LaplacesDemon {LaplacesDemon}

Laplace's Demon

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=10000,
     Status=100, Thinning=10, Algorithm="MWG", Specs=list(B=NULL),
     Debug=list(DB.chol=FALSE, DB.eigen=FALSE, DB.MCSE=FALSE,
     DB.Model=TRUE), LogFile="", ...)
LaplacesDemon.hpc(Model, Data, Initial.Values, Covar=NULL,
     Iterations=10000, Status=100, Thinning=10, Algorithm="MWG",
     Specs=list(B=NULL), Debug=list(DB.chol=FALSE, DB.eigen=FALSE,
     DB.MCSE=FALSE, DB.Model=TRUE), 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 (see dcrmrf for an alternative). 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 \times 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 an integer between 1 and the number of iterations, and indicates how often, in iterations, the user would like the status printed to the screen or log file. Usually, the following is reported: the number of iterations, the proposal type (for example, multivariate or componentwise, or mixture, or subset), and LP. 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, 800, and 1,000.

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.

Debug

This argument accepts a list of logical scalars that control whether or not errors or warnings are reported due to a try function or non-finite values. List components include DB.chol regarding chol, DB.eigen regarding eigen, DB.MCSE regarding MCSE, and DB.Model regarding the Model specification function. Errors and warnings should be investigated, but do not necessarily indicate a faulty Model specification function or a bug in the software. For example, a sampler may make a proposal that would result in a matrix that is not positive definite, when it should be. This kind of error or warning is acceptable, provided the sampler handles it correctly by rejecting the proposal, and provided the Model specification function is not causing the issue. Oftentimes, blockwise sampling with carefully chosen blocks will mostly or completely eliminate errors or warnings that occur otherwise in larger, multivariate proposals. Similarly, debugged componentwise algorithms tend to provide more information than multivariate algorithms, since usually the parameter and both its current and proposed values may be reported. If confident in the Model specification function, and errors or warnings are produced frequently that are acceptable, then consider setting DB.Model=FALSE for cleaner output and faster sampling. If the Model specification function is not faulty and there is a bug in LaplacesDemon, then please report it with a bug description and reproducible code on https://github.com/LaplacesDemonR/LaplacesDemon/issues.

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 (AIES)

• Componentwise Hit-And-Run Metropolis (CHARM)

• Delayed Rejection Adaptive Metropolis (DRAM)

• Delayed Rejection Metropolis (DRM)

• Differential Evolution Markov Chain (DEMC)

• Elliptical Slice Sampler (ESS)

• Gibbs Sampler (Gibbs)

• 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-Coupled Markov Chain Monte Carlo (MCMCMC)

• Metropolis-within-Gibbs (MWG)

• Multiple-Try Metropolis (MTM)

• No-U-Turn Sampler (NUTS)

• Oblique Hyperrectangle Slice Sampler (OHSS)

• Preconditioned Crank-Nicolson (pCN)

• Random Dive Metropolis-Hastings (RDMH)

• Random-Walk Metropolis (RWM)

• Reflective Slice Sampler (RSS)

• Refractive Sampler (Refractive)

• 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)

• Univariate Eigenvector Slice Sampler (UESS)

• 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 https://web.archive.org/web/20150206014000/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 used in AFSS, HMCDA, MALA, NUTS, OHSS, and UESS. In MALA, it is the maximum acceptable value of the Euclidean norm of the adaptive parameters mu and sigma, and the Frobenius norm of the covariance matrix. In AFSS, HMCDA, NUTS, OHSS, and UESS, it is the number of initial, adaptive iterations to be discarded as burn-in.

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

• alpha.star is the target acceptance rate in MALA and RAM, and is optional in CHARM and HARM. The recommended value for multivariate proposals is alpha.star=0.234, for componentwise proposals is alpha.star=0.44, and for MALA is alpha.star=0.574.

• 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, or an autoregressive parameter for pCN.

• 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 AFSS, AMM, AMWG, ESS, HARM, MWG, RAM, RWM, Slice, and UESS 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.

• Bounds is used in the Slice algorithm. It is a vector of length two with the lower and upper boundary of the slice. For continuous parameters, it is often set to negative and positive infinity, while for discrete parameters it is set to the minimum and maximum discrete values to be sampled. When blocks are used, this must be supplied as a list with the same number of list components as the number of blocks.

