Tools & the model
Free browser tools you can use today, and Kudos-1 - the reasoning model we are building for mathematics and machine learning.
Free tools
Browser-based tools that run entirely on your own machine. Nothing you open is uploaded anywhere.
Base-rate calculator
Enter a prevalence, a sensitivity, and a false-positive rate to see what a positive test result is actually worth - as a probability and as whole people out of ten thousand.
Entropy and information calculator
Edit two distributions side by side and read off entropy, cross entropy, and KL divergence in bits, with the identity H(p, q) = H(p) + KL(p ‖ q) held on screen.
Classifier metrics explorer
Move a decision threshold across an imbalanced population and watch precision, recall, F1, and the ROC point move with it - including the regime where accuracy looks excellent and the model is useless.
Attention weights explorer
Edit a query vector and see scaled dot-product attention resolve into softmax weights and a weighted sum, with temperature and the √dₖ scaling as switches you can throw.
Confidence-interval simulator
Draw thirty samples, form a confidence interval from each, and count how many cover the truth - the clearest way to see that the confidence level describes the procedure, not any single interval.
Bias-variance explorer
Sweep model complexity and sample size to watch training error fall monotonically while test error turns upward, decomposed into bias, variance, and irreducible noise.
Distribution explorer
Change the parameters of the distributions that keep appearing in the material and watch the shape, mean, and spread respond.
Maximum-likelihood explorer
Move a parameter along the log-likelihood curve of a fixed sample and watch the fitted distribution track it, with the peak sitting exactly at the maximum-likelihood estimate.
Study planner
Pick a goal - foundations, supervised methods, or modern AI - and get this site’s own ordered plan for reaching it: which learning path to take at each stage, what to read alongside it, and how long the lessons run.
Kudos-1
Most models will answer a mathematical question. Far fewer will show you the derivation, name the assumption they relied on, and tell you where it breaks. Kudos-1 is being built for the second thing: a model that reasons through a problem in steps you can check, cites the result it is invoking, and says plainly when a question falls outside what it can justify. It is trained on the same material this site teaches from, so an explanation it gives can be traced back to an article, an encyclopedia entry, or a book in the library.
Capabilities
Step-by-step derivations
Works a result through from stated assumptions to conclusion, showing intermediate algebra rather than asserting the final line.
Proof and argument checking
Reads a proposed argument and locates the first step that does not follow, instead of judging it right or wrong as a whole.
Model critique
Given a modelling setup, names the assumptions it depends on and the conditions under which its conclusions stop holding.
Code to mathematics
Reads an implementation and recovers the mathematics it encodes, so a training loop can be checked against the objective it claims to optimise.
Socratic tutoring
Diagnoses which prerequisite a learner is missing and teaches to that, rather than restating the original answer more slowly.
Foundations
- Verifiable reasoning. Every derivation is emitted as discrete, checkable steps rather than prose that merely sounds like a proof, so a reader can find the exact line where an argument goes wrong.
- Grounded in a real library. Answers are anchored to the reference texts behind this site and cite them, rather than reproducing half-remembered statements of a theorem.
- Calibrated uncertainty. A stated confidence should track how often the model is actually right. Saying "I cannot justify this" is treated as a correct answer, not a failure.
Principles
- Refusal over invention. A model that fabricates a plausible theorem is worse than one that declines. Kudos-1 is being trained to prefer an honest gap to a confident guess.
- Open evaluation. Benchmarks and failure cases are published alongside results. Claims about what the model can do should be reproducible by anyone who wants to check them.
- Teaching, not answering. The measure of success is whether the reader understands the result afterwards, not whether the answer was delivered quickly.