R: MCMC Sampling Algorithms

sampler_BASE {nimble} | R Documentation

## MCMC Sampling Algorithms

### Description

Details of the MCMC sampling algorithms provided with the NIMBLE MCMC engine

### Usage

    sampler_BASE()

    sampler_posterior_predictive(model, mvSaved, target, control)

    sampler_binary(model, mvSaved, target, control)

    sampler_categorical(model, mvSaved, target, control)

    sampler_RW(model, mvSaved, target, control)

    sampler_RW_block(model, mvSaved, target, control)

    sampler_RW_llFunction(model, mvSaved, target, control)

    sampler_slice(model, mvSaved, target, control)

    sampler_ess(model, mvSaved, target, control)

    sampler_AF_slice(model, mvSaved, target, control)

    sampler_crossLevel(model, mvSaved, target, control)

    sampler_RW_llFunction_block(model, mvSaved, target, control)

    sampler_RW_PF(model, mvSaved, target, control)

    sampler_RW_PF_block(model, mvSaved, target, control)

    sampler_RW_multinomial(model, mvSaved, target, control)

    sampler_RW_dirichlet(model, mvSaved, target, control)

    sampler_RW_wishart(model, mvSaved, target, control)

    sampler_CAR_normal(model, mvSaved, target, control)

    sampler_CAR_proper(model, mvSaved, target, control)

    sampler_RJ_fixed_prior(model, mvSaved, target, control)

    sampler_RJ_indicator(model, mvSaved, target, control)

    sampler_RJ_toggled(model, mvSaved, target, control)

    sampler_CRP_concentration(model, mvSaved, target, control)

    sampler_CRP(model, mvSaved, target, control)

    sampler_CRP_old(model, mvSaved, target, control)

### Arguments

`model` | (uncompiled) model on which the MCMC is to be run
`mvSaved` | `modelValues` object to be used to store MCMC samples
`target` | node(s) on which the sampler will be used
`control` | named list that controls the precise behavior of the sampler, with elements specific to `samplertype`. The default values for control list are specified in the setup code of each sampling algorithm. Descriptions of each sampling algorithm, and the possible customizations for each sampler (using the `control` argument) appear below.

### `sampler_base`

base class for new samplers

When you write a new sampler for use in a NIMBLE MCMC (see User Manual), you must include `contains = sampler_BASE`.

### binary sampler

The binary sampler performs Gibbs sampling for binary-valued (discrete 0/1) nodes. This can only be used for nodes following either a `dbern(p)` or `dbinom(p, size=1)` distribution.

The binary sampler accepts no control list arguments.

### RW sampler

The RW sampler executes adaptive Metropolis-Hastings sampling with a normal proposal distribution (Metropolis, 1953), implementing the adaptation routine given in Shaby and Wells, 2011. This sampler can be applied to any scalar continuous-valued stochastic node, and can optionally sample on a log scale.

The RW sampler accepts the following control list elements:

- log. A logical argument, specifying whether the sampler should operate on the log scale. (default = FALSE)

- reflective. A logical argument, specifying whether the normal proposal distribution should reflect to stay within the range of the target distribution. (default = FALSE)

- adaptive. A logical argument, specifying whether the sampler should adapt the scale (proposal standard deviation) throughout the course of MCMC execution to achieve a theoretically desirable acceptance rate. (default = TRUE)

- adaptInterval. The interval on which to perform adaptation. Every adaptInterval MCMC iterations (prior to thinning), the RW sampler will perform its adaptation procedure. This updates the scale variable, based upon the sampler's achieved acceptance rate over the past adaptInterval iterations. (default = 200)

- adaptFactorExponent. Exponent controling the rate of decay of the scale adaptation factor. See Shaby and Wells, 2011, for details. (default = 0.8)

- scale. The initial value of the normal proposal standard deviation. If adaptive = FALSE, scale will never change. (default = 1)

The RW sampler cannot be used with options log=TRUE and reflective=TRUE, i.e. it cannot do reflective sampling on a log scale.

