# 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