LaplacesDemon: Laplace's Demon

In benmarwick/LaplacesDemon: Complete Environment for Bayesian Inference

Description

The LaplacesDemon function is the main function of Laplace's Demon. Given data, a model specification, and initial values, LaplacesDemon maximizes the logarithm of the unnormalized joint posterior density with MCMC and provides samples of the marginal posterior distributions, deviance, and other monitored variables.

The LaplacesDemon.hpc function extends LaplacesDemon to parallel chains for multicore or cluster high performance computing.

Usage

LaplacesDemon(Model, Data, Initial.Values, Covar=NULL, Iterations=100000,
     Status=1000, Thinning=100, Algorithm="RWM", Specs=NULL, LogFile="",
     ...)
LaplacesDemon.hpc(Model, Data, Initial.Values, Covar=NULL,
     Iterations=100000, Status=1000, Thinning=100, Algorithm="RWM",
     Specs=NULL, LogFile="", Chains=2, CPUs=2, Type="PSOCK", Packages=NULL,
     Dyn.libs=NULL)

Arguments

Model	This required argument receives the model from a user-defined function that must be named Model. The user-defined function is where the model is specified. LaplacesDemon passes two arguments to the model function, parms and Data, and receives five arguments from the model function: LP (the logarithm of the unnormalized joint posterior), Dev (the deviance), Monitor (the monitored variables), yhat (the variables for posterior predictive checks), and parm, the vector of parameters, which may be constrained in the model function. More information on the Model specification function may be found in the "LaplacesDemon Tutorial" vignette, and the is.model function. Many examples of model specification functions may be found in the "Examples" vignette.
Data	This required argument accepts a list of data. The list of data must contain mon.names which contains monitored variable names, and must contain parm.names which contains parameter names. The as.parm.names function may be helpful for preparing the data, and the is.data function may be helpful for checking data.
Initial.Values	For LaplacesDemon, this argument requires a vector of initial values equal in length to the number of parameters. For LaplacesDemon.hpc, this argument also accepts a vector, in which case the same initial values will be applied to all parallel chains, or the argument accepts a matrix in which each row is a parallel chain and the number of columns is equal in length to the number of parameters. When a matrix is supplied for LaplacesDemon.hpc, each parallel chain begins with its own initial values that are preferably dispersed. For both LaplacesDemon and LaplacesDemon.hpc, each initial value will be the starting point for an adaptive chain or a non-adaptive Markov chain of a parameter. Parameters are assumed to be continuous, unless specified to be discrete (see dparm below), which is not accepted by all algorithms. If all initial values are set to zero, then Laplace's Demon will attempt to optimize the initial values with the LaplaceApproximation function. After Laplace's Demon finishes updating, it may be desired to continue updating from where it left off. To continue, this argument should receive the last iteration of the previous update. For example, if the output object is called Fit, then Initial.Values=as.initial.values(Fit). Initial values may be generated randomly with the GIV function.
Covar	This argument defaults to NULL, but may otherwise accept a K x K proposal covariance matrix (where K is the number of dimensions or parameters), a variance vector, or a list of covariance matrices (for blockwise sampling in some algorithms). When the model is updated for the first time and prior variance or covariance is unknown, then Covar=NULL should be used. Some algorithms require covariance, some only require variance, and some require neither. Laplace's Demon automatically converts the user input to the required form. Once Laplace's Demon has finished updating, it may be desired to continue updating where it left off, in which case the proposal covariance matrix from the last run can be input into the next run. The covariance matrix may also be input from the LaplaceApproximation function, if used.
Iterations	This required argument accepts integers larger than 10, and determines the number of iterations that Laplace's Demon will update the parameters while searching for target distributions. The required amount of computer memory will increase with Iterations. If computer memory is exceeded, then all will be lost. The Combine function can be used later to combine multiple updates.
Status	This argument accepts integers between 1 and the number of iterations, and indicates how often the user would like the status of the number of iterations and proposal type (for example, multivariate or componentwise, or mixture, or subset) printed to the screen. For example, if a model is updated for 1,000 iterations and Status=200, then a status message will be printed at the following iterations: 200, 400, 600, and 800 in LaplacesDemon. The LaplacesDemon.hpc function does not print the status during parallel processing.
Thinning	This argument accepts integers between 1 and the number of iterations, and indicates that every nth iteration will be retained, while the other iterations are discarded. If Thinning=5, then every 5th iteration will be retained. Thinning is performed to reduce autocorrelation and the number of marginal posterior samples.
Algorithm	This argument accepts the abbreviated name of the MCMC algorithm, which must appear in quotes. A list of MCMC algorithms appears below in the Details section, and the abbreviated name is in parenthesis.
Specs	This argument defaults to NULL, and accepts a list of specifications for the MCMC algorithm declared in the Algorithm argument. The specifications associated with each algorithm may be seen below in the examples, must appear in the order shown, and are described in the details section below.
LogFile	This argument is used to specify a log file name in quotes in the working directory as a destination, rather than the console, for the output messages of cat and stop commands. It is helpful to assign a log file name when using multiple cores, such as with LaplacesDemon.hpc. Doing so allows the user to check the progress in the log. A number of log files are created, one for each chain, and one for the overall process.
Chains	This argument is required only for LaplacesDemon.hpc, and indicates the number of parallel chains.
CPUs	This argument is required for parallel independent or interactive chains in LaplacesDemon or LaplacesDemon.hpc, and indicates the number of central processing units (CPUs) of the computer or cluster. For example, when a user has a quad-core computer, CPUs=4.
Type	This argument defaults to "PSOCK" and uses the Simple Network of Workstations (SNOW) for parallelization. Alternatively, Type="MPI" may be specified to use Message Passing Interface (MPI) for parallelization.
Packages	This optional argument is for use with parallel independent or interacting chains, and defaults to NULL. This argument accepts a vector of package names to load into each parallel chain. If the Model specification depends on any packages, then these package names need to be in this vector.
Dyn.libs	This optional argument is for use with parallel independent or interacting chain, and defaults to NULL. This argument accepts a vector of the names of dynamic link libraries (shared objects) to load into each parallel chain. The libraries must be located in the working directory.
...	Additional arguments are unused.

