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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

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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

Notes from bringing up a PCIe WiFi module on i.MX8MQ; symptoms and how to diagnose them. Phy link never came up — what does this mean? This message is typically seen in dmesg as shown below. [ 3.828121] imx6q-pcie 33800000.pcie: iATU: unroll T, 4 ob, 4 ib, align 64K, limit 4G [ 4.807241] imx6q-pcie 33c00000.pcie: Phy link never came up [ 4.841482] imx6q-pcie 33800000.pcie: Phy link never came up [ 5.821279] imx6q-pcie 33c00000.pcie: Phy link never came up [ 5.830481] imx6q-pcie 33c00000.pcie: PCI host bridge to bus 0001:00 [ 5.854997] imx6q-pcie 33800000.pcie: Phy link never came up [ 5.862205] imx6q-pcie 33800000.pcie: PCI host bridge to bus 0000:00 It means one of the following The PCIe peripheral is not powered up. PCIe reset is not deasserted, so the chip is in reset. This could be because the DTB is deasserting an incorrect GPIO. Reference clock is not enabled Using the incorrect PCIe controller in the device tree. As we can see that both the PCIe controllers can report this. So first determine which controller is the peripheral hooked to. More on this in the next section. Which PCIe controller is my device on? ( &pcie0 vs &pcie1 ) The rule here is to match by address and not by label/name. If the schematic calls out controllers as PCIE1 and PCIE2, and the device tree lists pcie0 and pcie1, understand the mapping. The DTS label is arbitrary - match by register base ( @address in the node name), which is the same in the DTS reg and the reference memory map. For definitive addresses look in the .dtsi , as sometimes the manuals are misleading. Given below is a mapping table for i.MX8MQ DTS. | ADDRESS (in .dtsi) | Silicon (RM) Label &pcie0 | 0x33800000 | PCIe1 &pcie1 | 0x33c00000 | PCIe2 The addresses are listed in the chip’s memory layout are from processor reference manual. Below is snapshot from the i.MX8MQ reference manual, where the layout for the core A-53 is listed. Start Address | End Address | Size | Description 3381_0000 | 3381_3FFF | 4MB | PCIe-2 << inco

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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 […]

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