BayesianTools (version 0.1.8)

WAIC: calculates the WAIC

Description

calculates the WAIC

Usage

WAIC(bayesianOutput, numSamples = 1000, ...)

Arguments

bayesianOutput

an object of class BayesianOutput. Must implement a log-likelihood density function that can return point-wise log-likelihood values ("sum" argument).

numSamples

the number of samples to calculate the WAIC

...

optional values to be passed on the the getSample function

Author

Florian Hartig

Details

The WAIC is constructed as $$WAIC = -2 * (lppd - p_{WAIC})$$

The lppd (log pointwise predictive density), defined in Gelman et al., 2013, eq. 4 as

$$lppd = \sum_{i=1}^n \log \left(\frac{1}{S} \sum_{s=1}^S p(y_i | \theta^s)\right)$$

The value of \(p_WAIC\) can be calculated in two ways, the method used is determined by the method argument.

Method 1 is defined as, $$p_{WAIC1} = 2 \sum_{i=1}^{n} (\log (\frac{1}{S} \sum_{s=1}^{S} p(y_i \ \theta^s)) - \frac{1}{S} \sum_{s = 1}^{S} \log p(y_i | \theta^s))$$ Method 2 is defined as, $$p_{WAIC2} = 2 \sum_{i=1}^{n} V_{s=1}^{S} (\log p(y_i | \theta^s))$$ where \(V_{s=1}^{S}\) is the sample variance.

References

Gelman, Andrew and Jessica Hwang and Aki Vehtari (2013), "Understanding Predictive Information Criteria for Bayesian Models," http://www.stat.columbia.edu/~gelman/research/unpublished/waic_understand_final.pdf.

Watanabe, S. (2010). "Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory", Journal of Machine Learning Research, https://www.jmlr.org/papers/v11/watanabe10a.html.

See Also

DIC, MAP, marginalLikelihood

Examples

bayesianSetup <- createBayesianSetup(likelihood = testDensityNormal, 
                                     prior = createUniformPrior(lower = rep(-10,2),
                                                                upper = rep(10,2)))

# likelihood density needs to have option sum = FALSE

testDensityNormal(c(1,1,1), sum = FALSE)
bayesianSetup$likelihood$density(c(1,1,1), sum = FALSE)
bayesianSetup$likelihood$density(matrix(rep(1,9), ncol = 3), sum = FALSE)

# running MCMC

out = runMCMC(bayesianSetup = bayesianSetup)

WAIC(out)