Details

LaplacesDemon offers numerous MCMC algorithms for numerical approximation in Bayesian inference. The algorithms are

• Adaptive Directional Metropolis-within-Gibbs (ADMG)

• Adaptive Griddy-Gibbs (AGG)

• Adaptive Hamiltonian Monte Carlo (AHMC)

• Adaptive Metropolis (AM)

• Adaptive Metropolis-within-Gibbs (AMWG)

• Adaptive-Mixture Metropolis (AMM)

• Affine-Invariant Ensemble Sampler (AIEC)

• Componentwise Hit-And-Run Metropolis (CHARM)

• Delayed Rejection Adaptive Metropolis (DRAM)

• Delayed Rejection Metropolis (DRM)

• Differential Evolution Markov Chain (DEMC)

• Elliptical Slice Sampling (ESS)

• Griddy-Gibbs (GG)

• Hamiltonian Monte Carlo (HMC)

• Hamiltonian Monte Carlo with Dual-Averaging (HMCDA)

• Hit-And-Run Metropolis (HARM)

• Independence Metropolis (IM)

• Interchain Adaptation (INCA)

• Metropolis-Adjusted Langevin Algorithm (MALA)

• Metropolis-within-Gibbs (MWG)

• No-U-Turn Sampler (NUTS)

• Random-Walk Metropolis (RWM)

• Reversible-Jump (RJ)

• Robust Adaptive Metropolis (RAM)

• Sequential Adaptive Metropolis-within-Gibbs (SAMWG)

• Sequential Metropolis-within-Gibbs (SMWG)

• Slice Sampler (Slice)

• Stochastic Gradient Langevin Dynamics (SGLD)

• Tempered Hamiltonian Monte Carlo (THMC)

• t-walk (twalk)

• Updating Sequential Adaptive Metropolis-within-Gibbs (USAMWG)

• Updating Sequential Metropolis-within-Gibbs (USMWG)

It is a goal for the documentation in the LaplacesDemon to be extensive. However, details of MCMC algorithms are best explored online at http://www.bayesian-inference.com/mcmc, as well as in the "LaplacesDemon Tutorial" vignette, and the "Bayesian Inference" vignette. Algorithm specifications (Specs) are listed below:

• A is the number of initial, adaptive iterations to be discarded as burn-in, and is used in HMCDA and NUTS.

• Adaptive is the iteration in which adaptation begins, and is used in AM, AMM, DRAM, and INCA. These algorithms adapt according to an observed covariance matrix, and should sample before beginning to adapt.

• alpha.star is the desired acceptance rate(s) in RAM, and is optional in CHARM and HARM. It is a scalar in CHARM or HARM, and in RAM it may be a scalar or a vector equal in length to the number of targets. The recommended value for multivariate proposals is alpha.star=0.234, or for componentwise proposals it is alpha.star=0.44.

