R: Generate Initial Values

GIV {LaplacesDemon} | R Documentation

## Generate Initial Values

### Description

The `GIV` function generates initial values for use with the `IterativeQuadrature`, `LaplaceApproximation`, `LaplacesDemon`, `PMC`, and `VariationalBayes` functions.

### Usage

    GIV(Model, Data, n=1000, PGF=FALSE)

### Arguments

`Model` | This required argument is a model specification function. For more information, see `LaplacesDemon`.
`Data` | This required argument is a list of data. For more information, see `LaplacesDemon`.
`n` | This is the number of attempts to generate acceptable initial values.
`PGF` | Logical. When `TRUE`, a Parameter-Generating Function (PGF) is required to be in `Data`, and `GIV` will generate initial values according to the user-specified PGF. This argument defaults to `FALSE`, in which case initial values are generated randomly without respect to a user-specified function.

### Details

Initial values are required for optimization or sampling algorithms. A user may specify initial values, or use the `GIV` function for random generation. Initial values determined by the user may fail to produce a finite posterior in complicated models, and the `GIV` function is here to help.

`GIV` has several uses. First, the `IterativeQuadrature`, `LaplaceApproximation`, `LaplacesDemon`, and `VariationalBayes` functions use `GIV` internally if unacceptable initial values are discovered. Second, the user may use `GIV` when developing their model specification function, `Model`, to check for potential problems. Third, the user may prefer to randomly generate acceptable initial values. Lastly, `GIV` is recommended when running multiple or parallel chains with the `LaplacesDemon.hpc` function (such as for later use with the `Gelman.Diagnostic`) for dispersed starting locations. For dispersed starting locations, `GIV` should be run once for each parallel chain, and the results should be stored per row in a matrix of initial values. For more information, see the `LaplacesDemon.hpc` documentation for initial values.

It is strongly recommended that the user specifies a Parameter-Generating Function (PGF), and includes this function in the list of data. Although the PGF may be specified according to the prior distributions (possibly considered as a Prior-Generating Function), it is often specified with a more restricted range. For example, if a user has a model with the following prior distributions

beta_j ~ N(0, 1000), j=1,…,5

sigma ~ HC(25)

then the PGF, given the prior distributions, is

`PGF <- function(Data) return(c(rnormv(5,0,1000),rhalfcauchy(1,25)))`

However, the user may not want to begin with initial values that could be so far from zero (as determined by the variance of 1000), and may instead prefer

`PGF <- function(Data) return(c(rnormv(5,0,10),rhalfcauchy(1,5)))`

When `PGF=FALSE`, initial values are attempted to be constrained to the interval [-100,100]. This is done to prevent numeric overflows with parameters that are exponentiated within the model specification function. First, `GIV` passes the upper and lower bounds of this interval to the model, and any changed parameters are noted.

At this point, it is hoped that a non-finite posterior is not found. If found, then the remainder of the process is random and without the previous bounds. This can be particularly problematic in the case of, say, initial values that are the elements of a matrix that must be positive-definite, especially with large matrices. If a random solution is not found, then `GIV` will fail.

If the posterior is finite and `PGF=FALSE`, then initial values are randomly generated with a normal proposal and a small variance at the center of the returned range of each parameter. As `GIV` fails to find acceptable initial values, the algorithm iterates toward its maximum number of iterations, `n`. In each iteration, the variance increases for the proposal.

Initial values are considered acceptable only when the first two returned components of `Model` (which are `LP` and `Dev`) are finite, and when initial values do not change through constraints, as returned in the fifth component of the list: `parm`.

If `GIV` fails to return acceptable initial values, then it is best to study the model specification function. When the model is complicated, here is a suggestion. Remove the log-likelihood, `LL`, from the equation that calculates the logarithm of the unnormalized joint posterior density, `LP`. For example, convert `LP <- LL + beta.prior` to `LP <- beta.prior`. Now, maximize `LP`, which is merely the set of prior densities, with any optimization algorithm. Replace `LL`, and run the model with initial values that are in regions of high prior density (preferably with `PGF=TRUE`. If this fails, then the model specification should be studied closely, because a non-finite posterior should (especially) never be associated with regions of high prior density.

### Value

The `GIV` function returns a vector equal in length to the number of parameters, and each element is an initial value for the associated parameter in `Data$parm.names`. When `GIV` fails to find acceptable initial values, each returned element is `NA`.

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

### See Also

`as.initial.values`, `Gelman.Diagnostic`, `IterativeQuadrature`, `LaplaceApproximation`, `LaplacesDemon`, `LaplacesDemon.hpc`, `PMC`, and `VariationalBayes`.

### Examples

    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 <- c("LP","sigma")
    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)
         }

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

[Package LaplacesDemon version 16.1.1 Index]