### RW_block sampler

The RW_block sampler performs a simultaneous update of one or more model nodes, using an adaptive Metropolis-Hastings algorithm with a multivariate normal proposal distribution (Roberts and Sahu, 1997), implementing the adaptation routine given in Shaby and Wells, 2011. This sampler may be applied to any set of continuous-valued model nodes, to any single continuous-valued multivariate model node, or to any combination thereof.

The RW_block sampler accepts the following control list elements:

- adaptive. A logical argument, specifying whether the sampler should adapt the scale (a coefficient for the entire proposal covariance matrix) and propCov (the multivariate normal proposal covariance matrix) throughout the course of MCMC execution. If only the scale should undergo adaptation, this argument should be specified as TRUE. (default = TRUE)

- adaptScaleOnly. A logical argument, specifying whether adaption should be done only for scale (TRUE) or also for provCov (FALSE). This argument is only relevant when adaptive = TRUE. When adaptScaleOnly = FALSE, both scale and propCov undergo adaptation; the sampler tunes the scaling to achieve a theoretically good acceptance rate, and the proposal covariance to mimic that of the empirical samples. When adaptScaleOnly = TRUE, only the proposal scale is adapted. (default = FALSE)

- adaptInterval. The interval on which to perform adaptation. Every adaptInterval MCMC iterations (prior to thinning), the RW_block sampler will perform its adaptation procedure, based on the past adaptInterval iterations. (default = 200)

- adaptFactorExponent. Exponent controling the rate of decay of the scale adaptation factor. See Shaby and Wells, 2011, for details. (default = 0.8)

- scale. The initial value of the scalar multiplier for propCov. If adaptive = FALSE, scale will never change. (default = 1)

- propCov. The initial covariance matrix for the multivariate normal proposal distribution. This element may be equal to the character string 'identity', in which case the identity matrix of the appropriate dimension will be used for the initial proposal covariance matrix. (default = 'identity')

### RW_llFunction sampler

Sometimes it is useful to control the log likelihood calculations used for an MCMC updater instead of simply using the model. For example, one could use a sampler with a log likelihood that analytically (or numerically) integrates over latent model nodes. Or one could use a sampler with a log likelihood that comes from a stochastic approximation such as a particle filter, allowing composition of a particle MCMC (PMCMC) algorithm (Andrieu et al., 2010). The RW_llFunction sampler handles this by using a Metropolis-Hastings algorithm with a normal proposal distribution and a user-provided log-likelihood function. To allow compiled execution, the log-likelihood function must be provided as a specialized instance of a nimbleFunction. The log-likelihood function may use the same model as the MCMC as a setup argument, but if so the state of the model should be unchanged during execution of the function (or you must understand the implications otherwise).

The RW_llFunction sampler accepts the following control list elements:

- adaptive. A logical argument, specifying whether the sampler should adapt the scale (proposal standard deviation) throughout the course of MCMC execution. (default = TRUE)

- adaptInterval. The interval on which to perform adaptation. (default = 200)

- scale. The initial value of the normal proposal standard deviation. (default = 1)

- llFunction. A specialized nimbleFunction that accepts no arguments and returns a scalar double number. The return value must be the total log-likelihood of all stochastic dependents of the target nodes – and, if includesTarget = TRUE, of the target node(s) themselves – or whatever surrogate is being used for the total log-likelihood. This is a required element with no default.

- includesTarget. Logical variable indicating whether the return value of llFunction includes the log-likelihood associated with target. This is a required element with no default.

### slice sampler

The slice sampler performs slice sampling of the scalar node to which it is applied (Neal, 2003). This sampler can operate on either continuous-valued or discrete-valued scalar nodes. The slice sampler performs a 'stepping out' procedure, in which the slice is iteratively expanded to the left or right by an amount sliceWidth. This sampler is optionally adaptive, governed by a control list element, whereby the value of sliceWidth is adapted towards the observed absolute difference between successive samples.