• delta is used in HMCDA, MALA, and NUTS. In HMCDA and NUTS, it is the target acceptance rate, and the recommended value is 0.65 in HMCDA and 0.6 in NUTS. In MALA, it is a constant in the bounded drift function, may be in the interval [1e-10,1000], and 1 is the default.

• 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 \times 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 used in AHMC, HMC, HMCDA, MALA, NUTS, SGLD, and THMC. It is the step-size in all algorithms except MALA. 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. In MALA, it is a vector of length two. The first element is the acceptable minimum of adaptive scale sigma, and the second element is added to the diagonal of the covariance matrix for regularization.

• FC is used in Gibbs and accepts a function that receives two arguments: the vector of all parameters and the list of data (similar to the Model specification function). FC must return the updated vector of all parameters. The user specifies FC to calculate the full conditional distribution of one or more parameters.

• 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 MALA and 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. For MALA, it is in the interval (1,Iterations), and defaults to 1.

• 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).

• K is a scalar number of proposals in MTM.

• 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 used in HMCDA and MCMCMC. In HMCDA, it is a scalar trajectory length. In MCMCMC, it is either a scalar that controls temperature spacing, or a vector of temperature spacings.

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

• m is used in the AFSS, AHMC, HMC, Refractive, RSS, Slice, THMC, and UESS algorithms. In AHMC, HMC, and THMC, it is a J \times J mass matrix for J initial values. In AFSS and UESS, it is a scalar, and is the maximum number of steps for creating the slice interval. In Refractive and RSS, it is a scalar, and is the number of steps to take per iteration. In Slice, it is either a scalar or a list with as many list components as blocks. It must be an integer in [1,Inf], and indicates the maximum number of steps for creating the slice interval.

• 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.

• MWG is used in Gibbs to specify a vector of parameters that are to receive Metropolis-within-Gibbs updates. Each element is an integer that indicates the parameter.

• 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.

• n is the number of previous iterations in ADMG, AFSS, AMM, AMWG, OHSS, RAM, and UESS.

• 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.

• r is a scalar used in the Refractive algorithm to indicate the ratio between r1 and r2.

• Periodicity specifies how often in iterations the adaptive algorithm should adapt, and is used by AHMC, AM, AMM, AMWG, DRAM, INCA, 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.

• Type is used in the Slice algorithm. It is either a scalar or a list with the same number of list components as blocks. This accepts "Continuous" for continuous parameters, "Nominal" for discrete parameters that are unordered, and "Ordinal" for discrete parameters that are ordered.

• w is used in AFSS, AMM, DEMC, Refractive, RSS, 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 Refractive algorithm, w is a scalar step size parameter. In AFSS, RSS, and the Slice algorithms, this is a step size interval for creating the slice interval. In AFSS and RSS, a scalar or vector equal in length the number of initial values is accepted. In Slice, a scalar or a list with a number of list components equal to the number of blocks is accepted.

• Z accepts a T \times J matrix or T \times J \times 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 \times 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 \times 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 Adaptive Griddy-Gibbs (AGG), Affine-Invariant Ensemble Sampler (AIES), Componentwise Hit-And-Run (CHARM), Delayed Rejection Metropolis (DRM), Elliptical Slice Sampling (ESS), Gibbs Sampler (Gibbs), Griddy-Gibbs (GG), Hamiltonian Monte Carlo (HMC), Hit-And-Run Metropolis (HARM), Independence Metropolis (IM), Metropolis-Coupled Markov Chain Monte Carlo (MCMCMC), Metropolis-within-Gibbs (MWG), Multiple-Try Metropolis, No-U-Turn Sampler (NUTS), Random Dive Metropolis-Hastings (RDMH), Random-Walk Metropolis (RWM), Reflective Slice Sampler (RSS), Refractive Sampler (Refractive), Reversible-Jump (RJ), Sequential Metropolis-within-Gibbs (SMWG), Slice Sampler (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., Silvere Vialet-Chabrand [e-mail]

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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.

Liu, J., Liang, F., and Wong, W. (2000). "The Multiple-Try Method and Local Optimization in Metropolis Sampling". Journal of the American Statistical Association, 95, p. 121–134.

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.

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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.

