Blocks

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Description

The Blocks function returns a list of N blocks of parameters, for use with some MCMC algorithms in the LaplacesDemon function. Blocks may be created either sequentially, or from a hierarchical clustering of the posterior correlation matrix.

Usage

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Blocks(Initial.Values, N, PostCor=NULL)

Arguments

Initial.Values

This required argument is a vector of initial values.

N

This optional argument indicates the desired number of blocks. If omitted, then the truncated square root of the number of initial values is used. If a posterior correlation matrix is supplied to PostCor, then N may be a scalar, or have length two. If N has length two, then the first element indicates the minimum number of blocks, and the second element indicates the maximum number of blocks, and the number of blocks is the maximum of the mean silhouette width for each hierarchical cluster solution.

PostCor

This optional argument defaults to NULL, in which case sequential blocking is performed. If a posterior correlation matrix is supplied, then blocks are created based on hierarchical clustering.

Details

Usually, there is more than one target distribution in MCMC, in which case it must be determined whether it is best to sample from target distributions individually, in groups, or all at once. Blockwise sampling (also called block updating) refers to splitting a multivariate vector into groups called blocks, and each block is sampled separately. A block may contain one or more parameters.

Parameters are usually grouped into blocks such that parameters within a block are as correlated as possible, and parameters between blocks are as independent as possible. This strategy retains as much of the parameter correlation as possible for blockwise sampling, as opposed to componentwise sampling where parameter correlation is ignored. The PosteriorChecks function can be used on the output of previous runs to find highly correlated parameters. See examples below.

Advantages of blockwise sampling are that a different MCMC algorithm may be used for each block (or parameter, for that matter), creating a more specialized approach (though different algorithms by block are not supported here), the acceptance of a newly proposed state is likely to be higher than sampling from all target distributions at once in high dimensions, and large proposal covariance matrices can be reduced in size, which is most helpful again in high dimensions.

Disadvantages of blockwise sampling are that correlations probably exist between parameters between blocks, and each block is updated while holding the other blocks constant, ignoring these correlations of parameters between blocks. Without simultaneously taking everything into account, the algorithm may converge slowly or never arrive at the proper solution. However, there are instances when it may be best when everything is not taken into account at once, such as in state-space models. Also, as the number of blocks increases, more computation is required, which slows the algorithm. In general, blockwise sampling allows a more specialized approach at the expense of accuracy, generalization, and speed. Blockwise sampling is offered in the following algorithms: Adaptive-Mixture Metropolis (AMM), Adaptive Metropolis-within-Gibbs (AMWG), Automated Factor Slice Sampler (AFSS), Elliptical Slice Sampler (ESS), Hit-And-Run Metropolis (HARM), Metropolis-within-Gibbs (MWG), Random-Walk Metropolis (RWM), Robust Adaptive Metropolis (RAM), Slice Sampler (Slice), and the Univariate Eigenvector Slice Sampler (UESS).

Large-dimensional models often require blockwise sampling. For example, with thousands of parameters, a componentwise algorithm must evaluate the model specification function once per parameter per iteration, resulting in an algorithm that may take longer than is acceptable to produce samples. Algorithms that require derivatives, such as the family of Hamiltonian Monte Carlo (HMC), require even more evaluations of the model specification function per iteration, and quickly become too costly in large dimensions. Finally, algorithms with multivariate proposals often have difficulty producing an accepted proposal in large-dimensional models. The most practical solution is to group parameters into N blocks, and each iteration the algorithm evaluates the model specification function N times, each with a reduced set of parameters.

The Blocks function performs either a sequential assignment of parameters to blocks when posterior correlation is not supplied, or uses hierarchical clustering to create blocks based on posterior correlation. If posterior correlation is supplied, then the user may specify a range of the number of blocks to consider, and the optimal number of blocks is considered to be the maximum of the mean silhouette width of each hierarchical clustering. Silhouette width is calculated as per the cluster package. Hierarchical clustering is performed on the distance matrix calculated from the dissimilarity matrix (1 - abs(PostCor)) of the posterior correlation matrix. With sequential assignment, the number of parameters per block is approximately equal. With hierarchical clustering, the number of parameters per block may vary widely. Creating blocks from hierarchical clustering performs well in practice, though there are many alternative methods the user may consider outside of this function, such as using factor analysis, model-based clustering, or other methods.

