I am trying to find the optimal set of parameters for my data in a probit model, which (according to literature) can be done through MLE.

So ind this case the log-likelihood function is described as:

And it consists of three parameters that I wish to estimate.

I know there probably are some optimization algorithms that can be used to do this, but you can also just brute force it, and choose a grid of several parameters for each, and then just calculate the log-likelihood for each parameter set for all patients (in this case). So I have about 500 patients, where I know whether y is equal to 0 or 1 (an incident). So basically I get one log-likelihood value when all 500 patients is used with one set of parameters. In turn, quite a few values depending on the size of the grid.

But I then end up with a huge grid/matrix of log-likelihood values, and the idea is then to find the largest value, and that value should then correspond to the most likely set of parameters for this group of patients. That's all and done.

My main problem is now: How do I define a confidence interval (fx 95%) for this most likely parameter set, given that I have the entire log-likelihood matrix with ALL log-likelihood values available ? Is it even possible, or...?