Bayesian Inference: Prior Distributions

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.