optimisation in Matlab: how to define constraints for parameters

Question

I am trying to estimate a Beta-Binomial model in Matlab (the data set is available here).

My likelihood function:

function  out= log_Lik(x,Data);

ms=Data(:,1);
xs=Data (:,2);

alpha=x(1)
beta=x(2)


P12=exp(gammaln(ms+1)-gammaln(xs+1)-gammaln(ms- xs+1)).*exp(gammaln(alpha+xs)+gammaln(beta+ms-xs)-gammaln(alpha+beta+ms));
P3= exp(gammaln(alpha+beta)-gammaln(alpha)-gammaln(beta));
P=P12.*P3;
Like=log(P);
out= -sum(Like);

My maximum likelihood estimation code:

Data=readtable('prob3.xls');

ms=Data.m_s;
xs=Data.x_s;

%% Data parsed to optimisation function
Data = [ms xs ];

f = @(x)log_Lik(x, Data);

%%
options = optimoptions('fminunc','Display','iter','Algorithm','quasi-  newton','MaxIter',10000,'TolX',10^-30,'TolFun',10^-30);
alpha0 = 1;
beta0=1;
x0 = [alpha0 beta0];
[x,fval,exitflag,output,grad,hessian] = fminunc(f,x0,options)

However, when I run the above code, I get an error saying:

Error using gammaln
Input must be nonnegative.

which I believe is coming from passing random negative values to alpha while optimising the likelihood function. Now I was wondering if there is any way to define a constraint for the values passed as my parameters (alpha and beta), when running the optimization command. Any hints would be highly appreciated.

Thanks Matthew! could you let me know where in my code and how this can be accommodated? Sorry I'm very new to matlab — AliCivil
– AliCivil, Commented Nov 5, 2016 at 10:46

Matthew Gunn · Accepted Answer · 2016-11-05 10:54:08Z

1

fminunc, the function you are currently using, finds a local minimum of an unconstrained optimization problem.

On the other hand, fmincon finds a local minimum of a constrained optimization problem.

It appears you have a constraint that alpha should be positive, so you should use fmincon and pass in the constraint.

fmincon operates similarly to fminunc. You should change "optimoptions('fminunc'..." to read "optimoptions('fmincon'..."

You want to set a lower bound of [0; -inf] to keep alpha non-negative. For the other inputs you don't care about, simply pass an empty array [].

The documentation for fmincon is going to be clearer than anything I have the time to write here.

answered Nov 5, 2016 at 10:54

Matthew Gunn

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