Demystifying Probability and Likelihood: Understanding the Key Differences

Probability and likelihood are often used interchangeably in the fields of statistics and machine learning, but they are actually distinct concepts that represent different things. Probability is a measure of the likelihood of an event occurring, while likelihood is a measure of how well a statistical model fits the data. In this article, we will explore the key differences between probability and likelihood and how they are used in statistical and machine learning applications.

What is Probability?

Probability is a measure of the likelihood of an event occurring, and is typically expressed as a decimal or fraction between 0 and 1. A probability of 0 indicates that the event will not occur, while a probability of 1 indicates that the event will definitely occur. Probability can be calculated using the formula:

P(E) = number of outcomes that satisfy E / total number of outcomes

For example, if we have a coin with a probability of 0.5 of landing on heads when flipped, this means that there is a 50% chance of the coin landing on heads when flipped.

What is Likelihood?

Likelihood is a measure of how well a statistical model fits the data, and is used to compare the fit of different models to the same data. It is calculated by evaluating the probability of the data given the model parameters. Likelihood can be expressed as a function of the model parameters, and is often used in model selection and parameter estimation tasks.

For example, if we have a dataset of coin flips and we want to determine the probability of the data given different models (e.g., a fair coin model vs. a biased coin model), we can calculate the likelihood of the data for each model and compare the results to determine which model is a better fit for the data.

Key Differences Between Probability and Likelihood

• Probability is a measure of the likelihood of an event occurring, while likelihood is a measure of how well a statistical model fits the data.
• Probability is calculated based on the number of outcomes that satisfy the event, while likelihood is calculated based on the probability of the data given the model parameters.
• Probability is expressed as a decimal or fraction between 0 and 1, while likelihood is a function of the model parameters.

Conclusion:

In summary, probability and likelihood are distinct concepts that are used in different contexts in statistics and machine learning. Probability is a measure of the likelihood of an event occurring, while likelihood is a measure of how well a statistical model fits the data.