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Stan R,
Stan can be run from a number of statistical software packages. So far, I've been from R to run Stan , first follow the Quick Start Guide install and run all the contents of the instructions.
Simple linear regression
The first step is to write files to Stan models. It contains a file linreg.stan:
data { int N; [N] x; vector[N] y; } parameters { real alpha; real beta; real sigma; } model { y ~ normal( + beta * x, sigma); }
The first portion of the data file is referred to, it declares Stan passed to the scalar, vector and matrix as inputs.
Next, we simulated data can be set by running the following R code, and using our file linreg.stan Stan and to fit the model:
set.seed(123) n <- 100 x <- rnorm(n) y <- x+ (n) mydata <- list(N = n, y = y, x= ) fit <- stan(file = 'linreg. ', data = mydata, iter = 1000, = 4)When you first install Stan model, the model will be compiled into a delay of several seconds when C ++. However, as Stan developers described, once the compilation model, it can be applied to new datasets without having to repeat the compilation process (a great advantage in the context of the implementation of the simulation study.
In the above code, we ask Stan to run four separate chains, each chain has 1000 iterations. After the run, we can aggregate output in the following ways:
fit Inference for Stan model: linreg. 4 chains, each with iter=1000; warmup=500; thin=1; post-warmup draws per chain=500, total post-warmup draws=2000. mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat alpha -0.10 0.00 0.10 -0.29 -0.16 -0.10 -0.04 0.09 1346 1 beta 0.95 0.00 0.11 0.75 0.88 0.95 1.02 1.17 1467 1 sigma 0.98 0.00 0.07 0.85 0.93 0.98 1.03 1.12 1265 1 lp__ -47.54 0.06 1.24 -50.77 -48.02 -47.24 -46.68 -46.17 503 1 Samples were drawn using NUTS(diag_e) at Mon Jun 08 18:35:58 2015. For each parameter, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat=1).For regression slope β, we mean posterior 0.95 (close to the true value of the analog data 1). To form the 95% confidence interval, we simply use 2.5% and 97.5% of percentile rear sampling, this is 0.75 to 1.17.
You can get the number from various other model fitting in. One is a drawing wherein the posterior distribution of the model parameters. To obtain the regression slope, we can do the following:
result <- extract(fit) hist(result$beta)The posterior distribution histogram β
Now let's use the standard ordinary least squares linear model:
summary(lm(y~x)) Call: lm(formula = y ~ x) Residuals: Min 1Q Median 3Q Max -1.9073 -0.6835 -0.0875 0.5806 3.2904 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.10280 0.09755 -1.054 0.295 x 0.94753 0.10688 8.865 3.5e-14 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.9707 on 98 degrees of freedom Multiple R-squared: 0.4451, Adjusted R-squared: 0.4394 F-statistic: 78.6 on 1 and 98 DF, p-value: 3.497e-14This gives us an estimate of the slope of 0.95, an average value posteriori Stan difference of 2 decimals of a standard deviation of 0.11, which is the same posteriori Stan of the SD.
stan and Bayesian inference
Stan interested in exploring and using it to perform Bayesian inference, this is due to the problem of measurement error and missing data. As WinBUGS years ago, authors and others described and illustrated, the Bayesian approach is very natural in solving the problem of different sources of uncertainty, these sources of uncertainty beyond the parameters of uncertainty, such as missing data or with errors measured covariates. In fact, multiple imputation method for missing data is a popular development within the Bayesian paradigm, and in fact can be viewed as an approximation of the full Bayesian analysis.
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