• at affects the traverse move in twalk. at=6 is recommended. It helps when some parameters are highly correlated, and the correlation structure may change through the state-space. The traverse move is associated with an acceptance rate that decreases as the number of parameters increases, and is the reason that n1 is used to select a subset of parameters each iteration. If adjusted, it is recommended to stay in the interval [2,10].

• aw affects the walk move in twalk, and aw=1.5 is recommended. If adjusted, it is recommended to stay in the interval [0.3,2].

• beta is a scale parameter for AIES, and defaults to 2.

• bin.n is the scalar size parameter for a binomial prior distribution of model size for the RJ algorithm.

• bin.p is the scalar probability parameter for a binomial prior distribution of model size for the RJ algorithm.

• B is a list of blocked parameters. Each component of the list represents a block of parameters, and contains a vector in which each element is the position of the associated parameter in parm.names. This function is optional in the AMM, ESS, HARM, and RWM algorithms. For more information on blockwise sampling, see the Blocks function.

• Begin indicates the time-period in which to begin updating (filtering or predicting) in the USAMWG and USMWG algorithms.

• delta is the target acceptance rate in HMCDA and NUTS. The recommended value is 0.65 in HMCDA and 0.6 in NUTS.

• Dist is the proposal distribution in RAM, and may either be Dist="t" for t-distributed or Dist="N" for normally-distributed.

• dparm accepts a vector of integers that indicate discrete parameters. This argument is for use with the AGG or GG algorithm.

• Dyn is a T x K matrix of dynamic parameters, where T is the number of time-periods and K is the number of dynamic parameters. Dyn is used by SAMWG, SMWG, USAMWG, and USMWG. Non-dynamic parameters are updated first in each sampler iteration, then dynamic parameters are updated in a random order in each time-period, and sequentially by time-period.

• epsilon is the step-size in AHMC, HMC, HMCDA, NUTS, SGLD, and THMC. It is a vector equal in length to the number of parameters in AHMC, HMC, and THMC. It is a scalar in HMCDA and NUTS. It is either a scalar or a vector equal in length to the number of iterations in SGLD. When epsilon=NULL in HMCDA or NUTS (only), a reasonable initial value is found.

• file is the quoted name of a numeric matrix of data, without headers, for SGLD. The big data set must be a .csv file. This matrix has Nr rows and Nc columns. Each iteration, SGLD will randomly select a block of rows, where the number of rows is specified by the size argument.

• Fit is an object of class demonoid in the USAMWG and USMWG algorithms. Posterior samples before the time-period specified in the Begin argument are not updated, and are used instead from Fit.

• gamma controls the step size in DEMC or the decay of adaptation in RAM. In DEMC, it is positive and defaults to 2.38/sqrt(2J) when NULL, where J is the length of initial values. For RAM, it is in the interval (0.5,1], and 0.66 is recommended.

• Grid accepts either a vector or a list of vectors of evenly-spaced points on a grid for the AGG or GG algorithm. When the argument is a vector, the same grid is applied to all parameters. When the argument is a list, each component in the list has a grid that is applied to the corresponding parameter. The algorithm will evaluate each continuous parameter at the latest value plus each point in the grid, or each discrete parameter (see dparm) at each grid point (which should be each discrete value).

• L is a scalar number of leapfrog steps in AHMC, HMC, and THMC. When L=1, the algorithm reduces to Langevin Monte Carlo (LMC).

• lambda is a scalar trajectory length in HMCDA.

• Lmax is a scalar maximum for L (see above) in HMCDA.

• m is a scalar (or vector equal in length to the number of initial values) integer in [1,Inf] in which each element indicates the maximum number of steps for creating the slice interval. It is used by the Slice algorithm, and defaults to infinity.

• mu is a vector that is equal in length to the initial values. This vector will be used as the mean of the proposal distribution, and is usually the posterior mode of a previously-updated LaplaceApproximation.

• Nc is either the number of (un-parallelized) parallel chains in DEMC (and must be at least 3) or the number of columns of big data in SGLD.

• Nr is the number of rows of big data in SGLD.

• n1 affects the size of the subset of each set of points to adjust, and is used in twalk. It relates to the number of parameters, and n1=4 is recommended. If adjusted, it is recommended to stay in the interval [2,20].

• parm.p is a vector of probabilities for parameter selection in the RJ algorithm, and must be equal in length to the number of initial values.

