Jeffreys Prior

Definition

Jeffreys prior is a type of non-informative prior used in Bayesian statistics that is derived from the likelihood function and is invariant under reparameterization. It provides a way to create priors that are objective and dependent only on the data, allowing for a more robust framework when prior information is not available. This prior is especially useful when dealing with parameters that are bounded or have constraints.

5 Must Know Facts For Your Next Test

1. Jeffreys prior is calculated as the square root of the determinant of the Fisher information matrix, making it particularly useful in cases involving multi-parameter models.
2. This prior is considered 'reference' because it does not favor any particular parameter values, making it a good choice when there is no clear prior information.
3. In situations where parameter estimates are derived from counts or proportions, Jeffreys prior can be equivalent to using a Beta distribution with parameters (1/2, 1/2).
4. The invariance property of Jeffreys prior means that transforming parameters will not change the form of the prior, which enhances its applicability across various models.
5. When using Jeffreys prior, it can lead to posterior distributions that are less sensitive to the choice of prior compared to other non-informative priors.

Review Questions

• How does Jeffreys prior relate to non-informative priors and what advantages does it provide?
  • Jeffreys prior is a specific type of non-informative prior designed to provide an objective approach to Bayesian analysis. Unlike general non-informative priors that may still be subjective, Jeffreys prior relies on the likelihood function and maintains invariance under parameter transformations. This makes it particularly advantageous as it offers a robust way to analyze data without imposing subjective beliefs, ensuring that conclusions drawn are largely dictated by the data itself.
• Discuss how the calculation of Jeffreys prior utilizes Fisher information and its implications in multi-parameter models.
  • Jeffreys prior is calculated using the square root of the determinant of the Fisher information matrix, which captures the amount of information that an observable random variable carries about an unknown parameter. In multi-parameter models, this calculation allows for each parameter's contribution to be properly accounted for. This ensures that Jeffreys prior remains objective and fair when assessing parameters collectively rather than individually, enhancing its utility in complex analyses.
• Evaluate the impact of using Jeffreys prior on posterior distributions and compare it to other forms of non-informative priors.
  • Using Jeffreys prior often results in posterior distributions that are more stable and less sensitive to arbitrary choices compared to other non-informative priors. This stems from its basis in the Fisher information and its reference nature, which leads to a more balanced influence from the data across various scenarios. In contrast, other non-informative priors might yield different results depending on their formulation or assumptions made, highlighting Jeffreys prior's strength in producing consistent outcomes in Bayesian inference.

Related terms

Non-informative Prior: A prior distribution that reflects minimal or no prior knowledge about the parameter being estimated, often used to let the data dominate in the analysis.

Likelihood Function: A function that measures how likely the observed data is given different values of the parameters, serving as a foundation for Bayesian inference.

Bayesian Inference: A statistical method that updates the probability for a hypothesis as more evidence or information becomes available, combining prior beliefs with new data.