Abstract :
We present a new test of hypothesis in which we seek the probability of the null conditioned
on the data, where the null is a simplification undertaken to counter the intractability of
the more complex model, that the simpler null model is nested within. With the more complex
model rendered intractable, the null model uses a simplifying assumption that capacitates the
learning of an unknown parameter vector given the data. Bayes factors are shown to be known
only up to a ratio of unknown data-dependent constants–a problem that cannot be cured using
prescriptions similar to those suggested to solve the problem caused to Bayes factor computation,
by non-informative priors. Thus, a new test is needed in which we can circumvent Bayes
factor computation. In this test, we undertake generation of data from the model in which the
null hypothesis is true and can achieve support in the measured data for the null by comparing
the marginalised posterior of the model parameter given the measured data, to that given such
generated data. However, such a ratio of marginalised posteriors can confound interpretation
of comparison of support in one measured data for a null, with that in another data set for a
different null. Given an application in which such comparison is undertaken, we alternatively
define support in a measured data set for a null by identifying the model parameters that are
less consistent with the measured data than is minimally possible given the generated data,
and realising that the higher the number of such parameter values, less is the support in the
measured data for the null. Then, the probability of the null conditional on the data is given
within an MCMC-based scheme, by marginalising the posterior given the measured data, over
parameter values that are as, or more consistent with the measured data, than with the generated
data. In the aforementioned application, we test the hypothesis that a galactic state space
bears an isotropic geometry, where the (missing) data comprising measurements of some components
of the state space vector of a sample of observed galactic particles, is implemented to
Bayesianly learn the gravitational mass density of all matter in the galaxy. In lieu of an assumption
about the state space being isotropic, the likelihood of the sought gravitational mass
density given the data, is intractable. For a real example galaxy, we find unequal values of the
probability of the null–that the host state space is isotropic–given two different data sets, implying
that in this galaxy, the system state space constitutes at least two disjoint sub-volumes
that the two data sets respectively live in. Implementation on simulated galactic data is also
undertaken, as is an empirical illustration on the well-known O-ring data, to test for the form
of the thermal variation of the failure probability of the O-rings.
