# mcgibbsit Example

    library(mcgibbsit)
    #> Loading required package: coda

    set.seed(42)        # for reproducibility
    tmpdir <- tempdir()

The `mcgibbsit` package provides an implementation of Warnes & Raftery’s MCGibbsit run-length diagnostic for a set of (not-necessarily independent) MCMC samplers. It combines the estimate error-bounding approach of Raftery and Lewis with the between chain variance verses within chain variance approach of Gelman and Rubin.

For a set of exchangeable1 MCMC simulations on the same data and model `mcgibbsit` computes:

- minimum run length: \(N_{min}\)

- required burn in length: \(M\)

- total run length: \(N\)

- run length inflation due to auto-correlation: \(I\)

- run length inflation due to between-chain correlation: \(R\)

These simulations need not be independent, such as those generated by the Normal Kernel Coupler adaptive CMC method (see G. R. Warnes (2000) or G. Warnes (2001)).

For

The normal usage is to perform an initial MCMC run of some pre-determined length (e.g. 300 iterations) for each of a set of \(k\) (e.g. 20) MCMC samplers. The output from these samplers is then read in to create an `mcmc.list` object and `mcgibbsit` is run specifying the desired accuracy of estimation for quantiles of interest. This will return the minimum number of iterations to achieve the specified error bound. The set of MCMC samplers is now run so that the total number of iterations exceeds this minimum, and `mcgibbsit` is again called. This should continue until the number of iterations already complete is less than the minimum number computed by `mcgibbsit`.

If the initial number of iterations in `data` is too small to perform the calculations, an error message is printed indicating the minimum pilot run length.

## Example

This basic example constructs a dummy set of files from an imaginary MCMC sampler and shows the results of running `mcgibbsit` with the default settings.

    # Define a function to generate the output of our imaginary MCMC sampler
    gen_samples <- function(run_id, nsamples=200)
    {
      x <- matrix(nrow = nsamples+1, ncol=4)
      colnames(x) <- c("alpha","beta","gamma", "nu")

      x[,"alpha"] <- exp(rnorm (nsamples+1, mean=0.025, sd=0.025))
      x[,"beta"]  <- rnorm (nsamples+1, mean=53,    sd=14)
      x[,"gamma"] <- rbinom(nsamples+1, 20,         p=0.15) + 1
      x[,"nu"]    <- rnorm (nsamples+1, mean=x[,"alpha"] * x[,"beta"], sd=1/x[,"gamma"])
    #'
      # induce serial correlation of 0.25
      x <- 0.75 * x[2:(nsamples+1),] + 0.25 * x[1:nsamples,]

      # induce ~50% acceptance rate
      accept <- runif(nsamples) > 0.50
      for(i in 2:nsamples)
        if(!accept[i]) x[i,] <- x[i-1,]

      write.table(
        x,
        file = file.path(
          tmpdir,
          paste("mcmc", run_id, "csv", sep=".")
          ),
        sep = ",",
        row.names = FALSE
      )
    }

First, we’ll generate and load only a 3 runs of length 200:

    # Generate and load 3 runs
    for(i in 1:3)
      gen_samples(i, 200)

    mcmc.3 <- read.mcmc(
      3,
      file.path(tmpdir, "mcmc.#.csv"),
      sep=",",
      col.names=c("alpha","beta","gamma", "nu")
      )

    # Trace and Density Plots
    plot(mcmc.3)

Now run `mcgibbsit` to determine the necessary total number of MCMC samples to to provide accurate 95% posterior confidence region estimates for all four of the parameters:

    # And check the necessary run length
    mcg.3 <- mcgibbsit(mcmc.3)
    print(mcg.3)
    #>                   Multi-Chain Gibbsit
    #>                   -------------------
    #>
    #> Call             = mcgibbsit(data = mcmc.3)
    #>
    #> Number of Chains = 3
    #> Per-Chain Length = 200
    #> Total Length     = 600
    #>
    #> Quantile (q)     = 0.025
    #> Accuracy (r)     = +/- 0.0125
    #> Probability (s)  = 0.95
    #>
    #>
    #>       Burn-in  Estimation Total Lower bound  Auto-Corr. Between-Chain
    #>       (M)      (N)        (M+N) (Nmin)       factor (I) Corr. factor (R)
    #>
    #> alpha 24       1301       1325  600          2.28       0.955
    #> beta  33       1830       1863  600          3.22       0.951
    #> gamma 27       1932       1959  600          2.74       1.180
    #> nu    33       1849       1882  600          3.22       0.961
    #>       -----    -----      ----- -----        -----      -----
    #>       33       1932       1959  600
    #>
    #> NOTE: The values for M, N, and Total are combined numbers of iterations
    #>       based on using 3 chains.

