Lecture_23_24 - Model Selection and Model Averaging in...

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Model Selection and Model Averaging in Phylogenetics Johan Nylander November 27, 2005 1 Key Concepts Model selection - to use the data to select a model - should be an integral part of inference [4]. The data generating or ”true” model ( f ) has an inFnite number of parameters and is un- reachable. The best approximate model ( g ): best descriptive model given the limited sample size. ±ind- ing the best g is (or can be) the goal of model selection. A more parameter-rich model has a higher potential than a less parameter rich model: less discrepancy due to approximation . However, a more parameter-rich model tends to perform farther below its potential than a less parameter rich model caused by the discrepancy due to estimation [20]. Parsimonious trade o² between error (decreases with additional parameters) and variance (increases with additional parameters). To help us with the trade o²: apply a model selection criterion. 2 Model Selection Criteria in Phylogenetics 2.1 Likelihood Changing the model changes the likelihood - (which is proportional to) the probability of data, given the parameters and the model [5]. 1
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Computational Evolutionary Biology Maximized log Likelihood is biased upward as an estimator of the target model. The bias is proportional to the number of parameters [4]. 2.2 Likelihood Ratio Testing [7] δ = - 2(ln L 0 - ln L 1 ) Basic idea: Is the increase in likelihood signi±cant? δ is asymptotically χ 2 n distributed, with df n , the di²erence in number of free parameters between models. Only for nested models (model L 0 must be a special case of L 1 ). Mixed χ 2 n distributions when one parameter is in its limit (e.g., GTR vs. GTR+G) [19]. Applications: Modeltest [13], MrModeltest2 [15] 2.3 AIC - Akaike Information Criterion [1, 4] AIC i = - 2 ln( L ) + 2 p L : Max. log Likelihood for model i , p : number of parameters. Estimates the expected Kullback-Leibler (K-L) distance: information lost when model g is used to approximate f . Min AIC is the best K-L model in the set of competing models. No accept or reject (not a strict test). Applies to nested and non-nested models. AIC c - takes sample size in to account. Must be based on the maximum likelihood - problematic(?).
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Lecture_23_24 - Model Selection and Model Averaging in...

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