4.6 Bayesian Statistics

• In Bayesian statistics, probability represents a degree of belief.


  True
• Steps to apply Bayes' Theorem

  1️⃣ Identify the prior probability P(A)

  2️⃣ Identify the likelihood P(B|A)

  3️⃣ Calculate the probability of evidence P(B)

  4️⃣ Apply Bayes' Theorem to find P(A|B)
• What is the probability of testing positive for a disease in the example provided?

  0.059
• Observing dark clouds might update your belief about rain from 60% to 80%
• What is the probability of having the disease after a positive test result in the example?

  0.161
• What does the prior distribution represent in Bayesian statistics?

  Initial belief about a parameter
• The posterior distribution incorporates both the prior distribution and the observed data.


  True
• Likelihood functions are used to update the prior distribution and obtain the posterior distribution.

  True
• Bayes' Theorem states that the posterior probability is proportional to the likelihood multiplied by the prior probability.
• In Bayesian statistics, parameter estimation is based solely on observed data.

  False
• What is parameter estimation based on in Bayesian statistics?

  Prior probability and data
• If a medical test has 95% accuracy, the probability of having the disease after a positive result is always 95%.

  False
• Bayesian statistics treats probabilities as degrees of belief rather than long-run frequencies.


  True
• What is the likelihood in Bayes' Theorem denoted as?

  P(B|A)
• The posterior distribution is calculated using Bayes' Theorem.
• What does the posterior distribution represent in Bayesian statistics?

  Updated belief about a parameter
• What does the likelihood function represent in Bayesian statistics?

  Probability of observing the data
• What does the prior probability represent in Bayesian statistics?

  Initial belief about a parameter
• Match the term with its definition:

  P(parameter|data) ↔️ Posterior probability

  P(data|parameter) ↔️ Likelihood function

  P(parameter) ↔️ Prior probability

  P(data) ↔️ Probability of the observed data
• Bayes' Theorem describes how to update the probability of a hypothesis based on new evidence.
• What does P(A|B) represent in Bayesian statistics?

  Posterior probability
• What does the likelihood function quantify in Bayesian statistics?

  Data probability
• What is the formula for Bayes' Theorem?

  $P(A|B) =$$\frac{P(B|A) \cdot P(A)}{P(B)}$
• What is the fundamental concept of Bayes' Theorem?

  Updating hypothesis probability
• A positive test result for a rare disease guarantees that the person has the disease.

  False
• The likelihood function is denoted as P(data|parameter)
• What does the prior probability represent in Bayesian statistics?

  Initial belief about a parameter
• Match the Bayesian terms with their descriptions:

  Posterior probability ↔️ Probability of parameter given data

  Likelihood function ↔️ Data supports parameter values

  Prior probability ↔️ Initial belief about a parameter

  P(data) ↔️ Probability of observed data
• In Bayesian inference, we start with a prior distribution, combine it with the likelihood function, and obtain the posterior distribution
• Steps of Bayesian inference in order:

  1️⃣ Start with a prior distribution

  2️⃣ Combine with the likelihood function

  3️⃣ Obtain the posterior distribution
• In Bayesian statistics, credible intervals are used to quantify the uncertainty around parameter estimates
• Bayesian statistics uses probability to quantify uncertainty
• Bayesian statistics provides a posterior probability to quantify uncertainty
• In Bayes' Theorem, P(A) represents the prior probability of hypothesis A
• What is the Bayesian interpretation of probability?

  Degrees of belief
• The posterior probability P(A|B) represents the updated belief about A after observing evidence B.

  True
• Match the concept with its description:

  Prior Distribution ↔️ Initial belief about a parameter

  Posterior Distribution ↔️ Updated belief after observing data

  Bayes' Theorem ↔️ Formula for updating beliefs
• The posterior distribution is calculated using Bayes' Theorem.
• The likelihood function is denoted as P(data|parameter).
• What does the posterior probability represent in Bayesian statistics?

  Updated belief about a parameter