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AI 资讯

Markov Chain Monte Carlo: Theoretical Foundations

Adapted from an appendix of my MS thesis. Markov Chain Monte Carlo Almost as soon as computers were invented, they were used for simulation. Markov chain Monte Carlo (MCMC) was invested as Los Alamos, Metropolis et al (1953) simulated a liquid in equilibrium with its gas phase. Their tour de force was the realization that they did not need to simulate the exact dynamics, they only needed to simulate some Markov chain with the same equilibrium distribution. The Metropolis algorithm was widely used by chemists and physicists, but was not widely known among statisticians until after 1990. Hastings (1970) generalized the Metropolis algorithm, and simulations following his scheme are said to use the Metropolis-Hastings (MH) algorithm [1]. A special case of the MH algorithm was introduced by Geman et al (1984) discussing optimization to find the posterior mode rather than simulation. Algorithms following their scheme are said to use the Gibbs sampler. It took some time for the spatial statistics community to understand that the Gibbs sampler simulated the posterior distribution, thus enabling full Bayesian inference of all kinds. Gelfand et al (1990) made the wider Bayesian community aware of the Gibbs sampler, and then it was rapidly realized that most Bayesian inference could be done using MCMC, whereas very little could be done without MCMC. Green (1995) generalized the MH algorithm as much as it could be generalized [1]. Theoretical Foundations A sequence X 1 ​ , X 2 ​ , … of random elements of some set is a Markov chain if the conditional distribution of X n + 1 ​ given X 1 ​ , … , X n ​ depends on X n ​ only. The set in which the X i ​ take values is called the state space of the Markov chain. A Markov chain has stationary transition probabilities if the conditional distribution of X n + 1 ​ given X n ​ does not depend on n . This is the main kind of Markov chain of interest in MCMC. The joint distribution of a Markov chain is determined by the following [1]. The ma

2026-07-11 原文 →
AI 资讯

Biot Number: How to Know When a Cooling Object Has a Single Temperature

Pull a hot steel bolt out of a furnace and quench it in oil, and a fair question is: does the bolt cool from the outside in, with a sharp temperature difference between its skin and its core, or does the whole thing drop in temperature more or less together? The answer is not obvious from the part itself. A thin copper washer and a thick ceramic block behave very differently in the same bath, even at the same starting temperature. The Biot number is the small calculation that settles this question before you commit to any heavy analysis. It tells you, in a single dimensionless figure, whether an object can be treated as having one uniform temperature or whether you must resolve a temperature gradient inside it. That distinction changes the math from a one-line exponential decay to a partial differential equation. Why this calculation matters Transient heating and cooling problems show up everywhere: heat-treating metal parts, quenching forgings, cooling electronics, baking or chilling food, warming up an engine block. In every one of these, the engineer wants to know how the temperature changes over time. The hard version of that question requires solving the heat conduction equation across the body, with position and time as variables. The easy version is the lumped-capacitance model, which treats the whole object as a single point at one temperature. It reduces the problem to a simple first-order exponential. The catch is that the lumped model is only valid when internal conduction is fast compared with surface convection. The Biot number is exactly the check that tells you whether that condition holds. Skip the check and apply the lumped model where it does not belong, and you can badly mispredict cooling times, residual stresses, and the risk of cracking from thermal gradients. The core formula The Biot number compares two thermal resistances. One is the resistance to conducting heat through the inside of the solid. The other is the resistance to carrying heat a

2026-07-11 原文 →
开发者

Empathy for the optimizers

This is Optimizer, a weekly newsletter sent from Verge senior reviewer Victoria Song that dissects and discusses the latest gizmos and potions that swear they're going to change your life. Opt in for Optimizer here. Bryan Johnson, best known as the man who wants to live forever, has an incurable autoimmune disease. The internet's most […]

2026-07-10 原文 →
AI 资讯

I tracked every trending AI repo's stars daily for 3 weeks. The growth is not where I expected

I run a small AI trends site, and three weeks ago I started doing something simple: every day, snapshot the star count of every repo that crosses my GitHub trending scan for AI. No judgment, no curation, just append-only rows in a database. 611 repos and 2,671 data points later (June 19 to July 10), the picture of what's actually growing looks pretty different from what my feeds told me was hot. Here's what the data says. Before publishing this I re-checked every number below against GitHub's live API. Star counts drift by the hour, so treat them as of July 10. The top 10 risers, by raw stars gained Repo Gained Window From → To calesthio/OpenMontage +30,253 21 days 5,899 → 36,152 DeusData/codebase-memory-mcp +20,483 19 days 7,516 → 27,999 mattpocock/skills +19,053 15 days 137,485 → 156,538 obra/superpowers +16,887 20 days 232,908 → 249,795 NousResearch/hermes-agent +14,896 21 days 197,297 → 212,193 Panniantong/Agent-Reach +14,334 14 days 34,780 → 49,114 usestrix/strix +13,243 12 days 26,363 → 39,606 addyosmani/agent-skills +12,685 21 days 63,156 → 75,841 asgeirtj/system_prompts_leaks +11,720 21 days 43,415 → 55,135 msitarzewski/agency-agents +11,055 10 days 118,241 → 129,296 Windows differ because I only hold snapshots for the days a repo appeared in my scan; each row states its own real window. Three things in this data genuinely surprised me. 1. "Skills" are eating agent frameworks Four of the top ten are not agent frameworks. They are collections of packaged expertise that plug into an existing agent: obra/superpowers (still compounding at roughly 840 stars a day on a 250k base), mattpocock/skills, addyosmani/agent-skills, msitarzewski/agency-agents. A year ago this table would have been full of new frameworks. Now the framework layer looks settled and the growth is in what you load INTO the agent. The moat moved from orchestration code to encoded judgment. 2. The sharpest climbs are applications, not infrastructure The steepest sustained climb from a newcomer in

2026-07-10 原文 →
AI 资讯

Microsoft’s carbon emissions went up 25 percent last year

Microsoft may once again be struggling to keep up with its own climate goals, according to its 2026 sustainability report. As reported by GeekWire, the report states that Microsoft's carbon emissions increased 25 percent in 2025, totalling 34 million metric tons "without select interventions." Microsoft says this was "driven primarily by the expansion of our […]

2026-07-10 原文 →
科技前沿

SpaceX is on track for record-setting Starlink deployments

SpaceX is currently ahead of last year's record-setting pace for Starlink satellite deployments. SpaceX launched 1,589 Starlink satellites into low-Earth orbit in the first half of 2026, according to launch data compiled by Jonathan McDowell's satellite tracker, compared to 1,489 satellites deployed at the same point in 2025. 2025 was already a record year for […]

2026-07-09 原文 →