Likelihood

Solution & Explanation

Understanding the difference between probability and likelihood is crucial in statistical modeling and inference, especially in the context of data science. While they are related concepts, they serve distinct purposes in statistical analysis.

Probability

• Definition: Probability is a measure of the likelihood that a given event will occur. It quantifies the uncertainty associated with a specific outcome within a defined set of possible outcomes.
• Expression: Probability is expressed as $P(A)$, where $A$ is an event.
• Context: It is used when the model (distribution) is known, and we want to predict the likelihood of observing particular data.
• Example: If we have a fair six-sided die, the probability of rolling a three is $\frac{1}{6}$.
• Mathematical Notation: For a random variable $X$ with parameter $\theta$, the probability of observing $x$ is denoted as $P(X = x | \theta)$.

Likelihood

• Definition: Likelihood is a measure of how well a particular set of parameters explains the observed data. It is not a probability, but rather a function of the parameters given the data.
• Expression: Likelihood is expressed as $L(\theta | x)$, where $\theta$ are the parameters and $x$ is the observed data.
• Context: It is used when the data is known, and we want to infer the most plausible model parameters.
• Example: If we observe that a die rolled a three several times, likelihood helps us determine whether the die is fair (i.e., whether each side has an equal chance of appearing).
• Mathematical Notation: For a random variable $X$ with parameter $\theta$, the likelihood of $\theta$ given $x$ is denoted as $L(\theta | x) = P(x | \theta)$.

Key Differences

1. Direction of Inference:

   • Probability: Starts with a known model to predict data.
   • Likelihood: Starts with observed data to infer the model.

2. Role in Statistical Modeling:

   • Probability: Used to make predictions about future data.
   • Likelihood: Used to estimate parameters of a model (e.g., Maximum Likelihood Estimation).

3. Mathematical Representation:

   • Probability: $P(data | model)$
   • Likelihood: $L(model | data)$

Conclusion

Probability and likelihood are foundational concepts in statistics and data science. Probability is used to determine the chance of an event given a model, while likelihood is used to evaluate the plausibility of a model given observed data. Understanding these concepts is essential for performing statistical inference and building predictive models.