In the folder "LabelSwitching" we have included all the code used in Subsection: Relabeling algorithms, Gibbs Sampler. This will enable readers to replicate all the analysis, simulations and

Markov chain Monte Carlo (MCMC) or the Metropolis-Hastings algorithm is a simulation algorithm that has made modern Bayesian statistical inference possible. Nevertheless, the efficiency of different Metropolis-Hastings proposal kernels has rarely been studied except for the Gaussian proposal. Here we propose a unique class of Bactrian kernels, which avoid proposing values that are very close to the current value, and compare their efficiency with a number of proposals for simulating different target distributions, with efficiency measured by the asymptotic variance of a parameter estimate. The uniform kernel is found to be more efficient than the Gaussian kernel, whereas the Bactrian kernel is even better. When optimal scales are used for both, the Bactrian kernel is at least 50% more efficient than the Gaussian. Implementation in a Bayesian program for molecular clock dating confirms the general applicability of our results to generic MCMC algorithms. Our results refute a previous claim that all proposals had nearly identical performance and will prompt further research into efficient MCMC proposals.Bayesian inference | mixing | convergence rate M arkov chain Monte Carlo (MCMC) algorithms can be used to simulate a probability distribution π(x) that is known only up to a factor, that is, with only πðyÞ πðxÞ known; they are especially important in Bayesian inference where π is the posterior distribution. In a Metropolis-Hastings (MH) algorithm (1, 2), a proposal density q(yjx), with x, y ∈ χ, is used to generate a new state y given the current state x. The proposal is accepted with probability α(x, y). If the proposal is accepted, the new state becomes y; otherwise it stays at x. The algorithm generates a discrete-time Markov chain with state space χ and transition law P having transition probability density pðx; yÞ = qð yjxÞ · αðx; yÞ; y ≠ x; 1 − Z χ qðyjxÞ · αðx; yÞdy; y = x:The acceptance probability α is chosen so that the detailed balance condition is satisfied: π(x)p(x, y) = π(y)p(y, x), for all x, y ∈ χ. The MH choice of α is αðx; yÞ = min & 1; πðyÞ πðxÞ × qðxjyÞ qðyjxÞThe proposal kernel q(yjx) can be very general; as long as it specifies an irreducible aperiodic Markov chain, the algorithm will generate a reversible Markov chain with stationary distribution π. Here we assume that χ is a subset of R k and both π(y) and q(yjx) are densities on χ.Given a sample x 1 , x 2 , . . ., x n simulated from P, the expectation of any function f(x) over πcan be approximated by the time average over the sampleĨThis is an unbiased estimate of I, and converges to I according to the central limit theorem, independent of the initial state x 0 (ref. 3, p. 99). The asymptotic variance ofĨ iswhere V f = var π {f(X)} is the variance of f(X) over π, ρ k = corr{f(x i ), f(x i + k )} is the lag-k autocorrelation (ref. 4, pp. 87-92), and the variance ratio E = V f /ν is the efficiency: an MCMC sample of size N is as informative about I as an independent sample of size NE. Thus, NE is known as the effective sample size. Given π an...

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Nickel biosorption ability was evaluated in two bacterial strains: Acinetobacter baumannii UCR-2971 and Pseudomonas aeruginosa UCR-2957, resulting in greatest adsorption at pH 4.5 and a residence time of 100 minutes. Biosorption isotherms showed that the process follows the Langmuir model. The maximum adsorption rates (N max ) were 8.8 and 5.7 mg·g -1 for A. baumannii and P. aeruginosa, respectively; however, affinity constants suggest that P. aeruginosa (K=1.28) has higher affinity for nickel than A. baumannii (K=0.68). It is suggested that both strains could be used for wastewater treatment, as long as the concentration of Ni 2+ is within the range of mg·L Nickel is an important environmental inorganic pollutant, with allowed levels under 0.04 mg·L -1 in human consumption water. Higher concentrations affect normal flora in ecosystems and are toxic for human beings.Conventional chemical methods for heavy metal removal from wastewater (precipitation, filtration, ion-exchange, reduction-oxidation) are expensive and ineffective, particularly when metal concentration is low (4,12,13). Thus, biotechnological methods such as biosorption are emerging as an interesting alternative. Since cells are metabolically inactive in non-viable biomass systems, metal interactions occur at the superficial level (14). Bacteria express a wide range of complex molecules on their cell wall, which confer anionic net charge to the cell surface at acidic pH values (13). In Gram negative bacteria, the lipopolysaccharide, a highly anionic structure, has been identified as the main binding site for metals (9). When the cell wall is in direct contact with the environment, negatively charged groups are able to attract and bind metallic cations based on electrostatic forces, without cellular energy consumption, an effect that is favored by the high surfacevolume ratio in bacteria (3,5).In this work, nickel biosorption ability was investigated using bacteria isolated from wastewater contaminated with heavy metals (34.9 ± 9.0 mg Ni 2+ ·L -1 ; 31.5 ± 4.0 mg Pb 2+ ·L -1 ). The sample was aseptically filtered (0.45 µm membrane); the residues were resuspended in 100 mL Trypticase Soybean Broth (TSB, Difco) and incubated at 25ºC for four days. Then, 0.1 mL from the TSB were inoculated in Blood Agar, Cetrimide Agar (Difco), and Mac Conkey Agar (Oxoid) plates and incubated for 48 hours at 25ºC. Isolated strains were identified using the automatic VITEK system (BioMèrieux, Inc); Acinetobacter baumannii (UCR-2971) and Pseudomonas aeruginosa (UCR-2957) were the selected strains. For biomass production, these strains were inoculated in TSB and agitated (80 rpm) in a thermal bath (Orbit 3540, Labline), for 72 hours at 28ºC. The TSB was centrifuged at 5000 rpm for 10 minutes and biomass was washed three times with sterile distilled water and dried at 56ºC for 48 hours. The inocula of each adsorption

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