and relentlessly useful. It's how I fill the gaps between the physics of a transistor and a board that actually works. The kind of reference that makes you want to go build something at 2am.
The Vital Question — Nick Lane. Why life is the way it is, argued from energy and membranes up. Lane makes the case that the proton gradient — chemiosmosis — is as central to life as the genetic code. It reframed biology for me as fundamentally a physics-and-chemistry problem. Bracing and original.
The Feynman Lectures on Physics — Richard P. Feynman. Physics taught by someone who clearly loved it. The famous opening — that everything is made of atoms in perpetual motion — is the sentence I'd save if I could keep only one. Dense to work through, but every chapter pays you back in intuition.
Superintelligence — Nick Bostrom. The careful version of the argument that getting advanced AI right matters enormously. You don't have to buy every scenario to take the core seriously: capability and alignment are different problems, and the gap between them is where the risk lives. Worth reading slowly.
Lehninger Principles of Biochemistry — Nelson & Cox. Where the molecules of life meet the thermodynamics that drive them. The chapters on enzyme kinetics and metabolic regulation are the connective tissue between the chemistry I love and the biology I study. A workhorse I expect to keep on the desk for years.
The Double Helix — James D. Watson. How the structure of DNA actually got solved — competitive, messy, and very human. Read it alongside the knowledge that Rosalind Franklin's data made it possible. A useful antidote to the myth that science is tidy; it's people, models, and a race to the right picture.