• Periodicity specifies how often in iterations the adaptive algorithm should adapt, and is used by AHMC, AM, AMM, AMWG, DRAM, INCA, RAM, SAMWG, and USAMWG. If Periodicity=10, then the algorithm adapts every 10th iteration. A higher Periodicity is associated with an algorithm that runs faster, because it does not have to calculate adaptation as often, though the algorithm adapts less often to the target distributions, so it is a trade-off. It is recommended to use the lowest value that runs fast enough to suit the user, or provide sufficient adaptation.

• selectable is a vector of indicators of whether or not a parameter is selectable for variable selection in the RJ algorithm. Non-selectable parameters are assigned a zero, and are always in the model. Selectable parameters are assigned a one. This vector must be equal in length to the number of initial values.

• selected is a vector of indicators of whether or not each parameter is selected when the RJ algorithm begins, and must be equal in length to the number of initial values.

• SIV stands for secondary initial values and is used by twalk. SIV must be the same length as Initial.Values, and each element of these two vectors must be unique from each other, both before and after being passed to the Model function. SIV defaults to NULL, in which case values are generated with GIV.

• size is the number of rows of big data to be read into SGLD each iteration.

• smax is the maximum allowable tuning parameter sigma, the standard deviation of the conditional distribution, in the AGG algorithm.

• Temperature is used in the THMC algorithm to heat up the momentum in the first half of the leapfrog steps, and then cool down the momentum in the last half. Temperature must be positive. When greater than 1, THMC should explore more diffuse distributions, and may be helpful with multimodal distributions.

• w is used in AMM, DEMC, and Slice. It is a mixture weight for both the AMM and DEMC algorithms, and in these algorithms it is in the interval (0,1]. For AMM, it is recommended to use w=0.05, as per Roberts and Rosenthal (2009). The two mixture components in AMM are adaptive multivariate and static/symmetric univariate proposals. The mixture is determined at each iteration with mixture weight w. In the AMM algorithm, a higher value of w is associated with more static/symmetric univariate proposals, and a lower w is associated with more adaptive multivariate proposals. AMM will be unable to include the multivariate mixture component until it has accumulated some history, and models with more parameters will take longer to be able to use adaptive multivariate proposals. In DEMC, it indicates the probability that each iteration uses a snooker update, rather than a projection update, and the recommended default is w=0.1. In the Slice algorithm, this argument may be a scalar or a vector equal in length the number of initial values, and each element indicates the step size used for creating the slice interval.

• Z accepts a T x J matrix or T x J x Nc array of thinned samples for T thinned iterations, J parameters, and Nc chains for DEMC. Z defaults to NULL. The matrix of thinned posterior samples from a previous run may be used, in which case the samples are copied across the chains.

Value

LaplacesDemon returns an object of class demonoid, and LaplacesDemon.hpc returns an object of class demonoid.hpc that is a list of objects of class demonoid, where the number of components in the list is the number of parallel chains. Each object of class demonoid is a list with the following components:

