Marginal likelihood

Definition

Marginal likelihood is the probability of observing the given data under a specific statistical model, integrating over all possible parameter values. It plays a crucial role in model comparison and Bayesian inference, allowing us to evaluate how well a model explains the observed data by incorporating uncertainty about the parameters. This concept is also essential for updating beliefs in Bayesian estimation and understanding the relationships between prior and posterior distributions.

5 Must Know Facts For Your Next Test

1. Marginal likelihood is calculated by integrating the product of the likelihood function and the prior distribution over all possible parameter values.
2. It helps determine the best-fitting model among competing models by comparing their marginal likelihoods, often referred to as model evidence.
3. In Bayesian analysis, marginal likelihood is crucial for calculating Bayes factors, which quantify the strength of evidence for one model over another.
4. Marginal likelihood can be computationally challenging to obtain, especially in complex models with high-dimensional parameter spaces.
5. Methods like Markov Chain Monte Carlo (MCMC) and variational inference are often used to approximate marginal likelihood in practice.

Review Questions

• How does marginal likelihood contribute to Bayesian inference and model selection?
  • Marginal likelihood is fundamental in Bayesian inference because it quantifies how well a specific model explains observed data while accounting for uncertainty in parameter values. By integrating over all possible parameters, it provides a way to compare different models through their respective marginal likelihoods. When selecting between models, the one with the highest marginal likelihood indicates stronger evidence for its fit to the data, aiding in decision-making about which model to use.
• Discuss the significance of integrating over parameter values when calculating marginal likelihood and its implications for understanding prior and posterior distributions.
  • Integrating over parameter values when calculating marginal likelihood allows us to account for all possible hypotheses regarding those parameters rather than selecting a single point estimate. This approach recognizes uncertainty in our parameter beliefs and ensures that the marginal likelihood captures this complexity. The resulting value is then used to update our beliefs in the form of posterior distributions, highlighting how prior knowledge interacts with new data to shape our understanding of a model's parameters.
• Evaluate how computational challenges associated with estimating marginal likelihood impact its use in machine learning and probabilistic models.
  • Estimating marginal likelihood can be computationally intensive, particularly in complex models with high-dimensional parameter spaces. This limitation affects its application in machine learning and probabilistic models because it may lead researchers to resort to approximations or alternative methods, such as MCMC or variational inference. As these methods strive to provide efficient estimates of marginal likelihood, they introduce their own assumptions and biases, which could affect the validity of model comparisons and conclusions drawn from the results.

Related terms

Bayesian inference:

A statistical method that updates the probability estimate for a hypothesis as more evidence or information becomes available, using prior distributions and marginal likelihood.

Prior distribution: The initial belief or knowledge about a parameter before observing any data, which is combined with the likelihood to form the posterior distribution.

Posterior distribution: The updated probability distribution of a parameter after observing the data, obtained by combining the prior distribution with the likelihood using Bayes' theorem.