## 2.2 Subjectivity is not the key

The concepts of subjectivity and objectivity indeed characterize both statistical paradigms in differing ways. Among Bayesians there are those who are immersed in subjective rationality (, , , ), but others who adopt objective prior distributions such as Jeffreys’, reference, empirical or robust (, , , ) to operationalize Bayes’ rule, and thereby weight quantitative (data-based) evidence. Among Frequentists, there are choices made about significance levels which, if not explicitly subjective, are typically not grounded in any objective and documented assessment of the relative losses of Type I and Type II errors.14 In addition, both Frequentist and Bayesian statisticians make decisions about the form of the data generating process, or “model,” which - if not subject to rigorous diagnostic assessment - retains a subjective element that potentially influences the final inferential outcome. Although we all know that by definition a model is a schematic and simplified approximation to reality,

“Since all models are wrong the scientist cannot obtain a correct one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena.”

We also know that “All models are wrong, but some are useful” , that is why model diagnostics are important. This task can be performed in both approaches. Particularly, the Bayesian framework can use predictive p–values for absolute testing (, ) or posterior odds ratios for relative statements (, ). This is because the marginal density, conditional on data, is interpreted as the likelihood of the prior distribution .

In addition, what is objectivity in a Frequentist approach? For example, why should we use a 5% or 1% significance level rather than any other value? As someone said, the apparent objectivity is really a consensus . In fact “Student” (William Gosset) saw statistical significance at any level as being “nearly valueless” in itself . But, this is not just a situation in the Frequentist approach. The cut-offs given to “establish” scientific evidence against a null hypothesis in terms of $$log_{10}$$ scale or $$log_{e}$$ scale are also ad hoc.

Although the true state of nature in Bayesian inference is expressed in “degrees of belief,” the distinction between the two paradigms does not reside in one being more, or less, subjective than the other. Rather, the differences are philosophical, pedagogical, and methodological.

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1. Type I error is rejecting the null hypothesis when this is true, and the Type II error is not rejecting the null hypothesis when this is false.↩︎