Acceptance.Rate	This is the acceptance rate of the MCMC algorithm, indicating the percentage of iterations in which the proposals were accepted. For more information on acceptance rates, see the AcceptanceRate function.
Algorithm	This reports the specific algorithm used.
Call	This is the matched call of LaplacesDemon.
Covar	This stores the K x K proposal covariance matrix (where K is the dimension or number of parameters), variance vector, or list of covariance matrices. If variance or covariance is used for adaptation, then this covariance is returned. Otherwise, the variance of the samples of each parameter is returned. If the model is updated in the future, then this vector, matrix, or list can be used to start the next update where the last update left off. Only the diagonal of this matrix is reported in the associated print function.
CovarDHis	This N x K matrix stores the diagonal of the proposal covariance matrix of each adaptation in each of N rows for K dimensions, where the dimension is the number of parameters or length of the initial values vector. The proposal covariance matrix should change less over time. An exception is that the AHMC algorithm stores an algorithm specification here, which is not the diagonal of the proposal covariance matrix.
Deviance	This is a vector of the deviance of the model, with a length equal to the number of thinned samples that were retained. Deviance is useful for considering model fit, and is equal to the sum of the log-likelihood for all rows in the data set, which is then multiplied by negative two.
DIC1	This is a vector of three values: Dbar, pD, and DIC. Dbar is the mean deviance, pD is a measure of model complexity indicating the effective number of parameters, and DIC is the Deviance Information Criterion, which is a model fit statistic that is the sum of Dbar and pD. DIC1 is calculated over all retained samples. Note that pD is calculated as var(Deviance)/2 as in Gelman et al. (2004).
DIC2	This is identical to DIC1 above, except that it is calculated over only the samples that were considered by the BMK.Diagnostic to be stationary for all parameters. If stationarity (or a lack of trend) is not estimated for all parameters, then DIC2 is set to missing values.
Initial.Values	This is the vector of Initial.Values, which may have been optimized with the IterativeQuadrature or LaplaceApproximation function.
Iterations	This reports the number of Iterations for updating.
LML	This is an approximation of the logarithm of the marginal likelihood of the data (see the LML function for more information). LML is estimated only with stationary samples, and only with a non-adaptive algorithm, including Componentwise Hit-And-Run (CHARM), Delayed Rejection Metropolis (DRM), Hamiltonian Monte Carlo (HMC), Hit-And-Run (HARM), Metropolis-within-Gibbs (MWG), No-U-Turn Sampler (NUTS), Random-Walk Metropolis (RWM), Slice Sampling (Slice), Stochastic Gradient Langevin Dynamics (SGLD), Tempered Hamiltonian Monte Carlo (THMC), or t-walk (twalk). LML is estimated with nonparametric self-normalized importance sampling (NSIS), given LL and the marginal posterior samples of the parameters. LML is useful for comparing multiple models with the BayesFactor function.
Minutes	This indicates the number of minutes that LaplacesDemon was running, and includes the initial checks as well as time it took the LaplaceApproximation function, assessing stationarity, effective sample size (ESS), and creating summaries.
Model	This contains the model specification Model.
Monitor	This is a vector or matrix of one or more monitored variables, which are variables that were specified in the Model function to be observed as chains (or Markov chains, if Adaptive=0), but that were not deviance or parameters.
Parameters	This reports the number of parameters.
Posterior1	This is a matrix of marginal posterior distributions composed of thinned samples, with a number of rows equal to the number of thinned samples and a number of columns equal to the number of parameters. This matrix includes all thinned samples.
Posterior2	This is a matrix equal to Posterior1, except that rows are included only if stationarity (a lack of trend) is indicated by the BMK.Diagnostic for all parameters. If stationarity did not occur, then this matrix is missing.
Rec.BurnIn.Thinned	This is the recommended burn-in for the thinned samples, where the value indicates the first row that was stationary across all parameters, and previous rows are discarded as burn-in. Samples considered as burn-in are discarded because they do not represent the target distribution and have not adequately forgotten the initial value of the chain (or Markov chain, if Adaptive=0).
Rec.BurnIn.UnThinned	This is the recommended burn-in for all samples, in case thinning will not be necessary.
Rec.Thinning	This is the recommended value for the Thinning argument according to the autocorrelation in the thinned samples, and it is limited to the interval [1,1000].
Specs	This is an optional list of algorithm specifications.
Status	This is the value in the Status argument.
Summary1	This is a matrix that summarizes the marginal posterior distributions of the parameters, deviance, and monitored variables over all samples in Posterior1. The following summary statistics are included: mean, standard deviation, MCSE (Monte Carlo Standard Error), ESS is the effective sample size due to autocorrelation, and finally the 2.5%, 50%, and 97.5% quantiles are reported. MCSE is essentially a standard deviation around the marginal posterior mean that is due to uncertainty associated with using MCMC. The acceptable size of the MCSE depends on the acceptable uncertainty associated around the marginal posterior mean. Laplace's Demon prefers to continue updating until each MCSE is less than 6.27% of each marginal posterior standard deviation (see the MCSE and Consort functions). The default IMPS method is used. Next, the desired precision of ESS depends on the user's goal, and Laplace's Demon prefers to continue until each ESS is at least 100, which should be enough to describe 95% boundaries of an approximately Gaussian distribution (see the ESS for more information).
Summary2	This matrix is identical to the matrix in Summary1, except that it is calculated only on the stationary samples found in Posterior2. If universal stationarity was not estimated for the parameters, then this matrix is set to missing values.
Thinned.Samples	This is the number of thinned samples that were retained.
Thinning	This is the value of the Thinning argument.

Author(s)

Statisticat, LLC [e-mail], Silvere Vialet-Chabrand [e-mail]

References

Atchade, Y.F. (2006). "An Adaptive Version for the Metropolis Adjusted Langevin Algorithm with a Truncated Drift". Methodology and Computing in Applied Probability, 8, p. 235–254.

Bai, Y. (2009). "An Adaptive Directional Metropolis-within-Gibbs Algorithm". Technical Report in Department of Statistics at the University of Toronto.

Craiu, R.V., Rosenthal, J., and Yang, C. (2009). "Learn From Thy Neighbor: Parallel-Chain and Regional Adaptive MCMC". Journal of the American Statistical Assocation, 104(488), p. 1454–1466.

