Expanding From Discrete To Continuous 
Estimation Of Distribution Algorithms: 
The IDEA 
Peter A.N. Bosman and Dirk Thierens 
Department of Computer Science, Utrecht University, 
P.O. Box 80.089, 3508 TB Utrecht, The Netherlands 
fpeterb, Dirk.Thierensg@cs.uu.nl 
Abstract. The direct application of statistics to stochastic optimiza- 
tion based on iterated density estimation has become more important 
and present in evolutionary computation over the last few years. The 
estimation of densities over selected samples and the sampling from the 
resulting distributions, is a combination of the recombination and muta- 
tion steps used in evolutionary algorithms. We introduce the framework 
named IDEA to formalize this notion. By combining continuous proba- 
bility theory with techniques from existing algorithms, this framework 
allows us to dene new continuous evolutionary optimization algorithms. 
1 Introduction 
Algorithms in evolutionary optimization guide their search through statistics 
based on a vector of samples, often called a population. By using this stochastic 
information, non{deterministic induction is performed in order to attempt to 
use the structure of the search space and thereby aid the search for the optimal 
solution. In order to perform induction, these samples are combined so as to gen- 
erate new solutions that will hopefully be closer to the optimum. As this process 
is iterated, convergence is intended to lead the algorithm to a nal solution. 
In the genetic algorithm [11, 14] and many variants thereof, values for problem 
variables are often exchanged and subsequently individually adapted. Another 
way of combining the samples is to regard them as being representative of some 
probability distribution. Estimating this probability distribution and sampling 
more solutions from it, is a global statistical type of inductive iterated search. 
Such algorithms have been proposed for discrete spaces [2{4, 12, 13, 15, 17, 19, 21], 
as well as in a limited way for continuous spaces [5, 10, 15, 22, 23]. An overview 
of this eld has been given by Pelikan, Goldberg and Lobo [20]. 
Our goal in this paper is to apply the search for good probability density 
models to continuous spaces. To this end, we formalize the notion of building 
and using probabilistic models in a new framework named IDEA. We show how 
we can adjust existing techniques to be used in the continuous case. We thereby 
dene new evolutionary optimization algorithms. Using a set of test functions, 
we validate their performance. 

The remainder of this paper is organized as follows. In section 2, we present 
the IDEA framework. In section 3 we describe a few existing algorithms that 
build and use probabilistic models. In section 4 we state some derivations of 
probability density functions (pdfs). We use the described algorithms and pdfs 
within the IDEA in our experiments in section 5. Topics for further research are 
discussed in section 6 and our nal conclusions are drawn in section 7. 
2 The IDEA 
We write a = (a 0 ; a 1 ; : : : ; a ja the 
elements in a vector is relevant. We assume to have l random variables available, 
meaning that each sample point is an l dimensional vector. We introduce the 
notation ahci = (a c 0 ; a c 1 ; : : : ; a c jc 
vector of l 
numbers and let = (Z 0 ; Z 1 ; : : : ; Z l 
h i), we 
might as well assume a uniform distribution over . Now denote a probability 
distribution that is uniformly distributed over all zh i with C(zh i)   and 
that has a probability of 0 otherwise, by P  ( ). In the discrete case we have: 
P  ( )(zh i) =