createPriorDensity: Fits a density function to a multivariate sample

In BayesianTools: General-Purpose MCMC and SMC Samplers and Tools for Bayesian Statistics

Fits a density function to a multivariate sample

Description

Fits a density function to a multivariate sample

Usage

createPriorDensity(
  sampler,
  method = "multivariate",
  eps = 1e-10,
  lower = NULL,
  upper = NULL,
  best = NULL,
  scaling = 1,
  ...
)

Arguments

sampler	an object of class BayesianOutput or a matrix
method	method to generate prior - default and currently only option is multivariate
eps	numerical precision to avoid singularity
lower	vector with lower bounds of parameter for the new prior, independent of the input sample
upper	vector with upper bounds of parameter for the new prior, independent of the input sample
best	vector with "best" values of parameter for the new prior, independent of the input sample
scaling	optional scaling factor for the covariance. If scaling > 1 will create a prior wider than the posterior, < 1 a prior more narrow than the posterior. Scaling is linear to the posterior width, i.e. scaling = 2 will create a prior that with 2x the sd of the original posterior.
...	parameters to pass on to the getSample function

Details

This function fits a density estimator to a multivariate (typically a posterior) sample. The main purpose is to summarize a posterior sample as a pdf, in order to include it as a prior in a new analysis, for example when new data becomes available, or to calculate a fractional Bayes factor (see marginalLikelihood).

The limitation of this function is that we currently only implement a multivariate normal density estimator, so you will have a loss of information if your posterior is not approximately multivariate normal, which is likely the case if you have weak data. Extending the function to include more flexible density estimators (e.g. gaussian processes) is on our todo list, but it's quite tricky to get this stable, so I'm not sure when we will have this working. In general, creating reliable empirical density estimates in high-dimensional parameter spaces is extremely tricky, regardless of the software you are using.

For that reason, it is usually recommended to not update the posterior with this option, but rather:

1. If the full dataset is available, to make a single, or infrequent updates, recompute the entire model with the full / updated data

2. For frequent updates, consider using SMC instead of MCMC sampling. SMC sampling doesn't require an analytical summary of the posterior.

Author(s)

Florian Hartig

See Also

createPrior
createBetaPrior
createTruncatedNormalPrior
createUniformPrior
createBayesianSetup

Examples

# the BT package includes a number of convenience functions to specify
# prior distributions, including createUniformPrior, createTruncatedNormalPrior
# etc. If you want to specify a prior that corresponds to one of these
# distributions, you should use these functions, e.g.:

prior <- createUniformPrior(lower = c(0,0), upper = c(0.4,5))

prior$density(c(2, 3)) # outside of limits -> -Inf
prior$density(c(0.2, 2)) # within limits, -0.6931472

# All default priors include a sampling function, i.e. you can create
# samples from the prior via
prior$sampler()
# [1] 0.2291413 4.5410389

# if you want to specify a prior that does not have a default function, 
# you should use the createPrior function, which expects a density and 
# optionally a sampler function:

density = function(par){
  d1 = dunif(par[1], -2,6, log =TRUE)
  d2 = dnorm(par[2], mean= 2, sd = 3, log =TRUE)
  return(d1 + d2)
}

sampler = function(n=1){
  d1 = runif(n, -2,6)
  d2 = rnorm(n, mean= 2, sd = 3)
  return(cbind(d1,d2))
}

prior <- createPrior(density = density, sampler = sampler, 
                     lower = c(-10,-20), upper = c(10,20), best = NULL)

# note that the createPrior supports additional truncation


# To use a prior in an MCMC, include it in a BayesianSetup 

set.seed(123)
ll <- function(x) sum(dnorm(x, log = TRUE)) # multivariate normal ll
bayesianSetup <- createBayesianSetup(likelihood = ll, prior = prior)

settings = list(iterations = 100)
out <- runMCMC(bayesianSetup = bayesianSetup, settings = settings)

# use createPriorDensity to create a new (estimated) prior from MCMC output

newPrior = createPriorDensity(out, method = "multivariate",
                              eps = 1e-10, lower = c(-10,-20), 
                              upper = c(10,20), best = NULL, scaling = 0.5)

BayesianTools documentation built on Feb. 16, 2023, 8:44 p.m.