PosteriorChecks: Posterior Checks
In LaplacesDemon: Complete Environment for Bayesian Inference

Description Usage Arguments Details Value Author(s) See Also Examples

Not to be confused with posterior predictive checks, this function provides additional information about the marginal posterior distributions of continuous parameters, such as the probability that each posterior coefficient of the parameters (referred to generically as theta), is greater than zero [p(theta > 0)], the estimated number of modes, the kurtosis and skewness of the posterior distributions, the burn-in of each chain (for MCMC only), integrated autocorrelation time, independent samples per minute, and acceptance rate. A posterior correlation matrix is provided only for objects of class demonoid or pmc.

For discrete parameters, see the Hangartner.Diagnostic.

1	PosteriorChecks(x, Parms)

x

This required argument accepts an object of class demonoid, iterquad, laplace, pmc, or vb.

Parms

This argument accepts a vector of quoted strings to be matched for selecting parameters. This argument defaults to NULL and selects every parameter. Each quoted string is matched to one or more parameter names with the grep function. For example, if the user specifies Parms=c("eta", "tau"), and if the parameter names are beta[1], beta[2], eta[1], eta[2], and tau, then all parameters will be selected, because the string eta is within beta. Since grep is used, string matching uses regular expressions, so beware of meta-characters, though these are acceptable: ".", "[", and "]".

PosteriorChecks is a supplemental function that returns a list with two components. Following is a summary of popular uses of the PosteriorChecks function.

First (and only for MCMC users), the user may be considering the current MCMC algorithm versus others. In this case, the PosteriorChecks function is often used to find the two MCMC chains with the highest IAT, and these chains are studied for non-randomness with a joint trace plot, via the joint.density.plot function. The best algorithm has the chains with the highest independent samples per minute (ISM).

Posterior correlation may be studied between model updates as well as after a model seems to have converged. While frequentists consider multicollinear predictor variables, Bayesians tend to consider posterior correlation of the parameters. Models with multicollinear parameters take more iterations to converge. Hierarchical models often have high posterior correlations. Posterior correlation often contributes to a lower effective sample size (ESS). Common remedies include transforming the predictors, re-parameterization to reduce posterior correlation, using WIPs (Weakly-Informative Priors), or selecting a different numerical approximation algorithm. An example of re-parameterization is to constrain related parameters to sum to zero. Another approach is to specify the parameters according to a multivariate distribution that is assisted by estimating a covariance matrix. Some algorithms are more robust to posterior correlation than others. For example, posterior correlation should generally be less problematic for twalk than AMWG in LaplacesDemon. Posterior correlation may be plotted with the plotMatrix function, and may be useful for blocking parameters. For more information on blockwise sampling, see the Blocks function.

After a user is convinced of the applicability of the current MCMC algorithm, and that the chains have converged, PosteriorChecks is often used to identify multimodal marginal posterior distributions for further study or model re-specification.

Although many marginal posterior distributions appear normally distributed, there is no such assumption. Nonetheless, a marginal posterior distribution tends to be distributed the same as its prior distribution. If a parameter has a prior specified with a Laplace distribution, then the marginal posterior distribution tends also to be Laplace-distributed. In the common case of normality, kurtosis and skewness may be used to identify discrepancies between the prior and posterior, and perhaps this should be called a 'prior-posterior check'.

Lastly, parameter importance may be considered, in which case it is recommended to be considered simultaneously with variable importance from the Importance function.

PosteriorChecks returns an object of class posteriorchecks that is a list with the following components:

Posterior.Correlation

This is a correlation matrix of the parameters selected with the Parms argument. This component is returned as NA for objects of classes "laplace" or "vb".

Posterior.Summary

This is a matrix in which each row is a parameter and there are eight columns: p(theta > 0), N.Modes, Kurtosis, Skewness, Burn-In, IAT, ISM, and AR. The first column, p(theta > 0), indicates parameter importance by reporting how much of the distribution is greater than zero. An important parameter distribution will have a result at least as extreme as 0.025 or 0.975, and an unimportant parameter distribution is centered at 0.5. This is not the importance of the associated variable relative to how well the model fits the data. For variable importance, see the Importance function. The second column, N.Modes, is the number of modes, estimated with the Modes function. Kurtosis and skewness are useful posterior checks that may suggest that a posterior distribution is non-normal or does not fit well with a distributional assumption, assuming a distributional assumption exists, which it may not. The burn-in is estimated for each chain (only for objects of class demonoid with the burnin function. The integrated autocorrelation time is estimated with IAT. The number of independent samples per minute (ISM) is calculated for objects of class "demonoid" as ESS divided by minutes. Lastly, the local acceptance rate of each MCMC chain is calculated with the AcceptanceRate function, and is set to 1 for objects of class iterquad, laplace, pmc, or vb.