Add function to convert posterior distribution into likelihood profile

[schuemie]

The computeBayesianMetaAnalysis() outputs a posterior distribution for the parameter of interest (i.e. effect size estimate). If we convert that to a likelihood profile we can use it for example to estimate a systematic error distribution.. For this, the output should be a data frame with two fields: point and value. We can use the adaptive profiling also implemented in Cyclops to do this.

[msuchard]

Convert the posterior samples to 1D density estimates right before here: https://github.com/OHDSI/EvidenceSynthesis/blob/aee99c04c8d6869fb0bf4d46a3662ea94a768b8a/R/BayesianMetaAnalysis.R#L273

stats::density does this nicely.

But, since the posterior is 2D (mu, tau), you might want to use the dependence between parameters. In which case, just return the samples themselves (or consider a 2D density estimator).

[schuemie]

For the particular use case I have in mind I only care about mu: In a network study, each site produces likelihood profiles for negative controls. For each negative control, we combine the per-site likelihood profiles using our Bayesian meta-analysis to produce a cross-network posterior distribution for the effect size of the negative control. I would then like to use these posterior distributions (without summarizing them as point estimates and standard errors) to fit an empirical null distribution. The fitNullNonNormalLl() function in the EmpiricalCalibration package already takes likelihood profiles as input, so it would be convenient if we summarize the posterior distributions as such.

I could also modify the fitNullNonNormalLl() to work with the samples, but at some point these samples will need to be converted into a density estimate, so why not in this package?

So perhaps a reasonable approach would be to use stats:density with many points, and then use the adaptive profiling procedure in Cyclops to reduce the number of points to the minimum required.

[msuchard]

For your use-case, stats::density will return a good approximation to the marginal distribution p(\mu | Y). Do not use Cyclops to down-sample as that procedure assumes the function is convex (concave) which your posterior is certainly not (as a mixture of normals).

You can use these posteriors as input to EmpiricalCalibration but keep in mind that your model does not fully follow from the rules of conditional probability; i.e. what you really want to is p(Y | \mu). An alternative approach that may not violate this small hitch is fitting a single, joint Bayesian model to all of the negative controls across all of the data sites at once. We could do this as well.