## Using Bayesian model averaging to calibrate forecast ensembles. Monthly Weather Review 133 (2005)

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@MISC{Raftery05usingbayesian,

author = {Adrian E. Raftery and Tilmann Gneiting and Fadoua Balabdaoui and Michael Polakowski},

title = {Using Bayesian model averaging to calibrate forecast ensembles. Monthly Weather Review 133},

year = {2005}

}

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### Abstract

Ensembles used for probabilistic weather forecasting often exhibit a spread-error correlation, but they tend to be underdispersive. This paper proposes a statistical method for postprocessing ensembles based on Bayesian model averaging (BMA), which is a standard method for combining predictive distributions from different sources. The BMA predictive probability density function (PDF) of any quantity of interest is a weighted average of PDFs centered on the individual bias-corrected forecasts, where the weights are equal to posterior probabilities of the models generating the forecasts and reflect the models ’ relative contributions to predictive skill over the training period. The BMA weights can be used to assess the usefulness of ensemble members, and this can be used as a basis for selecting ensemble members; this can be useful given the cost of running large ensembles. The BMA PDF can be represented as an unweighted ensemble of any desired size, by simulating from the BMA predictive distribution. The BMA predictive variance can be decomposed into two components, one corresponding to the between-forecast variability, and the second to the within-forecast variability. Predictive PDFs or intervals based solely on the ensemble spread incorporate the first component but not the second. Thus BMA provides a theoretical explanation of the tendency of ensembles to exhibit a spread-error correlation but yet

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- 1977
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- 1995
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- 1997
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- 1998
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- 1995
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- 1922
(Show Context)
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- 1993
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- 1978
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- 1999
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- Glahn, Lowry
- 1972
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- 1997
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- Anderson
- 1996
(Show Context)
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- 1997
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- Epstein
- 1969
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- 1997
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- 1995
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- 1999
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- 1998
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- Carter, Dallavalle, et al.
- 1989
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- 1993
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Generalized Linear Models. 2d ed
- McCullagh, Nelder
- 1989
(Show Context)
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- Raftery, Zheng
- 2002
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- Leith
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- 1996
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