More complicated extensions are associated with testing the conditions of encompassing. One of the necessary conditions for model P to encompass Q is that the economically relevant features (variance, confidence interval, quartile, etc) of the one-step forecast from P has to dominates the corresponding forecast from Q. How to test encomapsing of a combination need the knowledge of combining probability distributions. A relavant question is the combination of quantiles. By combining a pair of quantiles forecasts, for example the first quarter and the third quarter, to form a forecast interquantile range (Granger et al. 1989).
Combining probability distributions is more pertinent to the problem, which is, yet, basically ignored by economists but developed well by business and management school (Clemen 1989).In regards of no completely satisfactory combining technique in the literature, Granger proposed to a possible method. First, find the corresponding quantile function of each distribution function. And second, by inversing the combination of two quantile functions, to find a sensible combined function (Granger 1989).
4.2.3.1 Axiom approaches
However, on the topic of mathematical combination of probability distribution, it is inevitable to discuss axiom approaches, which focused on axiom-based aggregation
formulas (Clemen and Winkler 1999). Aggregation of probability distribution has been long developed in management science and risk analysis journals, two common approaches are ‘linear opinion pool’ and ‘logarithmic opinion pool’.
‘Linear opinion pool’ was proposed by Stone (1961) in the article “The opinion pool”, in which a weighted linear combination of the forecasters' probabilities had been proposed. It combines subjective probability distribution to get group consensus in a mathematical approach. Let ݂݅ሺߠሻrepresent the probability distribution for a parameter ߠ of subject ݅. A consensus of probability distribution, denoted as a single distribution
݂ሺߠሻ, can be written in a weighted average form. That is
݂ሺߠሻ ൌ ݓ݂ሺߠሻǡ
ݓ݄݁ݎ݁ݓ Ͳܽ݊݀ ݓ ൌ ͳǤ
Several weighting schemes were proposed for the method, including: simple average, weighted by ranking, weighted by self-rating, weighted according to the previous performance. Simply put, the weighs are determined subjectively (Clemen and Winkler 1999).
Another axiom approach is the logarithmic approach. The ‘logarithmic opinion pool’ is usually written using the geometric form, as
ς ݂௪ሺݕሻ
ς ݂௪ሺݕሻ ݀ݕǤ
With its own strength, logarithmic combination attracts scholars’ attention.
Logarithmic pooling method is convenient to manipulate. No matter first combine individual distributions, then update the combined distribution following Bayesian, or update individual distributions first, then combine, if with logarithmic pooling method, same results are derived; this property is said to satisfy the principle of external Bayesianity (Clemen and Winkler 1999, Wallis 2011).
4.2.3.2 Bayesian Approaches
Around 1980s, rising concerns about Bayesian approach shift attention from the axiomatic approach to the development of Bayesian combination models (Bunn 1989)4.
4 According to Bunn (1989), at the time he published Bunn (1975, 1977), there was an increasing acceptability of the “Bayesian approach to using multiple experts and different sources of evidence” (162), and this trend “reinforced the alternative idea of using multiple models for forecasting” (162).
Winkler (1968) and Morris (1974) have proposed a general Bayesian updating scheme to combine information and assess differential weights. Though some people give credit to Morris (1974) as the first establisher of Bayesian consensus model (Hall and Mitchell 2007), Winkler (1968) is probably the first researcher who proposed the primary framework. The Bayesian formwork was called "nature conjugate method", investigating the consensus of subjective probability distribution. He assumed that ݂ሺߠሻ represents a prior distribution, ߠ is the uncertain variable and ݅ is information, and defines Bayesian theorem in the form
݂ሺߠȁ݅ሻ ൌ ݂ሺߠሻ݈ሺ݅ȁߠሻ
݂ሺߠሻ ݈ሺ݅ȁߠሻ݀ߠǡ
where ݈ሺ݅ȁߠሻ is primitively interpreted as a sampling distribution or a likelihood function.
