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Bayesian analysis is definitely different from the classic statistical methods. Although, both of them use subjective ideas, it is used in the selection of models in the classic statistical methods, rather than as an explicit part in Bayesian models, which allows the combination of subjective ideas with the data collected, update the prior information and improve inferences. Drastic growth of Bayesian applications indicates it becomes more and more popular, because the advent of computational methods (e.g., MCMC) renders sophisticated analysis. In Bayesian framework, the flexibility and generality allows it to cope with very complex problems. One big obstacle in earlier Bayesian analysis is how to sample from the usually complex posterior distribution. With modern techniques and fast-developed computation capacity, we now have tools to solve this problem. We discuss Acceptance-Rejection sampling, importance sampling and then the MCMC methods. Metropolis-Hasting algorithm, as a very versatile, efficient and powerful simulation technique to construct a Markov Chain, borrows the idea from the well-known acceptance-rejection sampling to generate candidates that are either accepted or rejected, but then retains the current values when rejection takes place (1). A special case of Metropolis-Hasting algorithm is Gibbs Sampler. When dealing with high dimensional problems, Gibbs Sampler doesn’t require a decent proposal distribution. It generates the Markov Chain through univariate conditional probability distribution, which greatly simplifies problems. We illustrate the use of those approaches with examples (with R codes) to provide a thorough review. Those basic methods have variants to deal with different situations. And they are building blocks for more advanced problems. This report is not a tutorial for statistics or the software R. The author assumes that readers are familiar with basic statistical concepts and common R statements. If needed, a detailed instruction of R programming can be found in the Comprehensive R Archive Network (CRAN): http://cran.R-project.org / text

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In structural equation models, researchers often examine two types of causal effects: direct and indirect effects. Direct effects involve variables that "directly" influence other variables, whereas indirect effects are transmitted via intervening variables. While researchers have paid considerable attention to the distribution of sample direct effects, the distribution of sample indirect effects has only recently been considered. Using the (delta) method (Rao, 1973), Sobel (1982) derived the asymptotic distribution for estimators of indirect effects in recursive systems. Sobel (1986) then derived the asymptotic distribution for estimators of total indirect effects in the Joreskog covariance structure model (Joreskog, 1977). This study examined the applicability of the large sample theory described by Sobel (1986) in small samples. Monte Carlo methods were used to evaluate the behavior of estimated total indirect effects in sample sizes of 50, 100, 200, 400, and 800. Two models were used in the analysis. Model 1 was a nonrecursive model with latent variables, feedback, and functional constraints among the effects (Duncan, Haller, & Portes, 1968; Sobel, 1986). Model 2 was a recursive model with observable variables (Duncan, Featherman, & Duncan, 1972). In addition, variations in these models were studied by randomly increasing and decreasing model parameters. The principal findings of the study suggest certain guidelines for researchers who use Sobel's procedures to evaluate total indirect effects in structural equation models. In order for the behavior of the estimates to approximate the asymptotic properties, sample sizes of 400 or more are indicated for nonrecursive systems similar to Model 1, and for recursive systems such as Model 2, sample sizes of 200 or more are suggested. At these sample sizes, researchers can expect sample indirect effects to be accurate point estimators, and confidence intervals for the effects to behave as theory predicts. A caveat to the above guidelines is that, when the total indirect effects are "small" in magnitude, relative to the scale of the model, convergence to the asymptotic properties appears to be very slow. Under these conditions, sampling distributions for the "smaller" valued estimates were positively skewed. This caused estimates to be significantly different from true values, and confidence intervals to behave contrary to theoretical expectations.

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Las propiedades físicas de los materiales de importancia tecnológica se originan en las reacciones y procesos a los que han sido sometidos. En todos ellos la difusión atómica juega un rol clave, porque la difusión es relevante para la cinética de muchos cambios microestructurales que ocurren durante la preparación, procesamiento y tratamiento térmico de estos materiales. Las superaleaciones, como el FeAl, son materiales tecnológicamente importantes; pues, son resistentes a altas temperaturas, mantienen su estabilidad estructural, superficial y la estabilidad de sus propiedades físicas. Por todo esto profundizar el conocimiento científico acerca de la difusión es necesario. Por tal motivo en el presente trabajo la migración atómica, en la aleación binaria ordenada con estructura B2, es estudiada por medio de simulaciones Monte Carlo Cinético, en donde la migración atómica resulta del intercambio de posiciones de un átomo con una vacante en una red rígida. El modelo cinético atomístico usado se fundamenta en la teoría de la tasa de saltos y el algoritmo del tiempo de residencia. También, usamos interacciones de a par hasta segundos vecinos más próximo. Tomamos los valores que se usaron en simulaciones del diagrama de fases, el ordenamiento B2 y precipitación del FeAl[41]. Determinamos las constantes de difusión como función de la temperatura. Además, investigamos la movilidad de las fronteras antifase en las últimas etapas del proceso de ordenamiento. Finalmente, se calcula la función de autocorrelación, la cual nos revela que la vacante efectúa saltos altamente correlacionados en la red a bajas temperaturas y también que los átomos saltan a posiciones de su propia subred a temperaturas moderadas.