The slice sampler accepts the following control list elements:

- adaptive. A logical argument, specifying whether the sampler will adapt the value of sliceWidth throughout the course of MCMC execution. (default = TRUE)

- adaptInterval. The interval on which to perform adaptation. (default = 200)

- sliceWidth. The initial value of the width of each slice, and also the width of the expansion during the iterative 'stepping out' procedure. (default = 1)

- sliceMaxSteps. The maximum number of expansions which may occur during the 'stepping out' procedure. (default = 100)

- maxContractions. The maximum number of contractions of the interval that may occur during sampling (this prevents infinite looping in unusual situations). (default = 100)

- maxContractionsWarning. A logical argument specifying whether to warn when the maximum number of contractions is reached. (default = TRUE)

### ess sampler

The ess sampler performs elliptical slice sampling of a single node, which must follow a multivariate normal distribution (Murray, 2010). The algorithm is an extension of slice sampling (Neal, 2003), generalized to the multivariate normal context. An auxilliary variable is used to identify points on an ellipse (which passes through the current node value) as candidate samples, which are accepted contingent upon a likelihood evaluation at that point. This algorithm requires no tuning parameters and therefore no period of adaptation, and may result in very efficient sampling from multivariate Gaussian distributions.

The ess sampler accepts the following control list arguments.

- maxContractions. The maximum number of contractions of the interval that may occur during sampling (this prevents infinite looping in unusual situations). (default = 100)

- maxContractionsWarning. A logical argument specifying whether to warn when the maximum number of contractions is reached. (default = TRUE)

### AF_slice sampler

The automated factor slice sampler conducts a slice sampling algorithm on one or more model nodes. The sampler uses the eigenvectors of the posterior covariance between these nodes as an orthogonal basis on which to perform its 'stepping Out' procedure. The sampler is adaptive in updating both the width of the slices and the values of the eigenvectors. The sampler can be applied to any set of continuous or discrete-valued model nodes, to any single continuous or discrete-valued multivariate model node, or to any combination thereof. The automated factor slice sampler accepts the following control list elements:

- sliceWidths. A numeric vector of initial slice widths. The length of the vector must be equal to the sum of the lengths of all nodes being used by the automated factor slice sampler. Defaults to a vector of 1's.

- sliceAdaptFactorMaxIter. The number of iterations for which the factors (eigenvectors) will continue to adapt to the posterior correlation. (default = 15000)

- sliceAdaptFactorInterval. The interval on which to perform factor adaptation. (default = 1000)

- sliceAdaptWidthMaxIter. The maximum number of iterations for which to adapt the widths for a given set of factors. (default = 512)

- sliceAdaptWidthTolerance. The tolerance for when widths no longer need to adapt, between 0 and 0.5. (default = 0.1)

- sliceMaxSteps. The maximum number of expansions which may occur during the 'stepping out' procedure. (default = 100)

- maxContractions. The maximum number of contractions of the interval that may occur during sampling (this prevents infinite looping in unusual situations). (default = 100)

- maxContractionsWarning. A logical argument specifying whether to warn when the maximum number of contractions is reached. (default = TRUE)

### crossLevel sampler

This sampler is constructed to perform simultaneous updates across two levels of stochastic dependence in the model structure. This is possible when all stochastic descendents of node(s) at one level have conjugate relationships with their own stochastic descendents. In this situation, a Metropolis-Hastings algorithm may be used, in which a multivariate normal proposal distribution is used for the higher-level nodes, and the corresponding proposals for the lower-level nodes undergo Gibbs (conjugate) sampling. The joint proposal is either accepted or rejected for all nodes involved based upon the Metropolis-Hastings ratio.

The requirement that all stochastic descendents of the target nodes must themselves have only conjugate descendents will be checked when the MCMC algorithm is built. This sampler is useful when there is strong dependence across the levels of a model that causes problems with convergence or mixing.