(Morris (1974)) enriched the Bayes' interpretation by decomposing the components of information z into two parts: one is from expert (denoted as݁) and another from decision maker (denoted as ݀ ). Decision makers' prior probability assessment onߠ , ݂ሺߠȁ݀ሻ , will be altered upon reception of expert's probability assessment on ߠ, ݃ሺߠȁ݁ሻ. The likelihood function ݈ሺǤ ሻ therefore explains how the decision maker subjectively feels about the credibility of the expert's probability assessment. The posterior probability distribution of decision maker can be write as
݂ሺߠȁ݃ሺߠȁ݁ሻǡ ݀ሻ ൌ ݂ሺߠȁ݀ሻ݈ሾ݃ሺߠȁ݁ሻȁ݀ሿ
݂ሺߠȁ݀ሻ݈ሾ݃ሺߠȁ݁ሻȁ݀ሿ݀ߠǡ
where ݂ሺߠȁ݀ሻ݈ሾ݃ሺߠȁ݁ሻȁ݀ሿ݀ߠ is the aggregation of the probability assessment of both decision maker and expert. Due to this sophisticated reinterpretation of Winkler (1968), (Morris (1974)) was credited as the first theoretical paper which is wholly consistent with the Bayesian view of probability. One thing notably is that Morris (1974)was published in the same journal, Management Science, as Winkler (1968), which indicates their inheriting relation.
Decisions in the face of uncertainty should be based on all available information, requiring combination of information obtained from models and experts; however, in the real world, due to common training and experiences of experts, the fact that experts have some sort of dependence is inevitable. With regard to the issue, Winkler (1981)
presents a theoretical model which formally allows dependence among experts without requiring a prior for particular form of consensus density function. Normal results were presented and the sensitivity to the degree of dependence was found in the consensus distribution.
Inspired by Winkler (1981), Agnew (1985) extended further the Bayesian consensus model to the case in which dependent experts provide probability assessment on multiple unknown parameters. Moreover, it developed Bayesian sequential updating procedure, which uses experts' past performance to determine weights in each period.
The literature extended but frustrates from practical difficulties to find the likelihood function. Because of this, effort has gone into the practical models for aggregating single probabilities and probability distributions (Clemen and Winkler 1999).
4.2.3.2.1 Bayesian combinations of event probabilities
For Bayesian combinations of event probabilities there are independence approach, Genest and Schervish approach, Bernoulli approach, and Normal approach (Clemen and Winkler 1999).
4.2.3.2.2 Bayesian models for combining probability distributions On the other front, there are Bayesian models which have been developed for combining probability distributions for continuous occurrence probability of a certain event.
The normal model has been important in this field. According to Liang and Shih (1994), the typical minimum-variance model for combining forecasts (Bates and Granger 1969, Newbold and Granger 1974) is consistent with the normal model (Winkler 1981, Bordley 1982). Moreover, a rewritten regression model (Granger and Ramanathan 1984) is equivalent to the normal model as well (Bordley 1986).In brief, I show the relations in Exhibit 5.
Exhibit 5 Connection between Bayesian Models for Combining Probability Distributions in the Normal settings and forecast combinations
By setting up the some necessary assumption, Bayesian combinations of probability are equivalent to forecast combination (Liang and Shih 1994). The Bayesian model in Winkler (1981) is equivalent to Newbold and Granger (1974) by assuming Normal distributed prior, Normal distributed likelihood and location invariant.
Following Winkler (1981), the Bayesian model in Bordley (1982) also is in equivalence to Newbold and Granger (1974) by assuming uniform distributed prior, Normal distributed likelihood, known variance-covariance matrix, location invariant.
Although the prior in Bordley is not normal, but since the prior is unimodal and symmetric5, this is generally not a problem (Cleman and Winkler 1999). Besides, the mean of posterior density in (Bordley 1986) can be equivalent to the rewritten regression model in Granger and Ramanathan (1984) by assuming Normal distributed prior and Normal distributed likelihood. Summary is provided in Table 2.
Table 2 Bayesian equivalence of the theories of forecast combination
The theoretical evolution continues. Anandalingam and Chen (1989) generalized results of Winkler (1981), Bordley (1982, 1986), deriving their models respectively under different conditions. Liang and Shih (1994) relaxed further the assumption of unbiased decision maker’s prior.
5 Even if the unimodal prior is just roughly symmetric, that would not be a problem (Clemen and Winkler
Although the normal model has been popular, it has some shortcomings, the obvious one is that a normal prior is required. As a consequence, several extensions are proposed (Clemen and Winkler 1999).