Jeffreys' prior

Definition

Jeffreys' prior is a non-informative prior distribution used in Bayesian statistics that is derived from the likelihood function of a statistical model. It is designed to be invariant under reparameterization, making it particularly useful for inference where prior knowledge is minimal. This type of prior is often used to express uncertainty without biasing the results, thereby facilitating likelihood ratio tests and overall Bayesian inference processes.

5 Must Know Facts For Your Next Test

1. Jeffreys' prior is computed using the Fisher information, which measures the amount of information that an observable random variable carries about an unknown parameter.
2. It provides a balanced approach in Bayesian analysis, especially when there is no strong prior knowledge about the parameters being estimated.
3. One significant property of Jeffreys' prior is that it leads to posterior distributions that are invariant under transformations of the parameters.
4. The use of Jeffreys' prior can be particularly advantageous in likelihood ratio tests, as it aligns well with maximizing the likelihood function.
5. This prior can handle parameters on different scales, making it versatile in complex models where traditional priors might not perform well.

Review Questions

• How does Jeffreys' prior support likelihood ratio tests in Bayesian analysis?
  • Jeffreys' prior helps establish a baseline for inference by providing a non-informative starting point that does not bias results. In likelihood ratio tests, this prior allows for a more straightforward comparison between models by focusing on the likelihood functions. By using a Jeffreys' prior, statisticians ensure that the test results are primarily influenced by the data rather than subjective prior beliefs.
• Discuss the importance of invariance in Jeffreys' prior and how this property affects Bayesian inference.
  • Invariance in Jeffreys' prior means that it yields the same results regardless of how parameters are scaled or transformed. This property is crucial because it ensures consistency in Bayesian inference across different representations of the model. By employing a prior that maintains this invariance, researchers can confidently interpret their results without worrying about biases introduced by parameterization choices.
• Evaluate the role of Jeffreys' prior in addressing parameter estimation challenges in complex models.
  • Jeffreys' prior plays a vital role in parameter estimation by providing a framework that remains effective even when parameters are on different scales or when there is limited prior knowledge. Its formulation based on Fisher information allows it to adapt flexibly to varying complexities within models. By utilizing this prior, analysts can achieve more reliable and interpretable posterior distributions, thereby enhancing their understanding and decision-making processes related to complex statistical challenges.

Related terms

Bayesian Inference: A method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available.

Likelihood Ratio Test: A statistical test that compares the goodness of fit of two models, one of which is a special case of the other, using the ratio of their likelihoods.

Non-informative Prior: A type of prior distribution that provides minimal information about a parameter, allowing the data to have a more significant influence on the posterior distribution.