Christen, J.A. and Fox, C. (2010). "A General Purpose Sampling Algorithm for Continuous Distributions (the t-walk)". Bayesian Analysis, 5(2), p. 263–282.

Duane, S., Kennedy, A.D., Pendleton, B.J., and Roweth, D. (1987). "Hybrid Monte Carlo". Physics Letters, B, 195, p. 216–222.

Gelman, A., Carlin, J., Stern, H., and Rubin, D. (2004). "Bayesian Data Analysis, Texts in Statistical Science, 2nd ed.". Chapman and Hall, London.

Goodman J, and Weare, J. (2010). "Ensemble Samplers with Affine Invariance". Communications in Applied Mathematics and Computational Science, 5(1), p. 65–80.

Green, P.J. (1995). "Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination". Biometrika, 82, p. 711–732.

Haario, H., Laine, M., Mira, A., and Saksman, E. (2006). "DRAM: Efficient Adaptive MCMC". Statistical Computing, 16, p. 339–354.

Haario, H., Saksman, E., and Tamminen, J. (2001). "An Adaptive Metropolis Algorithm". Bernoulli, 7, p. 223–242.

Hoffman, M.D. and Gelman. A. (2012). "The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo". Journal of Machine Learning Research, p. 1–30.

Kass, R.E. and Raftery, A.E. (1995). "Bayes Factors". Journal of the American Statistical Association, 90(430), p. 773–795.

Lewis, S.M. and Raftery, A.E. (1997). "Estimating Bayes Factors via Posterior Simulation with the Laplace-Metropolis Estimator". Journal of the American Statistical Association, 92, p. 648–655.

Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., and Teller, E. (1953). "Equation of State Calculations by Fast Computing Machines". Journal of Chemical Physics, 21, p. 1087–1092.

Mira, A. (2001). "On Metropolis-Hastings Algorithms with Delayed Rejection". Metron, Vol. LIX, n. 3-4, p. 231–241.

Murray, I., Adams, R.P., and MacKay, D.J. (2010). "Elliptical Slice Sampling". Journal of Machine Learning Research, 9, p. 541–548.

Neal, R.M. (2003). "Slice Sampling" (with discussion). Annals of Statistics, 31(3), p. 705–767.

Ritter, C. and Tanner, M. (1992), "Facilitating the Gibbs Sampler: the Gibbs Stopper and the Griddy-Gibbs Sampler", Journal of the American Statistical Association, 87, p. 861–868.

Roberts, G.O. and Rosenthal, J.S. (2009). "Examples of Adaptive MCMC". Computational Statistics and Data Analysis, 18, p. 349–367.

Roberts, G.O. and Tweedie, R.L. (1996). "Exponential Convergence of Langevin Distributions and Their Discrete Approximations". Bernoulli, 2(4), p. 341–363.

Rosenthal, J.S. (2007). "AMCMC: An R interface for adaptive MCMC". Computational Statistics and Data Analysis, 51, p. 5467–5470.

Smith, R.L. (1984). "Efficient Monte Carlo Procedures for Generating Points Uniformly Distributed Over Bounded Region". Operations Research, 32, p. 1296–1308.

Ter Braak, C.J.F. and Vrugt, J.A. (2008). "Differential Evolution Markov Chain with Snooker Updater and Fewer Chains", Statistics and Computing, 18(4), p. 435–446.

Vihola, M. (2011). "Robust Adaptive Metropolis Algorithm with Coerced Acceptance Rate". Statistics and Computing. Springer, Netherlands.

Welling, M. and Teh, Y.W. (2011). "Bayesian Learning via Stochastic Gradient Langevin Dynamics". Proceedings of the 28th International Conference on Machine Learning (ICML), p. 681–688.

See Also

AcceptanceRate, as.initial.values, as.parm.names, BayesFactor, Blocks, BMK.Diagnostic, Combine, Consort, ESS, GIV, is.data, is.model, IterativeQuadrature, LaplaceApproximation, LaplacesDemon.RAM, LML, and MCSE.

