How to obtain a negative log-likelihood value for custom pdf similar to negloglik?

How to obtain a negative log-likelihood value for custom pdf similar to negloglik?

I used the code below to aproximate a nlogpdf, now a want to get negative log-likelihood value similar to negloglik. This will be useful for compute the Akaike infrmation criterion.

% Fit that using log(alpha) to avoid problems with alpha <= 0
[p,ci] = mle(deg,'nlogpdf',@mylogpdf,'start',1);
alphahat = exp(p);   
function logf = mylogpdf(x,logalpha,xmin)
alpha = exp(logalpha);
if nargin<3
    xmin = min(x);
end
% f = ((alpha-1)/xmin) * (x/xmin).^-alpha;
logf = log(alpha-1) - log(xmin) - alpha.*log(x/xmin);
logf(x<xmin) = -Inf;
end

Answer 1

[Abhishek Kumar Singh]

Hello Aldo,

I understand that you want to calculate negative log likelihood since ‘negloglik’ function doesn’t currently supports custom probability distribution function.

You can start by creating a custom probability distribution object that includes the necessary methods for calculating the negative log likelihood. Since you are using a power-law distribution, you've already implemented the logarithm of the probability density function (‘mylogpdf’).

Now, you need to create a custom probability distribution object and create a new custom ‘negloglik’ like function accordingly.

Here’s a reference code for you:

% Assuming you have alphahat and xmin from your fitting procedure
alphahat = exp(p);
xmin = %... assign the actual values

% Create a custom distribution object
customDistribution.logalpha = log(alphahat);
customDistribution.xmin = xmin;

% Specify censoring and frequency information (modify as needed)
cens = false(size(deg));
freq = ones(size(deg));

Then, you can use a function like defined below to work specifically with a custom PDF:

function nll = customnegloglik(pd, data, cens, freq)
    % CUSTOMNEGLOGLIK Custom negative log likelihood for a power-law distribution.

    % Extract data and information
    x = data;
    
    % Handle missing frequency and censoring information
    if isempty(freq)
        freq = ones(size(x));
    end
    if isempty(cens)
        cens = false(size(x));
    else
        cens = logical(cens);
    end
    
    % Calculate negative log likelihood
    nll = 0;
    
    % Implement custom calculation based on the properties of pd
    alpha = exp(pd.logalpha);
    xmin = pd.xmin;
    
    % Power-law distribution negative log likelihood calculation
    nll = nll - sum(freq(~cens) .* (log(alpha-1) - log(xmin) - alpha.*log(x(~cens)/xmin)));
    
    % Handle censoring and other calculations as needed
    % (Note: You may need to modify this part based on your specific requirements)
    if any(cens)
        nll = nll - sum(freq(cens) .* log(1 - cdf_customdist(pd, x(cens))));
    end
end

% Define a custom cumulative distribution function for censoring
function y = cdf_customdist(pd, x)
    alpha = exp(pd.logalpha);
    xmin = pd.xmin;
    y = 1 - (x./xmin).^(-alpha+1);
    y(x < xmin) = 0;
end

Further you can calculate the value using the function and display the results.

nll = customnegloglik(customDistribution, deg, cens, freq);

% Display results
disp(['Negative Log Likelihood: ', num2str(nll)]);

Please note that this is a sample code and you may need to modify it to work according to your specific requirements.

I hope it helps!