  Statistical inference as a formal scientific method to covert experience to
knowledge has proven to be elusively difficult. While frequentist and Bayesian
methodologies have been accepted in the contemporary era as two dominant
schools of thought, it has been a good part of the last hundred years to see
growing interests in development of more sound methods, both philosophically,
in terms of scientific meaning of inference, and mathematically, in terms of
exactness and efficiency. These include Fisher's fiducial argument, the
Dempster-Shafe theory of belief functions, generalized fiducial, Confidence
Distributions, and the most recently proposed inferential framework, called
Inferential Models. Since it is notoriously challenging to make exact and
efficient inference about the Cauchy distribution, this article takes it as an
example to elucidate different schools of thought on statistical inference. It
is shown that the standard approach of Inferential Models produces exact and
efficient prior-free probabilistic inference on the location and scale
parameters of the Cauchy distribution, whereas all other existing methods
suffer from various difficulties.
