The key lies not in the absolute size of the likelihood values but in their relative comparison and the mathematical principles underlying likelihood-based methods. The smallness of the likelihood is expected when dealing with continuous distributions and a product of many probabilities because you're essentially multiplying a lot of numbers that are less than 1.
The utility of likelihoods comes from their comparative nature, not their absolute values. When we compare likelihoods across different sets of parameters, we're looking for which parameters make the observed data "most likely" relative to other parameter sets, rather than looking for a likelihood that suggests the data is likely in an absolute sense.
The scale of likelihood values is often less important than how these values change relative to changes in parameters. This is why in many statistical methods, such as MLE, we're interested in finding the parameters that maximize the likelihood function, as these are considered the best estimates given the data.
Because likelihood values can be extremely small, in practice, statisticians often work with the log of the likelihood. This transformation turns products into sums, making the values more manageable and the optimization problems easier to solve, while preserving the location of the maximum.
set.seed(123)random_numbers <- rnorm(50, mean = 5, sd = 5)# Function to calculate log likelihood of a normal distributionlog_likelihood <- function(data, mean, sd) { sum(dnorm(data, mean, sd, log = TRUE))}# Calculating log likelihood for the correct parameterslog_likelihood_correct <- log_likelihood(random_numbers, 5, 5)print(log_likelihood_correct)[1] -147.4507# Calculating log likelihood for incorrect parameterslog_likelihood_incorrect <- log_likelihood(random_numbers, 6, 6)print(log_likelihood_incorrect)[1] -150.5959# Comparisonprint(log_likelihood_correct > log_likelihood_incorrect)[1] TRUE