Tibbits, M., Groendyke, C., Haran, M., Liechty, J. (2014). "Automated Factor Slice Sampling". Journal of Computational and Graphical Statistics, 23(2), p. 543–563.

Thompson, M.D. (2011). "Slice Sampling with Multivariate Steps". http://hdl.handle.net/1807/31955

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See Also

AcceptanceRate, as.initial.values, as.parm.names, BayesFactor, Blocks, BMK.Diagnostic, Combine, Consort, dcrmrf, 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])

#########################  Data List Preparation  #########################
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) {
     beta <- rnorm(Data$J)
     sigma <- runif(1)
     return(c(beta, sigma))
     }
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$pos.sigma] <- sigma
     ### Log-Prior
     beta.prior <- sum(dnormv(beta, 0, 1000, log=TRUE))
     sigma.prior <- dhalfcauchy(sigma, 25, log=TRUE)
     ### Log-Likelihood
     mu <- tcrossprod(Data$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)
     }
#library(compiler)
#Model <- cmpfun(Model) #Consider byte-compiling for more speed

set.seed(666)

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

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

####################  Automated Factor Slice Sampler  #####################
Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
     Algorithm="AFSS", Specs=list(A=Inf, B=NULL, m=100, n=0, w=1))
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(n=0, Periodicity=50))

########################  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=0.02, L=2, m=NULL,
#     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(B=NULL, n=0, 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, n=0,
#     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 Sampler  ########################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="ESS", Specs=list(B=NULL))

#############################  Gibbs Sampler  #############################
### NOTE: Unlike the other samplers, Gibbs requires specifying a
### function (FC) that draws from full conditionals.
#FC <- function(parm, Data)
#     {
#     ### Parameters
#     beta <- parm[Data$pos.beta]
#     sigma <- interval(parm[Data$pos.sigma], 1e-100, Inf)
#     sigma2 <- sigma*sigma
#     ### Hyperparameters
#     betamu <- rep(0,length(beta))
#     betaprec <- diag(length(beta))/1000
#     ### Update beta
#     XX <- crossprod(Data$X)
#     Xy <- crossprod(Data$X, Data$y)
#     IR <- backsolve(chol(XX/sigma2 + betaprec), diag(length(beta)))
#     btilde <- crossprod(t(IR)) %*% (Xy/sigma2 + betaprec %*% betamu)
#     beta <- btilde + IR %*% rnorm(length(beta))
#     return(c(beta,sigma))
#     }
##library(compiler)
##FC <- cmpfun(FC) #Consider byte-compiling for more speed
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="Gibbs", Specs=list(FC=FC, MWG=pos.sigma))


#############################  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=0.001, L=2, m=NULL))

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

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

##################  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(A=1e7, alpha.star=0.574, gamma=1,
#          delta=1, epsilon=c(1e-6,1e-7)))

#############  Metropolis-Coupled Markov Chain Monte Carlo  ###############
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="MCMCMC", Specs=list(lambda=1, CPUs=2, Packages=NULL,
#     Dyn.libs=NULL))

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

########################  Multiple-Try Metropolis  ########################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="MTM", Specs=list(K=4, CPUs=1, Packages=NULL, Dyn.libs=NULL))

##########################  No-U-Turn Sampler  ############################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=1, Thinning=1,
#     Algorithm="NUTS", Specs=list(A=500, delta=0.6, epsilon=NULL,
#     Lmax=Inf))

#################  Oblique Hyperrectangle Slice Sampler  ##################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="OHSS", Specs=list(A=Inf, n=0))

#####################  Preconditioned Crank-Nicolson  #####################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="pCN", Specs=list(beta=0.1))

######################  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, B=NULL, Dist="N",
#     gamma=0.66, n=0))

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

##########################  Refractive Sampler  ###########################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="Refractive", Specs=list(Adaptive=1, m=2, w=0.1, r=1.3))

###########################  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)

########################  Reflective Slice Sampler  #######################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="RSS", Specs=list(m=5, w=1e-5))

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

#################  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=0.001, L=2, m=NULL,
#     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))

#################  Univariate Eigenvector Slice Sampler  #################
#Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values,
#     Covar=NULL, Iterations=1000, Status=100, Thinning=1,
#     Algorithm="UESS", Specs=list(A=Inf, B=NULL, m=100, n=0))

##########  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