This algorithm seeks to identify parameter probability distributions using a rejection sampling procedure. It is based on the procedure described by Chen (2005). The algorithm samples from a set of truncated uniform distributions in search of parameter sets that are representative of the “true” posterior probability distribution. It represents a middle ground between informal GLUE-like procedures, including DDS-AU, and formal MCMC procedures. The following optional group will configure the Rejection Sampling algorithm and will be processed if [ProgramType] is set to “RejectionSampler”.

BeginRejectionSampler
SamplesPerIter    [nper]
NumDesired        [nsols]
BurnInSamples     [nburn]
MaxSamples        [nmax]
LikelihoodType    [ltype]
ShapingFactor     [shape]
TelescopeRate     [trate]
MinWSSE           [wsse]
EndRejectionSampler

BeginRejectionSampler
SamplesPerIter    10
NumDesired        10
BurnInSamples     0
MaxSamples        100
LikelihoodType    Stedinger
ShapingFactor     0.5
TelescopeRate     0
MinWSSE           1.00E308
EndRejectionSampler

Figure 1: General Format (left) and Example (right) of the Rejection Sampler Group

Where BeginRejectionSampler and EndRejectionSampler are parsing tags that wrap the following set of algorithm configuration variables:

• SamplesPerIter: This variable controls the frequency of output within the OSTRICH run record. The current best solution will be reported after every SamplesPerIter model evaluations. The current number of accepted solutions will also be reported. Samples that are accepted after the burn-in period is complete are understood to come from a posterior distribution. The default value is 10.
• NumDesired: The desired number of post burn-in samples (i.e. the number of samples that are desired from the posterior parameter distributions). The default value is 10.
• BurnInSamples: The number of accepted samples that should be discarded before assuming accepted samples are representative of a posterior distribution. The default value is 0 (i.e. no burn-in).
• MaxSamples: The maximum number of model evaluations that will be performed as part of the MCMC search. The default value is 100.
• LikelihoodType: Use this variable to select alternative formulations for computing the likelihood ratio. Two options are available, namely “Beven” and “Stedinger”. The Beven likelihood type is a pseudo-likelihood described by Beven and Binley (1992). The “Stedinger” likelihood type is a formal likelihood function described by Stedinger et al. (2008). If “Beven” is selected, the ShapingFactor (see below) will also be processed. The default setting is Stedinger.
• ShapingFactor: This variable is a correction exponent for the Beven pseudo-likelihood function. As described by Stedinger et al. (2008), adjusting the ShapingFactor can help remove bias when using the “Beven” approach to computing likelihood ratios. The default value is 0.5, corresponding to a root-mean-squared-error type of likelihood function.
• TelescopeRate: This variable is the fraction by which to constrict parameter bounds after each iteration. It can increase the acceptance rate by focusing the sampler on high-probability regions of the parameter space. The default value is 0 (i.e. no telescoping).
• MinWSSE: This variable is the calibrated (i.e. minimized) weighted sum of squared error objective function for the model. The default value is 1.00E308 (i.e. infinity). Users should replace this value with the results of a calibration algorithm.

References

Chen, Y. 2005. Another look at rejection sampling through importance sampling. Statistics & Probability Letters 72, 277-283.