Aside from sequentially-assigned blocks, or blocks based on posterior correlation, it is also common to group parameters with similar uses, such as putting regression effects parameters into one block, and autocorrelation parameters into another block. Another popular way to group parameters into blocks is by time-period for some time-series models. These alternative blocking strategies are unsupported in the Blocks function, and best left to user discretion.

Some MCMC algorithms that accept blocked parameters also require blocked variance-covariance matrices. The Blocks function does not return these matrices, because it may not be necessary, or when it is, the user may prefer identity matrices, scaled identity matrices, or matrices with explicitly-defined elements.

If the user is looking for a place to begin with blockwise sampling, then the recommended, default approach (when blocked parameters by time-period are not desired in a time-series) is to begin with a trial run of the adaptive, unblocked HARM algorithm (since covariance matrices are not required) for the purposes of obtaining a posterior correlation matrix. Next, create blocks with the Blocks function based on the posterior correlation matrix obtained from the trial run. Finally, run the desired, blocked algorithm with the newly created blocks (and possibly user-specified covariance matrices), beginning where the trial run ended.

If hierarchical clustering is used, then it is important to note that hierarchical clustering has no idea that the user intends to perform blockwise sampling in MCMC. If hierarchical clustering returns numerous small blocks, then the user may consider combining some or all of those blocks. For example, if several 1-parameter blocks are returned, then blockwise sampling will equal componentwise sampling for those blocks, which will iterate slower. Conversely, if hierarchical clustering returns one or more big blocks, each with enough parameters that multivariate sampling will have difficulty getting an accepted proposal, or an accepted proposal that moves more than a small amount, then the user may consider subdividing these big blocks into smaller, more manageable blocks, though with the understanding that more posterior correlation is unaccounted for.

Value

The Blocks function returns an object of class blocks, which is a list. Each component of the list is a block of parameters, and parameters are indicated by their position in the initial values vector.

Author(s)

Statisticat, LLC. software@bayesian-inference.com

See Also

LaplacesDemon and PosteriorChecks.

Examples

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library(LaplacesDemon)

### Create the default number of sequentially assigned blocks:
Initial.Values <- rep(0,1000)
MyBlocks <- Blocks(Initial.Values)
MyBlocks

### Or, a pre-specified number of sequentially assigned blocks:
#Initial.Values <- rep(0,1000)
#MyBlocks <- Blocks(Initial.Values, N=20)

### If scaled diagonal covariance matrices are desired:
#VarCov <- list()
#for (i in 1:length(MyBlocks))
#  VarCov[[i]] <- diag(length(MyBlocks[[i]]))*2.38^2/length(MyBlocks[[i]])

### Or, determine the number of blocks in the range of 2 to 50 from
### hierarchical clustering on the posterior correlation matrix of an
### object, say called Fit, output from LaplacesDemon:
#MyBlocks <- Blocks(Initial.Values, N=c(2,50),
#  PostCor=cor(Fit$Posterior1))
#lapply(MyBlocks, length) #See the number of parameters per block

### Or, create a pre-specified number of blocks from hierarchical
### clustering on the posterior correlation matrix of an object,
### say called Fit, output from LaplacesDemon:
#MyBlocks <- Blocks(Initial.Values, N=20, PostCor=cor(Fit$Posterior1))

### Posterior correlation from a previous trial run could be obtained
### with either method below (though cor() will be fastest because
### additional checks are not calculated for the parameters):
#rho <- cor(Fit$Posterior1)
#rho <- PosteriorChecks(Fit)$Posterior.Correlation

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