Examples

# The accompanying Examples vignette is a compendium of examples.
####################  Load the LaplacesDemon Library  #####################
library(LaplacesDemon)

##############################  Demon Data  ###############################
data(demonsnacks)
y <- log(demonsnacks$Calories)
X <- cbind(1, as.matrix(log(demonsnacks[,c(1,4,10)]+1)))
J <- ncol(X)
for (j in 2:J) X[,j] <- CenterScale(X[,j])
mon.names <- "LP"
parm.names <- as.parm.names(list(beta=rep(0,J), sigma=0))
pos.beta <- grep("beta", parm.names)
pos.sigma <- grep("sigma", parm.names)
PGF <- function(Data) return(c(rnormv(Data$J,0,10), rhalfcauchy(1,5)))
MyData <- list(J=J, PGF=PGF, X=X, mon.names=mon.names,
     parm.names=parm.names, pos.beta=pos.beta, pos.sigma=pos.sigma, y=y)

##########################  Model Specification  ##########################
Model <- function(parm, Data)
     {
     ### Parameters
     beta <- parm[Data$pos.beta]
     sigma <- interval(parm[Data$pos.sigma], 1e-100, Inf)
     parm[Data$pos.sigma] <- sigma
     ### Log of Prior Densities
     beta.prior <- sum(dnormv(beta, 0, 1000, log=TRUE))
     sigma.prior <- dhalfcauchy(sigma, 25, log=TRUE)
     ### Log-Likelihood
     mu <- tcrossprod(Data$X, t(beta))
     LL <- sum(dnorm(Data$y, mu, sigma, log=TRUE))
     ### Log-Posterior
     LP <- LL + beta.prior + sigma.prior
     Modelout <- list(LP=LP, Dev=-2*LL, Monitor=LP,
          yhat=rnorm(length(mu), mu, sigma), parm=parm)
     return(Modelout)
     }

set.seed(666)

############################  Initial Values  #############################
Initial.Values <- GIV(Model, MyData, PGF=TRUE)

###########################################################################
# Examples of MCMC Algorithms                                             #
###########################################################################

########################  Hit-And-Run Metropolis  #########################
Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
     Algorithm="HARM", Specs=NULL)
Fit
print(Fit)
#Consort(Fit)
#plot(BMK.Diagnostic(Fit))
#PosteriorChecks(Fit)
#caterpillar.plot(Fit, Parms="beta")
#BurnIn <- Fit$Rec.BurnIn.Thinned
#plot(Fit, BurnIn, MyData, PDF=FALSE)
#Pred <- predict(Fit, Model, MyData, CPUs=1)
#summary(Pred, Discrep="Chi-Square")
#plot(Pred, Style="Covariates", Data=MyData)
#plot(Pred, Style="Density", Rows=1:9)
#plot(Pred, Style="ECDF")
#plot(Pred, Style="Fitted")
#plot(Pred, Style="Jarque-Bera")
#plot(Pred, Style="Predictive Quantiles")
#plot(Pred, Style="Residual Density")
#plot(Pred, Style="Residuals")
#Levene.Test(Pred)
#Importance(Fit, Model, MyData, Discrep="Chi-Square")

#############  Adaptive Directional Metropolis-within-Gibbs  ##############
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="ADMG", Specs=list(Periodicity=1))

########################  Adaptive Griddy-Gibbs  ##########################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="AGG", Specs=list(Grid=GaussHermiteQuadRule(3)$nodes,
#     dparm=NULL, smax=Inf, CPUs=1, Packages=NULL, Dyn.libs=NULL))

##################  Adaptive Hamiltonian Monte Carlo  #####################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="AHMC", Specs=list(epsilon=rep(0.02, length(Initial.Values)),
#     L=2, Periodicity=10))

##########################  Adaptive Metropolis  ##########################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="AM", Specs=list(Adaptive=500, Periodicity=10))

###################  Adaptive Metropolis-within-Gibbs  ####################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="AMWG", Specs=list(Periodicity=50))

######################  Adaptive-Mixture Metropolis  ######################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="AMM", Specs=list(Adaptive=500, B=NULL, Periodicity=10,
#     w=0.05))

###################  Affine-Invariant Ensemble Sampler  ###################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="AIES", Specs=list(Nc=2*length(Initial.Values), Z=NULL,
#     beta=2, CPUs=1, Packages=NULL, Dyn.libs=NULL))

#################  Componentwise Hit-And-Run Metropolis  ##################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="CHARM", Specs=NULL)

###########  Componentwise Hit-And-Run (Adaptive) Metropolis  #############
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="CHARM", Specs=list(alpha.star=0.44))

#################  Delayed Rejection Adaptive Metropolis  #################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="DRAM", Specs=list(Adaptive=500, Periodicity=10))

#####################  Delayed Rejection Metropolis  ######################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="DRM", Specs=NULL)

##################  Differential Evolution Markov Chain  ##################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="DEMC", Specs=list(Nc=3, Z=NULL, gamma=NULL, w=0.1))