The crossLevel sampler accepts the following control list elements:

- adaptive. Logical argument, specifying whether the multivariate normal proposal distribution for the target nodes should be adaptived. (default = TRUE)

- adaptInterval. The interval on which to perform adaptation. (default = 200)

- scale. The initial value of the scalar multiplier for propCov. (default = 1)

- propCov. The initial covariance matrix for the multivariate normal proposal distribution. This element may be equal to the character string 'identity' or any positive definite matrix of the appropriate dimensions. (default = 'identity')

### RW_llFunction_block sampler

Sometimes it is useful to control the log likelihood calculations used for an MCMC updater instead of simply using the model. For example, one could use a sampler with a log likelihood that analytically (or numerically) integrates over latent model nodes. Or one could use a sampler with a log likelihood that comes from a stochastic approximation such as a particle filter, allowing composition of a particle MCMC (PMCMC) algorithm (Andrieu et al., 2010) (but see samplers listed below for NIMBLE's direct implementation of PMCMC). The `RW_llFunctionBlock` sampler handles this by using a Metropolis-Hastings algorithm with a multivariate normal proposal distribution and a user-provided log-likelihood function. To allow compiled execution, the log-likelihood function must be provided as a specialized instance of a nimbleFunction. The log-likelihood function may use the same model as the MCMC as a setup argument, but if so the state of the model should be unchanged during execution of the function (or you must understand the implications otherwise).

The RW_llFunctionBlock sampler accepts the following control list elements:

- adaptive. A logical argument, specifying whether the sampler should adapt the proposal covariance throughout the course of MCMC execution. (default is TRUE)

- adaptScaleOnly. A logical argument, specifying whether adaption should be done only for scale (TRUE) or also for provCov (FALSE). This argument is only relevant when adaptive = TRUE. When adaptScaleOnly = FALSE, both scale and propCov undergo adaptation; the sampler tunes the scaling to achieve a theoretically good acceptance rate, and the proposal covariance to mimic that of the empirical samples. When adaptScaleOnly = TRUE, only the proposal scale is adapted. (default = FALSE)

- adaptInterval. The interval on which to perform adaptation. (default = 200)

- adaptFactorExponent. Exponent controling the rate of decay of the scale adaptation factor. See Shaby and Wells, 2011, for details. (default = 0.8)

- scale. The initial value of the scalar multiplier for propCov. If adaptive = FALSE, scale will never change. (default = 1)

- propCov. The initial covariance matrix for the multivariate normal proposal distribution. This element may be equal to the character string 'identity', in which case the identity matrix of the appropriate dimension will be used for the initial proposal covariance matrix. (default = 'identity')

- llFunction. A specialized nimbleFunction that accepts no arguments and returns a scalar double number. The return value must be the total log-likelihood of all stochastic dependents of the target nodes – and, if includesTarget = TRUE, of the target node(s) themselves – or whatever surrogate is being used for the total log-likelihood. This is a required element with no default.

- includesTarget. Logical variable indicating whether the return value of llFunction includes the log-likelihood associated with target. This is a required element with no default.

### RW_PF sampler

The particle filter sampler allows the user to perform particle MCMC (PMCMC) (Andrieu et al., 2010), primarily for state-space or hidden Markov models of time-series data. This method uses Metropolis-Hastings samplers for top-level parameters but uses the likelihood approximation of a particle filter (sequential Monte Carlo) to integrate over latent nodes in the time-series. The `RW_PF` sampler uses an adaptive Metropolis-Hastings algorithm with a univariate normal proposal distribution for a scalar parameter. Note that samples of the latent states can be retained as well, but the top-level parameter being sampled must be a scalar. A bootstrap, auxiliary, or user defined particle filter can be used to integrate over latent states.

For more information about user-defined samplers within a PMCMC sampler, see the NIMBLE User Manual.

