Any expectation and credal interval depends on the exact prior that is chosen. In addition, there is a well-known, but no less grave drawback to the way in which the Bayesian conclusions are reached: we have to assume a prior probability assignment over the statistical hypotheses. As indicated in the discussion of Neyman-Pearson hypothesis testing, this difference remains problematic. In the Bayesian procedure, by contrast, the choice is expressed in a posterior probability assignment over the set of hypotheses. One difference is that in parameter estimation, and in classical statistics more generally, the choice for some hypothesis is an all-or-nothing affair: we accept or reject, we choose a single best estimate, and so on. Of course there are also large differences between the results of parameter estimation and the results of a Bayesian analysis. By contrast, a credal interval does allow for such an inferential reading. The latter only expresses how far an estimate is off the mark on average, while it does not warrant an inference about how far away the specific estimate that we have obtained, lies with respect to the true value of the parameter. We might argue that this expression is an improvement over the classical confidence interval of Equation (17). This set of values for θ is such that the posterior probability of the corresponding h θ jointly add up to 1−ε. Thus, for each set of possible observations, | s t ) = 1 − ε. And according to Equation (18) these relative frequencies are also the maximum likelihood estimators. In the Carnapian prediction rule of Equation (4), choosing λ = 0 will yield the observed relative frequency as predictions. Let me make this concrete by means of the example on red and green pears. So we can present the latter as a probability assignment over sample space, from which estimations can be derived by a non-ampliative inference. The estimation function θ ˆ by Fisher is thereby captured in a single probability function P. Where P θ ˆ ( s t ) refers to the probability function induced by the hypothesis h θ ˆ ( s t ). P ( q t + 1 k | s t ) = P θ ˆ ( s t ) ( q t + 1 k ) ,
0 Comments
Leave a Reply. |