#######################  Elliptical Slice Sampling  #######################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="ESS", Specs=list(B=NULL))

#############################  Griddy-Gibbs  ##############################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="GG", Specs=list(Grid=seq(from=-0.1, to=0.1, len=5),
#     dparm=NULL, CPUs=1, Packages=NULL, Dyn.libs=NULL))

#######################  Hamiltonian Monte Carlo  #########################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="HMC", Specs=list(epsilon=rep(0.02, length(Initial.Values)),
#     L=2))

#############  Hamiltonian Monte Carlo with Dual-Averaging  ###############
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="HMCDA", Specs=list(A=500, delta=0.65, epsilon=NULL,
#     Lmax=1000, lambda=0.1))

##################  Hit-And-Run (Adaptive) Metropolis  ####################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="HARM", Specs=list(alpha.star=0.234, B=NULL))

########################  Independence Metropolis  ########################
### Note: the mu and Covar arguments are populated from a previous Laplace
### Approximation.
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=Fit$Covar, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="IM",
#     Specs=list(mu=Fit$Summary1[1:length(Initial.Values),1]))

#########################  Interchain Adaptation  #########################
#Initial.Values <- rbind(Initial.Values, GIV(Model, MyData, PGF=TRUE))
#Fit <- LaplacesDemon.hpc(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="INCA", Specs=list(Adaptive=500, Periodicity=10),
#     LogFile="MyLog", Chains=2, CPUs=2, Type="PSOCK", Packages=NULL,
#     Dyn.libs=NULL)

################  Metropolis-Adjusted Langevin Algorithm  #################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="MALA", Specs=list(Periodicity=1))

#######################  Metropolis-within-Gibbs  #########################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="MWG", Specs=NULL)

##########################  No-U-Turn Sampler  ############################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=100, Status=10, Thinning=1,
#     Algorithm="NUTS", Specs=list(A=50, delta=0.6, epsilon=NULL))

######################  Robust Adaptive Metropolis  #######################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="RAM", Specs=list(alpha.star=0.234, Dist="N", gamma=0.66,
#     Periodicity=1))

###########################  Reversible-Jump  #############################
#bin.n <- J-1
#bin.p <- 0.2
#parm.p <- c(1, rep(1/(J-1),(J-1)), 1)
#selectable <- c(0, rep(1,J-1), 0)
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="RJ", Specs=list(bin.n=bin.n, bin.p=bin.p,
#          parm.p=parm.p, selectable=selectable,
#          selected=c(0,rep(1,J-1),0)))

########################  Random-Walk Metropolis  #########################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="RWM", Specs=NULL)

##############  Sequential Adaptive Metropolis-within-Gibbs  ##############
#NOTE: The SAMWG algorithm is only for state-space models (SSMs)
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="SAMWG", Specs=list(Dyn=Dyn, Periodicity=50))

##################  Sequential Metropolis-within-Gibbs  ###################
#NOTE: The SMWG algorithm is only for state-space models (SSMs)
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="SMWG", Specs=list(Dyn=Dyn))

#############################  Slice Sampler  #############################
#m <- Inf; w <- 1
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="Slice", Specs=list(m=m, w=w))

#################  Stochastic Gradient Langevin Dynamics  #################
#NOTE: The Data and Model functions must be coded differently for SGLD.
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=10, Thinning=10,
#     Algorithm="SGLD", Specs=list(epsilon=1e-4, file="X.csv", Nr=1e4,
#     Nc=6, size=10))

###################  Tempered Hamiltonian Monte Carlo  ####################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="THMC", Specs=list(epsilon=rep(0.05,length(Initial.Values)),
#     L=2, Temperature=2))

###############################  t-walk  #################################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="twalk", Specs=list(SIV=NULL, n1=4, at=6, aw=1.5))

##########  Updating Sequential Adaptive Metropolis-within-Gibbs  #########
#NOTE: The USAMWG algorithm is only for state-space model updating
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values, 
#     Covar=NULL, Iterations=100000, Status=100, Thinning=100,
#     Algorithm="USAMWG", Specs=list(Dyn=Dyn, Periodicity=50, Fit=Fit,
#     Begin=T.m))

##############  Updating Sequential Metropolis-within-Gibbs  ##############
#NOTE: The USMWG algorithm is only for state-space model updating
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values, 
#     Covar=NULL, Iterations=100000, Status=100, Thinning=100,
#     Algorithm="USMWG", Specs=list(Dyn=Dyn, Fit=Fit, Begin=T.m))

#End

benmarwick/LaplacesDemon documentation built on May 12, 2019, 12:59 p.m.