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== Description == Bayesian Methods cover a very broad class of methods based on Bayes equation in which the posterior probability of an outcome is proportional to the product of a prior probability and a likelihood measure evaluated with respect to some observation. Bayesian statistics is now the dominant paradigm in statistical inference, even for cases where prior probability distributions are poorly known and subjectively chosen. Bayes equation also provides a method of sequential updating as more observational data become available (see [Sequential Monte Carlo methods]). The [Generalized Likelihood Uncertainty Estimation (GLUE)] methodology can also be interpreted as a Bayesian method when used with a formal likelihood measure (noting that Bayes equation does not preclude the use of subjective likelihood measures, though the resulting posterior will then not provide a formal estimate of the probability of predicting an observed value conditional on the model). For simple cases (e.g. Gaussian prior distributions and a likelihood based on a Gaussian error model) Bayes equation can be applied analytically: for more complex cases, particularly where prior distributions about model inputs and parameters are processed by a nonlinear model function, approximate numerical methods must be used to calculate the posterior distribution. This is essentially a problem of estimating the shape of the likelihood surface in the multiple dimensions of the uncertain variables (model structures and parameters and/or inputs). Much current work on evaluating such likelihood surfaces is based on Monte Carlo Markov Chain or Importance Sampling techniques. To speed up this process for models that have very large numbers of parameters or long run times, there is research on model emulation techniques under development (e.g.Oakley and O'Hagan, 2004; Ratto et al., 2004; Van Oijen et al., 2005) The two critical components of a Bayesian method are the choice of the prior distributions and the choice of a likelihood function or measure. The likelihood function is usually based on a model of the error structure, which might include a component to allow for model structural error (e.g. Kennedy and O'Hagan, 2000). Given valid assumptions about the nature of the errors, application of Bayes equation will then provide an estimate of the probability of predicting an observation conditional on the model, which can be presented as prediction quantiles for an predicted variable. As more data become available, continued application of Bayes equation should refine these estimates. == Software == none == Advantages == The method results in an estimate of the probability of predicting an observation conditional on the model, if the error assumptions are correct. Given a residual error series, these assumptions can be tested. Integrating over the likelihood surface can also give marginal distributions for particular parameter values. Multiple model structures can be included within the Bayesian framework. == Disadvantages == The method requires prior distributions to be specified for all uncertain quantities, which are often poorly known (especially for "effective" parameter values required by a particular model structure). The method is often applied as if the input data and model were correct or with unrealistic assumptions about the nature of the residual errors. This can lead to overconfidence in the estimates of parameter values, or to model errors being compensated by large residual variances (see, for example, the application of a sequential Bayesian methodology to the calibration of a hydrological model in the paper of Thiemann et al. (2001) and the resulting discussion by Beven and Young(2003). In some circumstances it may be difficult to define an appropriate likelihood function, because the nature of the errors is complex (e.g. showing non-Gaussian behaviour, changing form of distribution, or non-stationary variances). Some of these problems might be overcome by suitable transformations, where, for example, non-stationarity can be related to some other variable. An exciting discussion on the subjectivity of Bayesian analysis has been published in [http://ba.stat.cmu.edu/vol01is03.php Bayesian Analysis, Vol 3(3), 385-472] == Case studies == * [Estimating design discharges (case study)] == References and Further Reading == Bernardo, J.M. and Smith, A.F.M., 1994. Bayesian theory. Wiley, Chichester, xiv,586p. pp. Beven, K.J. and Young, P., 2003. Comment on "Bayesian recursive parameter estimation for hydrologic models" by M. Thiemann, M. Trosset, H. Gupta, and S. Sorooshian. Water Resources Research, 39(5): art. no.-1116. Congdon, P., 2003. Applied Bayesian modelling. Wiley series in probability and statistics. Wiley, Chichester, xii, 457 p. pp. Howson, C. and Urbach, P., 1993. Scientific Reasoning: The Bayesian Approach. Open Court, Chicago, IL, 470 pp. Kennedy, M.C. and O'Hagan, A., 2000. Predicting the output from a complex computer code when fast approximations are available. Biometrika, 87(1): 1-13. Lee, P.M., 2004. Bayesian statistics: an introduction. Arnold, London, 352 p. pp. O'Hagan, A., 1994. Bayesian methods in Asset Management. In: V. Barnett and K.F. Turkman (Editors), Statistics for the Environment 2: Water Related Issues. Wiley & Sons, Chichester, pp. 235-248. Oakley, J.E. and O'Hagan, A., 2004. Probabilistic sensitivity analysis of complex models: a Bayesian approach. Journal Of The Royal Statistical Society Series B-Statistical Methodology, 66: 751-769. Ratto, M., Saltelli, A., Tarantola, A. and Young, P., 2004. Improved and accelerated sensitivity analysis using State Dependent Parameter models, Sensitivity Analysis of Model Output, Santa Fe, New Mexico, March 8-11. Romanowicz, R., Beven, K.J. and Tawn, J., 1994. Evaluation of predictive uncertainty in nonlinear hydrological models using a Bayesian approach. In: V.T. Barnett, K.F. (Editor), Statistics for the Environment. II. Water Reslated Issues. John Wiley, Chichester, pp.?-? Thiemann, M., Trosset, M., Gupta, H. and Sorooshian, S., 2001. Bayesian recursive parameter estimation for hydrologic models. Water Resources Research, 37(10): 2521-2535. Van Oijen, M., Rougier, J. and Smith, R., 2005. Bayesian calibration of process-based forest models: bridging the gap between models and data. Tree Physiology, 25(7): 915-927. From unknown Fri Jun 22 04:32:20 +0100 2007 From: Date: Fri, 22 Jun 2007 04:32:20 +0100 Subject: Bayesian linear regression Message-ID: <20070622043220+0100@www.floodrisknet.org.uk>
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