The `RW_PF` sampler accepts the following control list elements:

- adaptive. A logical argument, specifying whether the sampler should adapt the scale (proposal standard deviation) throughout the course of MCMC execution to achieve a theoretically desirable acceptance rate. (default = TRUE)

- adaptInterval. The interval on which to perform adaptation. Every adaptInterval MCMC iterations (prior to thinning), the RW sampler will perform its adaptation procedure. This updates the scale variable, based upon the sampler's achieved acceptance rate over the past adaptInterval iterations. (default = 200)

- scale. The initial value of the normal proposal standard deviation. If `adaptive = FALSE`, scale will never change. (default = 1)

- pfNparticles. The number of particles to use in the approximation to the log likelihood of the data (default = 1000).

- latents. Character vector specifying the nodes that are latent states over which the particle filter will operate to approximate the log-likelihood function.

- pfType. Character argument specifying the type of particle filter that should be used for likelihood approximation. Choose from `"bootstrap"` and `"auxiliary"`. Defaults to `"bootstrap"`.

- pfControl. A control list that is passed to the particle filter function. For details on control lists for bootstrap or auxiliary particle filters, see `buildBootstrapFilter` or `buildAuxiliaryFilter` respectively. Additionally, this can be used to pass custom arguments into a user-defined particle filter.

- pfOptimizeNparticles. A logical argument, specifying whether to use an experimental procedure to automatically determine the optimal number of particles to use, based on Pitt and Shephard (2011). This will override any value of `pfNparticles` specified above.

- pf. A user-defined particle filter object, if a bootstrap or auxiliary particle filter is not adequate.

### RW_PF_block sampler

The particle filter block sampler allows the user to perform particle MCMC (PMCMC) (Andrieu et al., 2010) for multiple parameters jointly, primarily for state-space or hidden Markov models of time-series data. This method uses Metropolis-Hastings block samplers for top-level parameters but uses the likelihood approximation of a particle filter (sequential Monte Carlo) to integrate over latent nodes in the time-series. The `RW_PF` sampler uses an adaptive Metropolis-Hastings algorithm with a multivariate normal proposal distribution. Note that samples of the latent states can be retained as well, but the top-level parameter being sampled must be a scalar. A bootstrap, auxiliary, or user defined particle filter can be used to integrate over latent states.

For more information about user-defined samplers within a PMCMC sampler, see the NIMBLE User Manual.

The `RW_PF_block` sampler accepts the following control list elements:

- adaptive. A logical argument, specifying whether the sampler should adapt the proposal covariance throughout the course of MCMC execution. (default = TRUE)

- adaptScaleOnly. A logical argument, specifying whether adaptation should be done only for `scale` (TRUE) or also for `provCov` (FALSE). This argument is only relevant when `adaptive = TRUE`. When `adaptScaleOnly = FALSE`, both `scale` and `propCov` undergo adaptation; the sampler tunes the scaling to achieve a theoretically good acceptance rate, and the proposal covariance to mimic that of the empirical samples. When `adaptScaleOnly = TRUE`, only the proposal scale is adapted. (default = FALSE)

- adaptInterval. The interval on which to perform adaptation. (default = 200)

- scale. The initial value of the scalar multiplier for `propCov`. If `adaptive = FALSE`, `scale` will never change. (default = 1)

- adaptFactorExponent. Exponent controling the rate of decay of the scale adaptation factor. See Shaby and Wells, 2011, for details. (default = 0.8)

- propCov. The initial covariance matrix for the multivariate normal proposal distribution. This element may be equal to the `'identity'`, in which case the identity matrix of the appropriate dimension will be used for the initial proposal covariance matrix. (default is `'identity'`)

- pfNparticles. The number of particles to use in the approximation to the log likelihood of the data (default = 1000).

- latents. Character vector specifying the nodes that are latent states over which the particle filter will operate to approximate the log-likelihood function.

- pfType. Character argument specifying the type of particle filter that should be used for likelihood approximation. Choose from `"bootstrap"` and `"auxiliary"`. Defaults to `"bootstrap"`.

- pfControl. A control list that is passed to the particle filter function. For details on control lists for bootstrap or auxiliary particle filters, see `buildBootstrapFilter` or `buildAuxiliaryFilter` respectively. Additionally, this can be used to pass custom arguments into a user defined particle filter.

