The recently developed approach of Bayesian predictive synthesis (BPS: McAlinn and West,2018; McAlinn et al.,2018) is explicitly founded on subjective Bayesian principles and theory, and defines an over-arching framework within which many existing density (and other) combination “rules” can be recognized as special cases. Critically, this pro- vides opportunity to understand the implicit Bayesian assumptions underlying special
∗Booth School of Business, University of Chicago,[email protected] †Norges Bank, BI Norwegian Business School,[email protected] ‡Department of Statistical Science, Duke University,[email protected]
cases. BPS links to past literature on subjective Bayesian “agent/expert opinion analy- sis” (e.g. Genest and Schervish,1985; West and Crosse,1992; West,1992) and provides a formal Bayesian framework that regards predictive densities from multiple models (or individuals, agencies, etc) as data to be used in prior-posterior updating by a Bayesian observer (see also West and Harrison, 1997, Sect 16.3.2). The approach allows for the integration of other sources of information and explicitly provides the ability to deal with M-incompleteness. A main theoretical component of BPS is a general theorem describing a subset of Bayesian analyses showing how densities can be “synthesized”. Special cases include traditional BMA, most existing forecast pooling rules, and– in terms of theoretical construction– the stacking approach in the article.
In McAlinn and West (2018) and McAlinn et al. (2018) BPS is developed for time series forecasting where the underlying Bayesian foundation defines a class of dynamic latent factor models. The sequences of predictive densities define time-varying priors for inherent latent factor processes linked to the time series of interest. BPS is able to learn and adapt to the biases, aspects of mis-calibration, and– critically– inter-dependences among predictive densities. A further practical key point is that BPS can– and should– be defined with respect to specific predictive goals; this is a point of wider import presaged in the earlier Bayesian macroeconomics literature and illustrated in McAlinn and West (2018) and McAlinn et al. (2018) through separate forecast combination models for multiple different goals (multiple-step ahead forecasting). Applications in macroeconomic forecasting in these papers demonstrate that a class of proposed BPS models can significantly improve over conventional methods (including BMA and other pooling/weighting schemes). Further, as BPS is a fully-specified Bayesian model within which the information from each of the sources generating predictive density are treated as (complicated) “covariates,” posterior inferences on (time-varying or otherwise) pa- rameters weighting and relating the sources provides direct access to inferences on their biases and inter-dependencies.
It is of interest to consider how the current stacking approach relates to BPS through an understanding of how the resulting rule for predictive density combination can be interpreted in BPS theory (see equation (1), and the discussion thereafter, in McAlinn and West2018). As with other combination rules, an inherent latent factor interpreta- tion is implied and this may provide opportunity for further development. In related work with BPS based on mixture models, Johnson and West (2018) highlight the oppor- tunities to improve both resulting predictions and generate insights about the practical impact of model inter-dependencies that are largely ignored by other approaches. This can be particularly important in dealing with larger numbers of predictive densities when the underlying models generating the densities are known or expected to have strong dependencies (a topic touched upon in Section. 5.3 in the article).
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