Multi-parameter optimization in R - r

Question

I'm trying to estimate parameters that will maximize the likelihood of a certain event. My objective function looks like that:
event_prob = function(p1, p2) {
x = ((1-p1-p2)^4)^67 *
((1-p1-p2)^3*p2)^5 *
((1-p1-p2)^3*p1)^2 *
((1-p1-p2)^2*p1*p2)^3 *
((1-p1-p2)^2*p1^2) *
((1-p1-p2)*p1^2*p2)^2 *
(p1^3*p2) *
(p1^4)
return(x)
}
In this case, I'm looking for p1 and p2 [0,1] that will maximize this function. I tried using optim() in the following manner:
aaa = optim(c(0,0),event_prob)
but I'm getting an error "Error in fn(par, ...) : argument "p2" is missing, with no default".
Am I using optim() wrong? Or is there a different function (package?) I should be using for multi-parameter optimization?

Answer 1

This problem can in fact be solved analytically.
The objective function simplifies to
F(p1,p2) = (1-p1-p2)^299 * p1^19 * p2^11
which is to be maximised over the region
C = { (p1,p2) | 0<=p1, 0<=p2, p1+p2<=1 }
Note that F is 0 if p1=0 or p2 =0 or p1+p2 = 1, while if none of those are true then F is positive. Thus the maximum of F occurs in the interior of C
Taking the log
f(p1,p2) = 299*log(1-p1-p2) + 19*log(p1) + 11*log(p2)
In fact it is as easy to solve the more general problem: maximise f over C where
f( p1,..pN) = b*log( 1-p1-..-pn) + Sum{ a[j]*log(p[j])}
where b and each a[j] is positive and
C = { (p1,..pN) | 0<pj, j=1..N and p1+p2+..pN<1 }
The critical point occurs where all the partial derivatives of f are zero, which is at
-b/(1-p1-..-pn) + a[j]/p[j] = 0 j=1..N
which can be written as
b*p[j] + a[j]*(p1+..p[N]) = a[j] j=1..N
or
M*p = a
where M = b*I + a*Ones', and Ones is a vector with each component 1
The inverse of M is
inv(M) = (1/b)*(I - a*Ones'/(b + Ones'*a))
Thus the unique critical point is
p^ = inv(M)*a
= a/(b + Sum{i|a[i]})
Since there is a maximum, and only one critical point, the critical point must be the maximum.

Answer 2

Based on Erwin Kalvelagen's comment: Redefine your function event_prob:
event_prob = function(p) {
p1 = p[1]
p2 = p[2]
x = ((1-p1-p2)^4)^67 *
((1-p1-p2)^3*p2)^5 *
((1-p1-p2)^3*p1)^2 *
((1-p1-p2)^2*p1*p2)^3 *
((1-p1-p2)^2*p1^2) *
((1-p1-p2)*p1^2*p2)^2 *
(p1^3*p2) *
(p1^4)
return(x)
}
You may want to set limits to ensure that p1 and p2 fulfill your constraints:
optim(c(0.5,0.5),event_prob,method="L-BFGS-B",lower=0,upper=1)