Understanding Likelihood Functions: Why They Cannot Have Negative Values

Can a likelihood function ever have negative values? The straight answer is no. This article explores why likelihood functions are inherently non-negative and the nuances around this concept in probability and statistics.

What is a Likelihood Function?

A likelihood function is a tool in statistics used to determine how likely a given set of observations is, given a specific parameter value. In simpler terms, it measures the probability of observing the given data under a particular statistical model. Probabilities, by definition, are always non-negative and range from 0 to 1. Therefore, a likelihood function, which is based on these probabilities, must also adhere to this non-negativity constraint.

Mathematical Perspective

Mathematically, if L_{theta} represents the likelihood function for a parameter theta, then:

L_{theta} geq 0

This inequality holds for all values of theta. It's important to note that while likelihoods can take values greater than 1, particularly in cases of large sample sizes or specific distributions, they never dip below zero.

Distribution-Specific Analysis

The non-negativity of likelihood functions is consistent across both discrete and continuous distributions:

Discrete Distributions

In the case of discrete distributions, the likelihood function is a probability mass function (PMF). By definition, PMFs cannot be negative. Therefore, a likelihood function based on a discrete distribution will also not have negative values.

Continuous Distributions

For continuous distributions, the likelihood function is a probability density function (PDF). Although PDFs represent the relative likelihood of a variable taking on a given value, they are similarly non-negative. Thus, a likelihood function based on a continuous distribution will also be non-negative.

The confusion often arises around the concept of "having" negative values. To clarify, the likelihood of an event is at least zero. An event can be described with a negative value, but this negative value would represent the answer to a likelihood function rather than the question it poses.

Expectation vs. Likelihood

It's common to confuse likelihood with expected value. While likelihood deals with the probability of an event occurring, the expected value is the long-term average of repetitions of the experiment it represents. Expected values can indeed have negative values.

For example, consider the MegaMillions lottery in the United States. The official probability table shows the likelihood of winning various prizes is non-negative. However, if we calculate the expected value of a lottery ticket, taking into account the cost of the ticket and the probability of winning, it can be negative, indicating that on average, players lose money.

Example: MegaMillions Lottery

Let's examine the MegaMillions lottery data:

Likelihood of Winning PrizeCost of Ticket Grand Prize1 in 302,575,350$2 Second Prize (Jackpot Split)1 in 11,688,053$2 Tier 3 Prize (Jackpot Split)1 in 3,515,300$2 Tier 4 Prize (Jackpot Split)1 in 586,275$2 Tier 5 Prize1 in 46,074$2 No Prize17 in 19 times$2

These likelihoods are all non-negative, but the expected value, when calculated by taking into account the cost of the ticket and the odds of winning, results in a negative value. This effectively means that the expected outcome of buying a lottery ticket is a loss.

Conclusion

In conclusion, a likelihood function cannot have negative values, but the outcome (expected value) can. Likelihood functions are based on probabilities and, by definition, probabilities are non-negative. Understanding this distinction is crucial for statistical analysis and modeling.

Keywords: likelihood function, probability density function, probability mass function