2.6 Model Comparison and Parameter Estimation
2.6.3 Computing Best-Fit Parameters: Galapagos
To find a best-fit GRB afterglow model, it is necessary to explore a rather large param- eter space. The full emission, absorption and extinction model contains well over 50 parameters. Most of these, however, are either constrained by priors on the parameter values themselves, or by empirical relationships by which a parameter is a function of one or more other parameters. These priors considerably reduce the volume of the parameter space we must explore. Our most general afterglow emission, extinction and absorption model consists of only 10-13 truly free parameters: 7-8 free emission param- eters, plus 3-5 free parameters describing the source frame dust extinction and neutral
and molecular hydrogen absorption (see Chapter 3). Still, it is a daunting challenge to explore this parameter space completely and efficiently. The best tool we have found is the genetic algorithm, implemented in our model-fitting software package, Galapagos. We have found genetic algorithms to be much more computationally efficient at ex- ploring large and complex parameter spaces, and at avoiding local minima, than other stochastic optimization schemes (e.g., Markov-Chain Monto-Carlo/simulated anneal- ing heuristics), as well as being easily scalable to parallel-processor machines (Foster & Reichart 2011, in preparation).
The user initially provides Galapagos with the observed data, a set of priors, and an initial “search box”, or guess of the numerical range that the value of each parameter may take. A single list of parameter values, or “genes”, is called a “chromosome”, and a given model parameterization, or “organism”, consists of one or more chromosomes.
Galapagos creates a set, or “population” of organisms, each of whose genes are selected randomly from the initial parameter search box ranges. The “fitness” (i.e., −2 lnL) of each organism is computed, and the population is sorted into a list of most- to least-fit. Then, some fraction (usually half) of the least fit organisms in the population is “killed off”; the remaining organisms are allowed to “breed”, randomly exchanging genes, creating a new population of “offspring” organisms to replace those that were killed off. The process continues: each generation’s organisms breed and their fitnesses are ranked, with the least fit being killed off and the most fit allowed to breed again, until the fitnesses of the organisms in the population differ from one another only by some small value – typically, this results in a population of identical organisms, or “clones”. Since the true best-fit genes may not have been created in the original, randomized population – or, if they were, may not have survived the selection process – a new randomized population is generated, with search boxes recentered on the previous step’s best-fit parameter values (the search box is either reduced or expanded by some
factor from its previous width, depending on how close the best-fit parameter value is to the center of the previous search box). The selection and breeding process begins again, until a new best-fit/clone population is reached. The entire process iterates until the fitness of successive best-fit/clone populations converges on the true global minimum of−2 lnL, providing the best-fit parameter values.
It is often the case that a single set of model parameter values is inadequate to describe the entire range of the observed data. In the case of GRB afterglows, this most often occurs when emission or absorption characteristics change discontinuously with time (frequently during a gap between two nights’ observations). For this reason, we refer to subsets of the data, together with their list of model parameter values, as “time slices” (though, generally speaking, the data can be subgrouped in any number of ways, according to criteria other than time). Galapagos has the flexibility to allow any number of time slices, and can constrain, or “link”, certain parameters so that they always have the same value across all time slices (or among any arbitrary subset of time slices), while allowing other parameters to vary freely from slice to slice. In the genetic language introduced above, each time slice is a single chromosome in a multi-chromosome organism. For example, the redshift of the burst, zGRB, will always be linked across all time slices, as will parameters describing the Lyαforest absorption and Milky Way dust extinction along the burst’s line of sight. But it may be that the burst experiences, say, a discontinuous re-brightening due to energy injection, or that absorbing material local to the burst is modified over the duration of the after- glow, in which case we may want to allow certain emission, extinction and absorption parameters to vary independently from slice to slice. For example, in the analysis of the afterglow of GRB 090313 in §4, we will explore the relative likelihood of models for which different subsets of the model parameters are linked between the first and second night of observation and find, in this case, that the observed discontinuity in
-3 -2 -1 0 1 -8
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Figure 2.20 GRB 090313 best-fit light curve, with two time slices. The two time slices are divided at logt = −0.7. Solid curves show model fits to data in the first time slice, dashed curves to data in the second time slice, for each photometric bandpass, from X-ray (violet) to NIR K-band (brown). All model parameters except for those describing intrinsic flux normalization are linked across both slices. See §4 for more details.
brightness is better explained by an intrinsic rebrightening than by variable extinction or absorption (Figure 2.20).