School of Computing. Dublin City University.
Let us have n items. The "fitness" score of item i is f(i). Let the probability of picking item i, p(i), be defined as:
Then by varying the parameter T, we can vary the selection from picking a random item (T infinite) to: having higher probabilities for items with higher fitness (T small finite) to: strictly picking the item with the best fitness (T goes to 0).
T is called "temperature" - analogy to cooling. See Simulated Annealing.
T exactly = 0 will cause divide by 0 error.
Also, in software:
Have to figure out safe T ranges and f ranges for your software system.
We will be keeping T > 0 and looking at T -> 0
If T > 0, and all f(i) are finite (positive or negative),
then f(i)/T is finite (positive or negative).
all e(f(i)/T) > 0
all probabilities are in range:
0 < p(i) < 1
i.e. This is a probability distribution.
As T -> 0
a probability can -> 0
but as long as T > 0 all probabilities are > 0
If there are m joint best items, we pick them with probability 1/m, all others with probability 0.
To see what happens as T -> 0 consider 2 items:
i 1 2 f(i) 1 2If T=1, probabilities are:
If T=1/20, probabilities are:
e20 / (e20 + e40)
e40 / (e20 + e40)
i.e. pretty much 0 and 1.
Let us have any two items where one has a higher fitness value:
i 1 2 f(i) c dwhere d > c
p(1) = ecN / ( ecN + edN )
= 1 / ( 1 + e(d-c)N )
as N → ∞
since (d-c) > 0
So p(1) -> 0 as T -> 0