3.3.1 Probability statements and the role of data
In frequentist (“classical”) statistics, dataD are compared to a modelM using the prob- ability of observing the data given a particular model, p(D|M), and models are fitted to data by maximising this likelihood. The actual observed values D are seen as one among many po- tential values of D, and to infer the adequacy of model M in explaining the data frequentist statistics relies on tools such as hypothesis testing and p-values, which build on the concept of repeatability of observing the data D given M. That is, by repeatedly assuming certain fixed values for model parameters and calculating the frequency of observing dataD under repeated,
identical circumstances, a frequentist judges whether a model using these fixed values adequately represents the observed data D (Efron 1986). In chapter 1 of his textbook, D’Agostini (2003) reviews these fundamental assumptions of frequentist statistics in more detail.
In Bayesian statistics, a probability is a mathematical representation of the degree of belief in an event happening or a statement being true. In Bayesian statistics, data D are compared to a set of K models by calculating, for each modelk, the probability p(Mk|D) that the model Mk is true, given the observed dataD. Bayes’ theorem
ppost(Mk|D) = pprior(Mk)×p(D|Mk) Pk j=1p(D|Mj)p(Mj) (3.1) = pprior(Mk)×p(D|Mk) p(D)
is used to combine the prior subjective belief in model Mk and the likelihood p(D|Mk) of
observing D according to model Mk. p(D) is a normalising constant. Bayes’ theorem updates
a prior subjective belief in the truth of Mk to give a posterior subjective belief in the truth of Mk that is consistent with the observed data D. In Bayesian statistics, modelMk is treated as
unknown whereas the dataDare taken as fixed and known. Bayes’ theorem states the posterior probabilities of Mk directly. Because Bayesian statistics gives the subjective probability of a particular modelMk(Efron 1986), no further tools need to be invoked to draw conclusions about
Mk. Howson (1997) gives an extensive introduction to the meaning of “probability” in Bayesian
statistics, and many textbooks devote a chapter to defining the foundations of Bayesian statistics (e. g. chapter 2 in D’Agostini (2003)).
3.3.2 Prior information
Conceptually, the most striking advantage of the Bayesian paradigm and also its greatest weak- ness is the use of prior knowledge. In a Bayesian model, prior information about all quantities of interest is combined with the observed data to give an updated posterior (e. g. Punt and Hilborn (1997)). Models thus no longer exist in isolation from previous studies (Prato 2005). Instead, Bayesian modelling explicitly represents the scientific process of repeatedly collecting data and updating the level of knowledge about a system. Accordingly, sample size of each single experiment becomes less of an issue (Ellison (2004), Ghazoul and McAllister (2003)). The ability of a Bayesian model fit to combine old and new data (Ellison 2004) makes Bayesian modelling particularly suitable for adaptive management (Prato 2005), i. e. management in which model results are reviewed whenever new data become available.
Additionally, where disagreement exists about the admission of prior evidence or where stake- holders have different prior opinions, a Bayesian model can easily be calculated for each prior in turn (D’Agostini 2003). This exchangeability of prior information provides different stakeholders with an established method for calculating the posterior opinions that they should rationally subscribe to, given their own priors and the common data that enter a model. Reconcilia- tion between different stakeholders’ points of view can thus centre on the posterior beliefs of each stakeholder, which are based on common data and thus less divergent than their prior beliefs. The exchangeability of prior information in Bayesian models thus introduces additional transparency into the modelling process and can facilitate agreement between stakeholders. However, with its use of prior information, a Bayesian model fit is based not only on the available data, but also on the choice of prior. For this reason many scientists see Bayesian statistics as being at odds with the principle of scientific objectivity (Howson 1997), whereas the frequentist model is free from subjective judgments (apart from the decision on which hypotheses to test (Ellison 1996), which is a subjective decision common to all scientists).
Among Bayesian statisticians, different points of view prevail on this issue of objectivity. Sub- jective Bayesiansare happy to dismiss the idea of objectivity and welcome the fact that priors are essentially subjective judgments that may be employed to enter prior knowledge into the model, for example where earlier studies or common knowledge can inform a subset of the pa- rameters in the current model, or where experts have been asked for their opinions on particular model quantities. Objective Bayesians challenge the interpretation of Bayesian probabili- ties as subjective by defining reference priors that avoid the need of subective judgments (e. g. George and McCulloch (1993)). In this approach, criteria of objectivity are used to choose a particular prior (e. g. Jeffreys priors, see Box and Tiao (1973)). Usually, these criteria include the resilience of a prior to the scale of the model parameters. An objective prior should not influence model fit, regardless of the parameterisation chosen for a model. Another criterion is that a reference prior should favour simpler models over more complex ones, if both describe the data adequately.
