Converting a Likelihood Function into R Code

Introduction

In statistics and machine learning, the likelihood function is a fundamental concept used to measure the probability of observing certain data given a set of parameters. In this post, we will explore how to convert a likelihood function from a mathematical equation into R code.

Background

The likelihood function is based on Bayes’ theorem, which states that the posterior distribution of a parameter is proportional to the product of the prior distribution and the likelihood function. The likelihood function is used in maximum likelihood estimation (MLE), which is a widely used method for estimating parameters from data.

In this post, we will focus on converting the likelihood function into R code using optimization techniques.

Mathematical Equation

The given likelihood equation can be written as:

L(r) = ∑[i=1 to j] Ni * log(1 - exp(-r * Di))

Where:

L(r) is the likelihood function
r is the parameter we want to estimate
i is the individual data point
Ni is the number of individuals exposed to dose Di
Yi is the number of individuals that became infected

R Code

The R code provided in the question uses an optimization function from the optim package to minimize the negative log-likelihood function.

log_Lik &lt;- function(r, D, N, Y)
{
    return(sum(Y * log(1 - exp(-r * D))) - r * sum(D * (N - Y)))
}

optimise(f = log_Lik, lower = 0, upper = 5, maximum = TRUE, D = D, N = N, Y = Y)

However, let’s break down this code step by step and understand what each part does.

Step 1: Define the Log-Likelihood Function

The log_Lik function is defined as:

log_Lik &lt;- function(r, D, N, Y)
{
    return(sum(Y * log(1 - exp(-r * D))) - r * sum(D * (N - Y)))
}

This function takes four arguments: the parameter r, and three data vectors D, N, and Y. It returns the log-likelihood value of the model given these parameters.

Step 2: Optimization

The optimization is performed using the optim package, which provides an interface to various optimization algorithms. In this case, we use the maximize function with a specified range for the parameter r.

optimise(f = log_Lik, lower = 0, upper = 5, maximum = TRUE, D = D, N = N, Y = Y)

The arguments to optimise are:

f: The function to optimize. In this case, it’s the negative log-likelihood function (log_Lik).
lower and upper: The lower and upper bounds of the range for the parameter r.
maximum = TRUE: Indicates that we want to maximize the function (i.e., find the maximum likelihood estimate).

Step 3: Data Preparation

The data vectors D, N, and Y are used in the optimization process. These data vectors represent:

Di: The doses administered
Ni: The number of individuals exposed to each dose
Yi: The number of individuals that became infected

These data vectors are assumed to be already loaded into R.

Example Usage

Here’s an example usage of the code in this post:

# Load necessary libraries
library(optim)

# Define the log-likelihood function
log_Lik &lt;- function(r, D, N, Y)
{
    return(sum(Y * log(1 - exp(-r * D))) - r * sum(D * (N - Y)))
}

# Define the data vectors
D &lt;- c(4.2,8,10.6,11.9,12.2,12.6,13,13.2,28.3,28.7,68.2,69,71.1,83,91.7,95.5,106.5,136.5,171.2,186.9,309.3,1557.1)
N &lt;- c(1,1,1,1,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
Y &lt;- c(0,0,1,1,2,1,0,1,0,1,1,0,1,1,1,1,1,1,1,1,1,1)

# Perform optimization
result &lt;- optimise(f = log_Lik, lower = 0, upper = 5, maximum = TRUE, D = D, N = N, Y = Y)

Conclusion

In this post, we explored how to convert a likelihood function from a mathematical equation into R code using optimization techniques. We defined the log-likelihood function and optimized it using the optim package.

We also provided an example usage of the code in this post, which demonstrates how to load necessary libraries, define the data vectors, and perform optimization.

The key takeaway is that likelihood functions can be used to measure the probability of observing certain data given a set of parameters. By converting these functions into R code using optimization techniques, we can estimate the maximum likelihood estimates for our parameters.

Limitations

There are several limitations to this approach:

Assumptions: The likelihood function assumes that the data is independently and identically distributed (i.i.d.). In reality, data may not always be i.i.d.
Non-convexity: The log-likelihood function can be non-convex, which means it has multiple local optima. This makes optimization more challenging.
Computational complexity: Optimization algorithms can be computationally intensive and require significant resources.

Future Work

In future work, we can explore:

Regularization techniques: Regularization techniques, such as L1 or L2 regularization, can help improve the stability of the optimization process.
Approximation methods: Approximation methods, such as Monte Carlo integration, can be used to estimate the likelihood function in cases where it is difficult or impossible to compute exactly.

We hope this post has provided a comprehensive introduction to converting likelihood functions into R code using optimization techniques.

Last modified on 2024-02-17