- pfOptimizeNparticles. A logical argument, specifying whether to automatically determine the optimal number of particles to use, based on Pitt and Shephard (2011). This will override any value of `pfNparticles` specified above.

- pf. A user-defined particle filter object, if a bootstrap or auxiliary particle filter is not adequate.

### RW_multinomial sampler

This sampler is designed for sampling multinomial target distributions. The sampler performs a series of Metropolis-Hastings steps between pairs of groups. Proposals are generated via a draw from a binomial distribution, whereafter the proposed number density is moved from one group to another group. The acceptance or rejection of these proposals follows a standard Metropolis-Hastings procedure. Probabilities for the random binomial proposals are adapted to a target acceptance rate of 0.5.

The `RW_multinomial` sampler accepts the following control list elements:

- adaptive. A logical argument, specifying whether the sampler should adapt the binomial proposal probabilities throughout the course of MCMC execution. (default = TRUE)

- adaptInterval. The interval on which to perform adaptation. A minimum value of 100 is required. (default = 200)

### RW_dirichlet sampler

This sampler is designed for sampling non-conjugate Dirichlet distributions. The sampler performs a series of Metropolis-Hastings updates (on the log scale) to each component of a gamma-reparameterization of the target Dirichlet distribution. The acceptance or rejection of these proposals follows a standard Metropolis-Hastings procedure.

The `RW_dirichlet` sampler accepts the following control list elements:

- adaptive. A logical argument, specifying whether the sampler should independently adapt the scale (proposal standard deviation, on the log scale) for each componentwise Metropolis-Hasting update, to achieve a theoretically desirable acceptance rate for each. (default = TRUE)

- adaptInterval. The interval on which to perform adaptation. Every adaptInterval MCMC iterations (prior to thinning), the sampler will perform its adaptation procedure. (default = 200)

- adaptFactorExponent. Exponent controling the rate of decay of the scale adaptation factor. See Shaby and Wells, 2011, for details. (default = 0.8)

- scale. The initial value of the proposal standard deviation (on the log scale) for each component of the reparameterized Dirichlet distribution. If adaptive = FALSE, the proposal standard deviations will never change. (default = 1)

### RW_wishart sampler

This sampler is designed for sampling non-conjugate Wishart and inverse-Wishart distributions. More generally, it can update any symmetric positive-definite matrix (for example, scaled covaraiance or precision matrices). The sampler performs block Metropolis-Hastings updates following a transformation to an unconstrained scale (Cholesky factorization of the original matrix, then taking the log of the main diagonal elements.

The `RW_wishart` sampler accepts the following control list elements:

- adaptive. A logical argument, specifying whether the sampler should adapt the scale and proposal covariance for the multivariate normal Metropolis-Hasting proposals, to achieve a theoretically desirable acceptance rate for each. (default = TRUE)

- adaptInterval. The interval on which to perform adaptation. Every adaptInterval MCMC iterations (prior to thinning), the sampler will perform its adaptation procedure. (default = 200)

- adaptFactorExponent. Exponent controling the rate of decay of the scale adaptation factor. See Shaby and Wells, 2011, for details. (default = 0.8)

- scale. The initial value of the scalar multiplier for the multivariate normal Metropolis-Hastings proposal covariance. If adaptive = FALSE, scale will never change. (default = 1)

### CAR_normal sampler

The CAR_normal sampler operates uniquely on improper (intrinsic) Gaussian conditional autoregressive (CAR) nodes, those with a `dcar_normal` prior distribution. It internally assigns one of three univariate samplers to each dimension of the target node: a posterior predictive, conjugate, or RW sampler; however these component samplers are specialized to operate on dimensions of a `dcar_normal` distribution.