The results from `mcgibbsit` indicate that the required number of samples is 1,959, which is less than we’ve generated so far.

Lets generate 7 more runs, each of length 200, for a total of 2,000 samples:

    # Generate and load 7 more runs
    for(i in 3 + (1:7))
      gen_samples(i, 200)

    mcmc.10 <- read.mcmc(
      10,
      file.path(tmpdir, "mcmc.#.csv"),
      sep=",",
      col.names=c("alpha","beta","gamma", "nu")
      )

    # Trace and Density Plots
    plot(mcmc.10)

Now run `mcgibbsit` to determine the necessary number of MCMC samples:

    # And check the necessary run length
    mcg.10 <- mcgibbsit(mcmc.10)
    print(mcg.10)
    #>                   Multi-Chain Gibbsit
    #>                   -------------------
    #>
    #> Call             = mcgibbsit(data = mcmc.10)
    #>
    #> Number of Chains = 10
    #> Per-Chain Length = 200
    #> Total Length     = 2000
    #>
    #> Quantile (q)     = 0.025
    #> Accuracy (r)     = +/- 0.0125
    #> Probability (s)  = 0.95
    #>
    #>
    #>       Burn-in  Estimation Total Lower bound  Auto-Corr. Between-Chain
    #>       (M)      (N)        (M+N) (Nmin)       factor (I) Corr. factor (R)
    #>
    #> alpha 90       1534       1624  600          2.76       0.933
    #> beta  110      1743       1853  600          3.11       0.940
    #> gamma 90       1820       1910  600          3.19       0.955
    #> nu    100      1569       1669  600          2.79       0.942
    #>       -----    -----      ----- -----        -----      -----
    #>       110      1820       1910  600
    #>
    #> NOTE: The values for M, N, and Total are combined numbers of iterations
    #>       based on using 10 chains.

`mcgibbsit` now estimates that a total of required number of samples is 1,9102 Since we we have already generated 2,000 samples, we do not need to perform any additional runs.

We can now calculate the posterior confidence regions for each of the parameters.

    summary(mcmc.10)
    #>
    #> Iterations = 1:200
    #> Thinning interval = 1
    #> Number of chains = 10
    #> Sample size per chain = 200
    #>
    #> 1. Empirical mean and standard deviation for each variable,
    #>    plus standard error of the mean:
    #>
    #>         Mean       SD  Naive SE Time-series SE
    #> alpha  1.026  0.02033 0.0004546       0.000787
    #> beta  53.012 11.10534 0.2483230       0.433342
    #> gamma  4.024  1.24685 0.0278804       0.046729
    #> nu    54.391 11.50530 0.2572663       0.438652
    #>
    #> 2. Quantiles for each variable:
    #>
    #>          2.5%    25%    50%    75% 97.5%
    #> alpha  0.9892  1.012  1.025  1.039  1.07
    #> beta  31.1137 45.587 52.633 60.664 73.74
    #> gamma  1.7500  3.000  4.000  4.750  6.75
    #> nu    32.3646 46.484 54.455 61.896 76.44

Warnes, GR. 2001. “The Normal Kernel Coupler: An Adaptive MCMC Method for Efficiently Sampling from Multi-Modal Distributions.” Technical Report 395, Dept. of Statistics, University of Washington. https://stat.uw.edu/sites/default/files/files/reports/2001/tr395.pdf.
Warnes, Gregory R. 2000. “The Normal Kernel Coupler: An Adaptive MCMC Method for Efficiently Sampling from Multi-Modal Distributions.” PhD thesis. https://digital.lib.washington.edu/researchworks/handle/1773/9541.

1. MCMC Simulations with the same data, prior, and posterior model.↩︎

2. This is slightly fewer than before because the the larger number of samples allowed more accurate estimates of the variances and correlations.↩︎