Howson (1997) shows that the objective Bayesian approach does not fully answer the objection against subjectivity in Bayesian models, because reference priors that have been defined for common models often fail to result in proper prior distributions and thus do not guarantee that the posterior model distributions will exist. While it is possible to use reference priors as a starting-point even when these are improper, the need to approximate them by proper prior distributions (for example by truncating their support to a range of parameter values that are deemed “realistic”) involves a subjective judgement that makes them no longer objective (Howson 1997). Efron (1986) claims that “subjective Bayes” is the only philosophically coherent standpoint within the realm of Bayesian statistics.
3.3.3 Model specification
Bayesian methods allow more flexibility in the model design and specification (Punt and Hilborn 1997). For example, Bayesian model fitting is capable of fitting overspecified models (Nielsen and Lewy 2002), which is very difficult to do in frequentist statistics (Omlin and Reichert 1999). Bayesian methods also facilitate the use of complex models that are designed and fine-tuned for each individual purpose, whereas frequentist teaching can look back on a long history of well- established models. The novelty of many Bayesian models and the trade-off between complexity and transparency thus often makes Bayesian models appear opaque to non-Bayesians. However, whereas the flexibility of designing Bayesian models to fit each individual application is an advantage compared to off-the-shelf frequentist methods, “Bayesian theory requires a great deal of thought about the given situation to apply sensibly” (Efron 1986), and this can be seen as a hindrance. Because Bayesian models are often highly complex, they are also more computer-intensive to analyse, which can impede the verification and independent review of model implementation and make some Bayesian models computationally too expensive to be useful (Punt and Hilborn 1997).
Structural uncertainty can be addressed by model selection and model averaging. Model selection aims to identify one model that best explains the data, whereas model averaging is a method for assigning weights to different models based on how well they fit the data. Within the Bayesian paradigm, a good overview of methods in model selection and model averaging is Hoeting et al. (1999), whereas a good reference text for addressing structural uncertainty in frequentist statistics is Burnham and Anderson (1998). Model averaging over all competing models results in better average predictive ability than using only the “best” model for making predictions (Hoetinget al. 1999).
In Bayesian model averaging, the posterior likelihood of each potential model is used to weight the contribution of this model in making predictions (Draper (1995), Hoeting et al. (1999)). Frequentist statistics does not assign probabilities to a model, because in frequentist statistics it is the data that are assumed to be random, not the model (see section 3.3.1). Thus, in frequentist statistics the weights used to average competing models need to be taken from other goodness-of-fit measurements, such as the AIC (see Bucklandet al. (1997)). This leads Hodges (1987) to conclude that structural uncertainty can be addressed more straightforwardly in a Bayesian modelling context than in a frequentist one.
3.3.4 Presenting the model results
According to the theoretical distinctions between Bayesian and frequentist statistics (see section 3.3.1), statements about the probability of particular events occurring or of model parameters taking values in a particular interval can be made immediately within the Bayesian paradigm, but to express model results within the frequentist paradigm the assumption on repeatability and the use of hypothesis testing andp-values, with associated levels of significance, are required. Output on a particular quantity (a rate of reproduction m, say) from a Bayesian model is sum- marised in a posterior distribution, which can be presented graphically or which can be used to make statements such as “There is a probability of 81% that m <2”, and the meaning of such statements is clear to anyone who understands the concept of subjective probability. Following a frequentist modelling approach, however, results on the same quantitymcan only be expressed by statements such as “At a 95% level of significance, a confidence interval for m is [1.8, 5]”, and to make sense of this information, knowledge of both the frequentist interpretation of prob- ability (as the frequency of observing an event under repeated, identical conditions) and of the means of a particular level of significance is required. Likewise, any graphical representation of output from a frequentist model has to be understood within the frequentist concept of proba- bility, whereas a graph of a Bayesian posterior distributions immediately gives the probability distribution of interest.