The CAR_normal sampler accepts the following control list elements:

- `carUseConjugacy`. A logical argument, specifying whether to assign conjugate samplers for conjugate components of the target node. If `FALSE`, a RW sampler would be assigned instead. (default = TRUE)

- `adaptive`. A logical argument, specifying whether any component RW samplers should adapt the scale (proposal standard deviation), to achieve a theoretically desirable acceptance rate. (default = `TRUE`)

- `adaptInterval`. The interval on which to perform adaptation for any component RW samplers. Every `adaptInterval` MCMC iterations (prior to thinning), component RW samplers will perform an adaptation procedure. This updates the `scale` variable, based upon the sampler's achieved acceptance rate over the past `adaptInterval` iterations. (default = 200)

- `scale`. The initial value of the normal proposal standard deviation for any component RW samplers. If `adaptive = FALSE`, `scale` will never change. (default = 1)

### CAR_proper sampler

The CAR_proper sampler operates uniquely on proper Gaussian conditional autoregressive (CAR) nodes, those with a `dcar_proper` prior distribution. It internally assigns one of three univariate samplers to each dimension of the target node: a posterior predictive, conjugate, or RW sampler, however these component samplers are specialized to operate on dimensions of a `dcar_proper` distribution.

The CAR_proper sampler accepts the following control list elements:

- `carUseConjugacy`. A logical argument, specifying whether to assign conjugate samplers for conjugate components of the target node. If `FALSE`, a RW sampler would be assigned instead. (default = `TRUE`)

- `adaptive`. A logical argument, specifying whether any component RW samplers should adapt the scale (proposal standard deviation), to achieve a theoretically desirable acceptance rate. (default = `TRUE`)

- `adaptInterval`. The interval on which to perform adaptation for any component RW samplers. Every adaptInterval MCMC iterations (prior to thinning), component RW samplers will perform an adaptation procedure. This updates the scale variable, based upon the sampler's achieved acceptance rate over the past adaptInterval iterations. (default = 200)

- `scale`. The initial value of the normal proposal standard deviation for any component RW samplers. If `adaptive = FALSE`, `scale` will never change. (default = 1)

### CRP sampler

The CRP sampler is designed for fitting models involving Dirichlet process mixtures. It is exclusively assigned by NIMBLE's default MCMC configuration to nodes having the Chinese Restaurant Process distribution, `dCRP`. It executes sequential sampling of each component of the node (i.e., the cluster membership of each element being clustered). Internally, either of two samplers can be assigned, depending on conjugate or non-conjugate structures within the model. For conjugate and non-conjugate model structures, updates are based on Algorithm 2 and Algorithm 8 in Neal (2000), respectively.

### CRP_concentration sampler

The CRP_concentration sampler is designed for Bayesian nonparametric mixture modeling. It is exclusively assigned to the concentration parameter of the Dirichlet process when the model is specified using theChinese Restaurant Process distribution, `dCRP`. This sampler is assigned by default by NIMBLE's default MCMC configuration and is and can only be used when the prior for the concentration is a gamma distribution. The assigned sampler is an augmented beta-gamma sampler as discussed in Section 6 in Escobar and West (1995).

### posterior_predictive sampler

The posterior_predictive sampler is only appropriate for use on terminal stochastic nodes. Note that such nodes play no role in inference but have often been included in BUGS models to accomplish posterior predictive checks. NIMBLE allows posterior predictive values to be simulated independently of running MCMC, for example by writing a nimbleFunction to do so. This means that in many cases where terminal stochastic nodes have been included in BUGS models, they are not needed when using NIMBLE.

The posterior_predictive sampler functions by calling the simulate() method of relevant node, then updating model probabilities and deterministic dependent nodes. The application of a posterior_predictive sampler to any non-terminal node will result in invalid posterior inferences. The posterior_predictive sampler will automatically be assigned to all terminal, non-data stochastic nodes in a model by the default MCMC configuration, so it is uncommon to manually assign this sampler.

The posterior_predictive sampler accepts no control list arguments.

