Posterior

Background

The concept of ‘posterior’ in Bayesian econometrics stems from the broader field of Bayesian statistics. It’s an integral part of Bayesian inference, which combines prior knowledge or beliefs with new evidence to update the probability estimate for a hypothesis.

Historical Context

The foundation for posterior stems from the 18th-century work of Thomas Bayes, who formulated Bayes’ Theorem. This theorem became a cornerstone of Bayesian economics and econometrics in the 20th and 21st centuries, thanks to advancements in computing power that allowed for more complex calculations and models.

Definitions and Concepts

Posterior: In Bayesian econometrics, the posterior is the revised belief or distribution of a parameter after considering new sample data. It is obtained by updating the prior (the previously assumed distribution) using Bayes’ Theorem.

Major Analytical Frameworks

Classical Economics

Classical economics, with its focus on equilibrium and deterministic models, generally does not incorporate the concept of subjective probability updates, making the use of posteriors outside its framework.

Neoclassical Economics

While neoclassical economics employs deterministic models, the incorporation of Bayesian methods, particularly posterior distributions, has penetrated in microeconomic modeling and forecasting as a way to incorporate evolving information.

Keynesian Economics

Though originally outside the Bayesian framework, modern Keynesian models can integrate Bayesian posteriors, especially in the adaptive expectations and the updating of macroeconomic parameters and forecasts.

Marxian Economics

Marxian economics typically does not use Bayesian methods directly. However, the probabilistic approach in data analysis related to economic classes and dynamics could utilize posterior distributions theoretically.

Institutional Economics

Institutional economics might use Bayesian statistics and posterior distributions to revise theories or parameters when new institutional data or evidence becomes available, allowing for a cyclic update of models.

Behavioral Economics

Behavioral economists can use posterior distributions to understand how individuals update their beliefs and make decisions based on new information, fitting seamlessly with cognitive biases studies.

Post-Keynesian Economics

Similar to Keynesian economics, Post-Keynesian streams have the potential to incorporate Bayesian updating to refine models based on evolving economic data and structural shifts.

Austrian Economics

Austrian economics values subjective interpretations of economic phenomena and could theoretically employ posterior distributions to describe updates in subjective beliefs.

Development Economics

In studying the dynamics of developing economies, posterior distributions can help incorporate new data to update models on growth, inequality, and other developmental parameters.

Monetarism

Monetarists can use Bayesian updating, particularly in the realm of expectations and policies to understand how new data about money supply and velocity influences revised beliefs about macroeconomic outcomes.

Comparative Analysis

Posterior distributions provide Bayesian econometrics a distinctive strength in dynamically updating models based on new data. Unlike traditional frequentist approaches which rely on fixed parameter estimates, posteriors offer a continuous and flexible updating mechanism.

Case Studies

Case studies in which posterior distributions have been crucial include financial market predictions, healthcare economics for updating disease control parameters, and policy studies where new economic indicators necessitate model updates.

Suggested Books for Further Studies

“Bayesian Theory” by José M. Bernardo and Adrian F. M. Smith
“Bayesian Data Analysis” by Andrew Gelman et al.
“Bayesian Econometric Methods” by Gary Koop, Dale J. Poirier, and Justin L. Tobias

Prior: The initial distribution of a parameter before any new data is considered in Bayesian analysis.
Bayesian Inference: A method of statistical inference in which Bayes’ Theorem is used to update the probability estimate for a hypothesis as more evidence or information becomes available.

Quiz

### What is a posterior distribution in Bayesian econometrics? - [x] The updated probability distribution of a parameter after considering new evidence. - [ ] The ordered sequence of observed data points. - [ ] The same as the likelihood function before adjustment. - [ ] A frequentist measure of central tendency. > **Explanation:** The posterior distribution represents the revised belief or probability distribution of a parameter after integrating new evidence into prior beliefs using Bayesian updating. ### What's the key difference between prior and posterior distributions? - [ ] There is no difference. - [x] Prior reflects initial beliefs; posterior incorporates new evidence. - [ ] Prior includes data adjustments; posterior doesn't. - [ ] Both are calculated with the same method and interpretation. > **Explanation:** The key difference is that the prior represents initial assumptions about a parameter, while the posterior incorporates new data to update those beliefs. ### Posterior is a result of combining which of the following? - [x] Prior and likelihood - [ ] Data and sample mean - [ ] Median and mode - [ ] Expectation and variance > **Explanation:** The posterior distribution results from combining the prior distribution and the likelihood, reflecting updated beliefs after observing new data. ### True or False: Bayesian inference only uses prior information to make inferences. - [ ] True - [x] False > **Explanation:** In Bayesian inference, inferences are made by using prior information and updating it with newly available data (likelihood) to achieve posterior distributions. ### Which best describes Bayesian updating? - [ ] Ignoring new data for fixed prior. - [x] Revising prior beliefs in light of new evidence. - [ ] Selecting the highest data point. - [ ] Eliminating prior inferences once new data is analyzed. > **Explanation:** Bayesian updating involves revising prior beliefs by incorporating new evidence, thus adjusting our understanding of the parameter. ### In Bayesian terms, the function representing the probability of observed data given certain parameter values is called? - [ ] Posterior - [ ] Prior - [x] Likelihood - [ ] Exponential > **Explanation:** The likelihood represents the probability of observing the given data under certain parameter values and is pivotal in updating the prior to form the posterior distribution. ### How does a posterior distribution differ from a likelihood function? - [x] Posterior includes prior beliefs; likelihood doesn't. - [ ] They are the same; no difference. - [ ] Likelihood includes prior beliefs; posterior uses new data only. - [ ] One uses Bayesian inference methods, and the other doesn't. > **Explanation:** The posterior distribution incorporates prior beliefs and updates them with new evidence, whereas the likelihood function solely represents the probability of observed data given specific parameter values. ### Which statistical theorem is used in Bayesian inference to obtain the posterior distribution? - [ ] Central Limit Theorem - [ ] Z-score Formula - [x] Bayes' Theorem - [ ] Law of Large Numbers > **Explanation:** Bayes' Theorem is the cornerstone for updating our prior knowledge to obtain the posterior distribution in Bayesian inference. ### True or False: The posterior distribution is commonly used in hypothesis testing in frequentist statistics. - [ ] True - [x] False > **Explanation:** The posterior distribution is a concept specific to Bayesian statistics and is not used in the frequentist approach of hypothesis testing, which relies on p-values and confidence intervals. ### What does the term "posterior" signify in its literal etymology? - [x] Later or subsequent - [ ] Prior assumptions - [ ] Central tendency - [ ] Variable observation > **Explanation:** The term "posterior" originates from the Latin word meaning later or subsequent, reflecting its nature of being an updated distribution after considering new data.