A likelihood method is a measure of how well a particular model fits the data; They explain how well a parameter (θ) explains the observed data. The logarithms of likelihood, the

**log likelihood function**, does the same job and is usually preferred for a few reasons:

- The log likelihood function in maximum likelihood estimations is usually
**computationally simpler**[1]. - Likelihoods are often tiny numbers (or large products) which makes them
**difficult to graph**. Taking the natural (base e) logarithm results in a better graph with large sums instead of products. - The log likelihood function is usually (not always!)
**easier to optimize**.

For example, let’s say you had a set of iid observations x_{1}, x_{2}…x_{n} with individual probability density function f_{X}(x). Their joint density function is:

f_{X1}, _{X2}…_{Xn}(x_{1}, x_{2}…x_{n}) = f_{X}(x_{1}) * f_{X}(x_{2}) * … * f_{X}(x_{n}) =

Where:

- Π = product (multiplication).

The log of a product is the sum of the logs of the multiplied terms, so we can rewrite the above equation with summation instead of products:

ln[f_{X}(x_{1}) * f_{X}(x_{2}) * … * f_{X}(x_{n})] =

The above relationship leads directly to the log likelihood function[2]:

l(Θ) = ln[*L*(Θ)].

Although log-likelihood functions are mathematically easier than their multiplicative counterparts, they can be **challenging to calculate by hand**. They are usually calculated with software.

## Formulation of Log-Likelihood Function

The value of the parameter that maximizes the probability of observing data is called a **maximum-likelihood estimate**. Let’s say we had a set of data *d* made up of random variables D. We want to use the data to estimate a parameter θ.

## Uses of the Log-Likelihood Function

The log likelihood function frequently pops up in financial risk forecasting and probability and statistics—especially in regression analysis / model fitting. For example:

- Akaike’s Information Criterion chooses the best model from a set. The basic AIC formula AIC = -2(log-likelihood) + 2K
**Likelihood-ratio test****: a hypothesis test to choose the best model between two nested models.**

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## References

[1] Robinson, E. (2016). Introduction to Likelihood Statistics. Retrieved April 16, 2021 from: https://hea-www.harvard.edu/AstroStat/aas227_2016/lecture1_Robinson.pdf

[2] Edge, M. (2021). Statistical Thinking from Scratch: A Primer for Scientists.

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