Documentation

Negative Loglikelihood Functions

Negative loglikelihood functions for supported Statistics and Machine Learning Toolbox™ distributions all end with `like`, as in `explike`. Each function represents a parametric family of distributions. Input arguments are lists of parameter values specifying a particular member of the distribution family followed by an array of data. Functions return the negative loglikelihood of the parameters, given the data.

To find maximum likelihood estimates (MLEs), you can use a negative loglikelihood function as an objective function of the optimization problem and solve it by using the MATLAB® function `fminsearch` or functions in Optimization Toolbox™ and Global Optimization Toolbox. These functions allow you to choose a search algorithm and exercise low-level control over algorithm execution. By contrast, the `mle` function and the distribution fitting functions that end with `fit`, such as `normfit` and `gamfit`, use preset algorithms with options limited to those set by the `statset` function.

Likelihoods are conditional probability densities. You can specify a parametric family of distributions by using a function f(x,a), where x and a represent a random variable and parameters, respectively. When a is fixed, the function is the probability density function (pdf) at x, f(x|a). When x is fixed, the function is the likelihood of the parameters a, f(a|x). The joint likelihood of the parameters over an independent random sample data set X is

`$L\left(a\right)=\prod _{x\in X}f\left(a|x\right).$`

Given X, MLEs maximize L(a) over all possible a. Numerical algorithms find MLEs that (equivalently) maximize the loglikelihood function, log(L(a)). The logarithm transforms the product of potentially small likelihoods into a sum of logs, which is easier to distinguish from 0 in computation. For convenience, Statistics and Machine Learning Toolbox negative loglikelihood functions return the negative of this sum because optimization algorithms typically search for minima rather than maxima.

Find MLEs Using Negative Loglikelihood Function

This example shows how to find MLEs by using the `gamlike` and `fminsearch` functions.

Use the `gamrnd` function to generate a random sample from a specific Gamma Distribution.

```rng default; % For reproducibility a = [1,2]; X = gamrnd(a(1),a(2),1e3,1);```

Visualize the likelihood surface in the neighborhood of `a` given `X` by using the `gamlike` function.

```mesh = 50; delta = 0.5; a1 = linspace(a(1)-delta,a(1)+delta,mesh); a2 = linspace(a(2)-delta,a(2)+delta,mesh); logL = zeros(mesh); % Preallocate memory for i = 1:mesh for j = 1:mesh logL(i,j) = gamlike([a1(i),a2(j)],X); end end [A1,A2] = meshgrid(a1,a2); surfc(A1,A2,logL)```

Search for the minimum of the likelihood surface by using the `fminsearch` function.

```LL = @(u)gamlike([u(1),u(2)],X); % Likelihood given X MLES = fminsearch(LL,[1,2])```
```MLES = 1×2 0.9980 2.0172 ```

Compare `MLES` to the estimates returned by the `gamfit` function.

`ahat = gamfit(X)`
```ahat = 1×2 0.9980 2.0172 ```

The difference of each parameter between `MLES` and `ahat` is less than `1e-4`.

Add the MLEs to the surface plot.

```hold on plot3(MLES(1),MLES(2),LL(MLES),'ro','MarkerSize',5,'MarkerFaceColor','r') view([-60 40]) % Rotate to show the minimum```