Bayesian Estimate of Mineral Distributions in Crushed Ore
Vladimir Nedeljkovic and Neil Pendock
Department of Computational and Applied Mathematics
University of the Witwatersrand
Johannesburg, South Africa
Abstract
Ore may be classified into various size classes of rock. The
distribution of a mineral of interest is typically not uniform over
all size classes. To extract the mineral, the ore is crushed and one
question of interest is ``what is the distribution of the mineral in
the crushed ore?'' Four positive additive distributions are involved:
- P = {p_i} the size class distribution for the
coarse (uncrushed) ore
- Q = {q_i} the size class distribution for the
fine (crushed) ore
- U = {u_i} the mineral distribution in the coarse
ore
- V = {v_i} the mineral distribution in the fine
ore
The first three distributions are known and we wish to use them to
infer the V distribution. Distributions
Q and P and V and
U are related by a transfer matrix A
with entries a_{kj} determining the proportion of
size classes k that moves to the size class
j in the crushing process (Q = AP and
V = AU). In spite of some obvious constraints on
A (for example a_{kj} = 0 if j
> k and \sum_k a_{kj} = 1 for all
j), A is underdetermined by the
available data. Various realizations of V may be
generated by varying A for fixed P,
Q and U. Bayesian methods provide us
with a means for generating honest estimates for V. We
start by choosing an entropic prior distribution for U
in terms of P and for V in terms of
Q and an exponential likelihood using Jaynes'
principle of maximum entropy since the measured data are means from
the U distribution. The posterior distribution for
V given U, P and
Q is then the product of the various prior and
likelihood distributions according to Bayes' theorem.
We sample the inference distribution for V using
simulated annealing and compare our solutions to those from a linear
programming formulation and a least-squares solution, for several test
data sets.
MaxEnt 94 Abstracts / mas@mrao.cam.ac.uk