Bayesian Inference: Prior Distributions

Study Notes

Bayesian Inference

Prior Distributions

A prior distribution is a probability distribution that represents our beliefs about a parameter or hypothesis before observing any data.
It is a subjective probability that reflects our prior knowledge or uncertainty about the parameter.
Common types of prior distributions:
- Uniform prior: assumes equal probability for all possible values of the parameter.
- Normal prior: assumes a normal distribution for the parameter.
- Informed prior: based on prior knowledge or data.

Posterior Distributions

A posterior distribution is a probability distribution that represents our updated beliefs about a parameter or hypothesis after observing data.
It is a combination of the prior distribution and the likelihood of the data given the parameter.
The posterior distribution is proportional to the product of the prior distribution and the likelihood function.
Posterior distributions can be used to make predictions, estimate parameters, and perform inference.

Bayes' Theorem

Bayes' theorem is a mathematical formula that describes how to update our beliefs about a parameter or hypothesis based on new data.
It is a fundamental principle in Bayesian inference.
The formula for Bayes' theorem is:

P(θ|D) = P(D|θ) * P(θ) / P(D)

Where: + P(θ|D) is the posterior distribution of the parameter θ given the data D. + P(D|θ) is the likelihood function of the data D given the parameter θ. + P(θ) is the prior distribution of the parameter θ. + P(D) is the marginal probability of the data D.

Note: Bayes' theorem is often used to update the prior distribution to obtain the posterior distribution.

Bayesian Inference

Prior Distributions

Prior distributions represent beliefs about a parameter or hypothesis before observing data, reflecting prior knowledge or uncertainty.
Common types of prior distributions include:
- Uniform prior: assumes equal probability for all possible values of the parameter.
- Normal prior: assumes a normal distribution for the parameter.
- Informed prior: based on prior knowledge or data.

Posterior Distributions

Posterior distributions represent updated beliefs about a parameter or hypothesis after observing data.
The posterior distribution is a combination of the prior distribution and the likelihood of the data given the parameter.
Posterior distributions are proportional to the product of the prior distribution and the likelihood function.
They can be used to make predictions, estimate parameters, and perform inference.

Bayes' Theorem

Bayes' theorem is a mathematical formula that updates our beliefs about a parameter or hypothesis based on new data.
It is a fundamental principle in Bayesian inference.
The formula is: P(θ|D) = P(D|θ) * P(θ) / P(D)
Where:
- P(θ|D) is the posterior distribution of the parameter θ given the data D.
- P(D|θ) is the likelihood function of the data D given the parameter θ.
- P(θ) is the prior distribution of the parameter θ.
- P(D) is the marginal probability of the data D.

Description

This quiz covers the concept of prior distributions in Bayesian inference, including types of priors such as uniform and normal distributions. Test your understanding of prior knowledge and uncertainty in parameter estimation.