Bootstrap Bayesian

Verdonck FAM, Jaworska J, Thas O, Vanrolleghem PA. 2001. Determining environmental standards using bootstrapping, Bayesian and maximum likelihood techniques a comparative study. Anal Chim Acta 446 429-438. 

Fig. 5.2. Phylogeny of monopisthocotylean Monogenea based on SSU rDNA. The tree topology is from a Bayesian analysis with nodal support indicated, from top to bottom, for maximum likelihood (bootstrap%, n = 100), maximum parsimony (bootstrap%, n = 1000) and Bayesian inference (posterior probabilities). Figure from Matejusova etal. (2003).

Fig. 2.11 Phylogeny of Class II of the peroxidase-catalase superfamily. Sequences coding for secretory fungal peroxidases lignin peroxidase (LiP), manganese peroxidase (MnP), and versatile peroxidase (VP) were used for this reconstruction. One of nine equally parsimonious trees is presented. Bootstrap values are indicated before slash, and Bayesian posterior probability values are indicated after the slash. With kind permission from Springer Science Business Media Morgenstem et al. [32], Fig. 2...

Step 3. Apply the basic model to each of the 100 bootstrap data sets and determine the individual Bayesian PM parameters. 

One hundred bootstrap samples are generated and the appropriate structural model that best describes the data from each sample is determined. This is done to ensure that the model that best describes the bootstrap data is not different from the basic structural model used for developing the population PK model for the data before bootstrapping. With the right structural model POSTHOC individual Bayesian estimates are generated and the data subjected to GAM. 

The Bayesian bootstrap was introduced by Rubin (26) in 1981 and subsequently used by Rubin and Schenker (29) for multiple imputation in missing-data problems. The Bayesian bootstrap is not covered because its application is for multiple imputation of missing data and this is addressed in Chapter 9. 

NONMEM was used to estimate the parameters for each bootstrap data set. Individual Bayesian parameters were generated. These estimates along with covariates formed a new data set. 

Since simulation is involved, multiple imputation shares with other simulation-based estimation procedures, such as the bootstrap and Markov chain Monte Carlo Bayesian approaches, the problem that it could conceivably produce different answers on different occasions even though the same data are involved. This might lead to some regulatory reluctance, although regulators have accepted the result of multiple imputations on occasion. 

Obtain posterior distributions of all fixed and random parameters, which typically requires a full Bayesian method as with POPKAN or BUGS, but can be accomplished with NONMEM under certain conditions and assumptions. Alternatively, posterior distributions can be simulated using a parametric bootstrap. 

Ensemble methods comprise bootstrap aggregating (bagging), Bayesian model combination, bucket of models, stacking, or boosting. 

Of the available options, ratio-based cutoffs are the simplest and most widely used but also probably the least meaningful, due to the arbitrary nature of the cutoff and the insensitivity to noise at low signal levels. Better methods include bootstrapped r-tests, Bayesian probability methods, and ANOVA analysis. However, all these depend on the design of the experiment. Generally speaking, outliers identified in replicate hybridizations provide a good starting point for further validation (6). 

Stefanovic S, Olmstead RG (2004) Testing the phylogenetic position of a parasitic plant (Cuscuta, Convolvulaceae, Asteridae) Bayesian inference and the parametric bootstrap on data drawn from three genomes. Syst Biol 53 384-399... 

In most trees resulting from the MP analyses (e.g.. Figure 4.24) and the trees resulting from the Bayesian inference analyses (Figure 4.25), the Hypopterygiaceae is presented as being mono-phyletic. However, support for this is weak (bootstrap 59%, PP 76%), which is also demonstrated by the strict consensus tree of the MP analyses (Figure 4.22). 

Sampling-importance-resampling is sometimes called the Bayesian bootstrap since we are resampling. However, here we are resampling from the sample of parameters using the importance weights, not resampling from the data. 

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