Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. On a 20% test split of the UCI digits, 360 images, logistic regression, an RBF SVM and 3-NN score 0.967, 0.986 and 0.986, and one test set’s 95% binomial interval is about +-0....
School Maths 1: The Only Difference Between These Two Models Is the Order of the Lines
Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. Two character transformers, identical in architecture, parameters, seed, steps, schedule and training problems. The one manipulated variable is the order of the lines in the training file. Packed in generation order, every reversed sum lands directly beneath the forward form of the same sum - 5,076 of 5,076, which is all of them - and ‘reverse the line above’ answers the question with no arithmetic....
Coverage 1: 90% Coverage Is a Promise About Averages, Not About You
Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. Split conformal at alpha = 0.1 on 24,576 held-out predictions split in half delivers 0.8961 coverage against a finite-sample guarantee of 0.9001, and over 200 calibration splits a mean of 0.8999. Exactly what was promised....
Gradients 1: Backprop Is the Chain Rule, and the Chain Rule Has Hypotheses
Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. relu, abs, clamp, hardtanh and a vector norm all return 0 at their kinks; maximum, minimum and max split the tie and return 0.5; and sqrt(x·x), which is abs(x), returns nan where abs returns 0....
Headline Statistics 1: Korea's Fertility Rebound Needs Nobody to Have More Children
Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. Give every cohort the same completed fertility and let each one postpone by d years relative to the last. The period TFR is then exactly Q/(1+d), the period mean age of childbearing rises at d/(1+d) rather than at d, and dividing the first by one minus the second returns Q to five decimal places — the Bongaarts-Feeney adjustment, which on this schedule is an identity rather than a correction....
Topology 1: H0 Is Single-Linkage Clustering, Bit for Bit
Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. Persistent homology arrives with a large vocabulary and, in dimension zero, a very small computation: walk the edges shortest first and record the length of each one that joins two components. That is the single-linkage dendrogram’s merge heights, and not approximately — the two return arrays that are equal float for float, on four designs from 12 to 60 points and 1 to 30 dimensions....
Numerical Analysis 1: Your Learning Rate Is a Step Size
Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. The optimiser everyone uses is an ODE solver, and that is an identity rather than an analogy: gradient descent is forward Euler on the gradient flow. So the learning rate is a step size, and step sizes have stability limits....
Linear Algebra 1: The Condition Number Is the Error Bar on Your Solve
Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. The residual is the check everybody runs after a solve, and it is the one check that cannot detect the most common way a solve goes wrong. The condition number can: it converts the precision of the inputs into an error bar on the output, before any statistics are involved....
ML 2: Cross-Validation Estimates the Error of a Model You Did Not Train
Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. On 2,000 simulated least-squares datasets of 100 rows, where the exact test error of each fitted model is known, the correlation between the 10-fold CV estimate and that error is -0.01. The CV estimates spread 2....
Coverage 2: Your Calibration Error Is Mostly Your Bin Count
Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. A model calibrated by construction - labels drawn from its own probabilities - scores an ECE of 0.085 on 200 predictions, 0.035 on 1,000 and 0.008 on 20,000 at 15 bins, and the fitted slope against bins/n is 0....