### RJ_fixed_prior sampler

This sampler proposes addition/removal for variable of interest in the framework of variable selection using reversible jump MCMC, with a specified prior probability of inclusion. A normal proposal distribution is used to generate proposals for the addition of the variable. This is a specialized sampler used by `configureRJ` function, when the model code is written without using indicator variables. See `help{configureRJ}` for details. It is not intended for direct assignment.

### RJ_indicator sampler

This sampler proposes transitions of a binary indicator variable, corresponding to a variable of interest, in the framework of variable selection using reversible jump MCMC. This is a specialized sampler used by `configureRJ` function, when the model code is written using indicator variables. See `help{configureRJ}` for details. It is not intended for direct assignment.

### RJ_toggled sampler

This sampler operates in the framework of variable selection using reversible jump MCMC. Specifically, it conditionally performs updates of the target variable of interest using the originally-specified sampling configuration, when variable is "in the model". This is a specialized sampler used by `configureRJ` when adding a reversible jump MCMC . See `help{configureRJ}` for details. It is not intended for direct assignment.

### Author(s)

Daniel Turek

### References

Andrieu, C., Doucet, A., and Holenstein, R. (2010). Particle Markov Chain Monte Carlo Methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(3), 269-342.

Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953). Equation of State Calculations by Fast Computing Machines. The Journal of Chemical Physics, 21(6), 1087-1092.

Neal, Radford M. (2003). Slice Sampling. The Annals of Statistics, 31(3), 705-741.

Murray, I., Prescott Adams, R., and MacKay, D. J. C. (2010). Elliptical Slice Sampling. arXiv e-prints, arXiv:1001.0175.

Pitt, M.K. and Shephard, N. (1999). Filtering via simulation: Auxiliary particle filters. Journal of the American Statistical Association 94(446), 590-599.

Roberts, G. O. and S. K. Sahu (1997). Updating Schemes, Correlation Structure, Blocking and Parameterization for the Gibbs Sampler. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 59(2), 291-317.

Shaby, B. and M. Wells (2011). Exploring an Adaptive Metropolis Algorithm. 2011-14. Department of Statistics, Duke University.

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

Escobar, M. D., and West, M. (1995). Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association, 90(430), 577-588.

Neal, R. M. (2000). Markov chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9(2), 249-265.

### See Also

`configureMCMC` `addSampler` `buildMCMC` `runMCMC`

### Examples

    ## y[1] ~ dbern() or dbinom():
    # mcmcConf$addSampler(target = 'y[1]', type = 'binary')

    # mcmcConf$addSampler(target = 'a', type = 'RW',
    #    control = list(log = TRUE, adaptive = FALSE, scale = 3))
    # mcmcConf$addSampler(target = 'b', type = 'RW',
    #    control = list(adaptive = TRUE, adaptInterval = 200))
    # mcmcConf$addSampler(target = 'p', type = 'RW',
    #    control = list(reflective = TRUE))

    ## a, b, and c all continuous-valued:
    # mcmcConf$addSampler(target = c('a', 'b', 'c'), type = 'RW_block')

    # mcmcConf$addSampler(target = 'p', type = 'RW_llFunction',
    #    control = list(llFunction = RllFun, includesTarget = FALSE))

    # mcmcConf$addSampler(target = 'y[1]', type = 'slice',
    #    control = list(adaptive = FALSE, sliceWidth = 3))
    # mcmcConf$addSampler(target = 'y[2]', type = 'slice',
    #    control = list(adaptive = TRUE, sliceMaxSteps = 1))

    # mcmcConf$addSampler(target = 'x[1:10]', type = 'ess')   ## x[1:10] ~ dmnorm()

    # mcmcConf$addSampler(target = 'x[1:5]', type = 'RW_multinomial')   ## x[1:5] ~ dmulti()

    # mcmcConf$addSampler(target = 'p[1:5]', type = 'RW_dirichlet')   ## p[1:5] ~ ddirch()

    ## y[1] is a posterior predictive node:
    # mcmcConf$addSampler(target = 'y[1]', type = 'posterior_predictive')

[Package nimble version 0.9.1 Index]