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. Three small arithmetic transformers answered 300 problems on each of three tasks, and each judged all three models’ answers by its own per-character likelihood, with a threshold set so its overall pass rate was right, up to ties....

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 Gaussian clusters in ten dimensions, one ten times as wide as the other, came out of t-SNE with a radius ratio between 0.99 and 1.01 at perplexities 5, 30 and 100; three clusters with one gap ten times the other came out with a gap ratio between 0....

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. Gaussian mixtures fitted to a single two-dimensional cluster: for a Gaussian one, the BIC chose one component at every size from 200 to 20,000 points; for a uniform square it chose 2, 7, 14, 18; for a heavy-tailed cluster 2, 3, 4, 9....

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 two half-moons DBSCAN scored an adjusted Rand index of 0.95 or more for radii between 0.071 and 0.206, and 1.00 from 0.116, where k-means scored 0.25; the radius at the knee of the k-distance plot, 0....

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. An isolation forest on simulated data with 0.5%, 2% or 10% planted anomalies ranked them with AUCs of 0.99 or more, and flagged exactly the share set by its contamination parameter; the ‘auto’ setting flagged 10....

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. k-means stability, the agreement between clusterings of two random 80% subsamples, chose k = 2 for one elongated Gaussian (stability 0.92) and k = 4 for a uniform square (0.94), neither of which has clusters....