Marginal likelihood

Definition

Marginal likelihood refers to the probability of the observed data under a specific model, integrating over all possible parameter values. It plays a crucial role in Bayesian analysis, as it helps in model comparison and selection, serving as a normalization constant in the Bayes theorem. Understanding marginal likelihood is essential for determining how well a model explains the data, influencing various aspects such as the likelihood principle, the derivation of posterior distributions, and the computation of posterior odds.

5 Must Know Facts For Your Next Test

1. Marginal likelihood can be computed by integrating the product of the likelihood and prior distribution over all parameter values.
2. It acts as a normalizing factor when deriving posterior distributions, ensuring that they sum to one across all possible parameter values.
3. In model comparison, models with higher marginal likelihoods are generally preferred as they indicate better fit to the data.
4. Marginal likelihood is often challenging to compute directly, leading to the use of techniques like Markov Chain Monte Carlo (MCMC) or Approximate Bayesian Computation (ABC).
5. The concept is vital for understanding posterior odds, as it allows for comparing different models by calculating their respective odds ratios based on their marginal likelihoods.

Review Questions

• How does marginal likelihood relate to the process of deriving posterior distributions in Bayesian statistics?
  • Marginal likelihood is integral to deriving posterior distributions as it acts as a normalization constant. When applying Bayes' theorem, the posterior distribution is calculated as proportional to the product of the prior distribution and the likelihood. The marginal likelihood ensures that this product is properly normalized over all possible parameter values, thus allowing for a valid probability distribution that reflects updated beliefs after observing data.
• Discuss the importance of marginal likelihood in model comparison and how it affects decision-making in Bayesian analysis.
  • Marginal likelihood serves a key role in model comparison by providing a measure of how well each model explains the observed data. In Bayesian analysis, we often encounter multiple competing models, and those with higher marginal likelihoods are preferred as they indicate a better fit. This decision-making process is facilitated by calculating posterior odds from marginal likelihoods, enabling researchers to weigh evidence for different hypotheses or models before making conclusions.
• Evaluate the challenges associated with computing marginal likelihood and discuss potential solutions employed in Bayesian statistics.
  • Computing marginal likelihood can be challenging due to its dependence on integrating over potentially high-dimensional parameter spaces. Direct calculation often leads to computational difficulties, prompting researchers to employ methods like Markov Chain Monte Carlo (MCMC) or Approximate Bayesian Computation (ABC). These techniques help estimate marginal likelihood by sampling from the posterior distribution or approximating complex integrals, thus making it feasible to work with complex models while maintaining valid comparisons between them.

Related terms

Bayes Theorem: A fundamental theorem that describes how to update the probability of a hypothesis based on new evidence, using prior probabilities and likelihoods.

Prior Distribution:

The probability distribution that represents the uncertainty about a parameter before observing any data.

Model Evidence: Another term for marginal likelihood, often used in the context of model selection to assess how well a model explains the observed data.