STA360 FInal

Which prior should you choose for a Binomial likelihood, and why?

You should choose a Beta prior because it is conjugate to the Binomial likelihood. This allows for an analytically tractable posterior distribution.

Using Monte Carlo integration, describe how to approximate E[y~∣y1,...,yn] if you can sample from p(θ∣y1,...,yn).

1. Sample θ from p(θ∣y1,...,yn). 2. For each sampled θ, sample y~ from p(y~∣θ). 3. Compute y~ for each iteration. 4. Take the sample mean of all y~ values

What does it mean for a prior to be "conjugate" to a likelihood?

A prior is conjugate to a likelihood if the posterior distribution is in the same family as the prior distribution after applying Bayes' rule.

A ______ prior is used when the parameter of interest is strictly bounded between 0 and 1.

Beta

The ______ distribution is often used as a likelihood function for modeling proportions or binary outcomes.

Binomial

Match the following likelihoods to their conjugate priors: Binomial Poisson Normal (known variance) Normal (known mean)

Binomial → Beta Poisson → Gamma Normal (known variance) → Normal Normal (known mean) → Inverse-Gamma

Why is it important to choose a conjugate prior in Bayesian inference?

Choosing a conjugate prior simplifies the computation of the posterior distribution because the posterior has the same form as the prior. This makes the calculations more tractable.

True or False: In Bayesian analysis, the prior distribution must always be conjugate to the likelihood.

False. While conjugate priors simplify computation, non-conjugate priors can be used with numerical methods like MCMC.

A ______ prior is used when modeling a parameter that must be positive, such as a standard deviation or rate.

Gamma

The ______ prior is conjugate to the parameter of an exponential likelihood.

Gamma

What is the posterior distribution when a Beta prior is combined with a Binomial likelihood?

If the prior is Beta(α,β) and the likelihood is Binomial with x successes in n trials, the posterior is: Beta(α+x, β+n−x)

What is the posterior distribution when a Gamma prior is combined with a Poisson likelihood?

If the prior is Gamma(α,β) and the likelihood is Poisson with observed total x over n observations, the posterior is: Gamma(α+x, β+n)

The ______ prior is conjugate to the covariance matrix in a multivariate normal likelihood.

Inverse-Wishart

Fill in the blank with a named distribution that makes the posterior convenient: The ______ prior is conjugate to the variance in a normal likelihood. The ______ prior is conjugate to the probability of success in a binomial likelihood. The ______ prior is conjugate to the mean in a normal likelihood. The ______ prior is conjugate to the rate in a Poisson likelihood.

Inverse-gamma Beta Normal Gamma

The posterior predictive distribution is obtained by integrating the ______ distribution and the posterior distribution.

Likelihood

Provide an example of a situation where a Poisson likelihood with a Gamma prior might be used.

Modeling the number of customer arrivals at a store in a given time period. The Poisson likelihood represents the observed count, and the Gamma prior represents prior knowledge about the average arrival rate.

Provide an example of a situation where a Binomial likelihood with a Beta prior might be used.

Modeling the success rate of a marketing campaign where the number of successes and total trials (e.g., emails opened out of emails sent) are observed. The Beta prior represents prior knowledge about the success probability.

How does Monte Carlo integration approximate expectations in Bayesian analysis?

Monte Carlo integration uses random samples from the posterior distribution to estimate expectations. The expectation E[g(θ)] is approximated by the sample mean of g(θ) over the sampled values.

The ______ distribution is used as a likelihood for continuous data with a known mean and variance.

Normal

When using a ______ prior with a normal likelihood, the posterior distribution for the mean is also normal.

Normal

The ______ distribution is often used as a likelihood function for modeling the number of events occurring in a fixed interval of time or space.

Poisson

The Beta prior is conjugate to which likelihood?

The Beta prior is conjugate to the Binomial likelihood. It is used for the probability parameter p in the Binomial distribution.

The Gamma prior is conjugate to which likelihood?

The Gamma prior is conjugate to the Poisson likelihood. It is used for the rate parameter λ in the Poisson distribution.

The Inverse-Gamma prior is conjugate to which likelihood?

The Inverse-Gamma prior is conjugate to the Normal likelihood when the mean is known. It is used for the variance parameter σ^2.

The Normal prior is conjugate to which likelihood?

The Normal prior is conjugate to the Normal likelihood when the variance is known. It is used for the mean parameter μ

How does the choice of prior affect the posterior distribution?

The choice of prior determines the initial belief about the parameter and influences the posterior, especially when the data size is small. As the data size increases, the posterior is dominated by the likelihood and becomes less sensitive to the prior.

Why is the normalizing constant often not computed directly in Bayesian analysis?

The normalizing constant involves an integral that is often computationally expensive or intractable to solve. Instead, numerical methods like Monte Carlo simulation are used to approximate posterior distributions.

What is a posterior predictive distribution used for in Bayesian analysis?

The posterior predictive distribution combines the uncertainty about the parameters (from the posterior) with the likelihood to predict new data.

A ______ prior is often used when no prior knowledge is available about a parameter (i.e., a non-informative prior).

Uniform

Write down the form of a posterior predictive distribution p(y~∣y1,...,yn).

p(y~​∣y1​,...,yn​)=∫p(y~​∣θ)p(θ∣y1​,...,yn​)dθ

he Beta prior has parameters α and β. What do these parameters represent?

α: The prior number of successes. β: The prior number of failures. These parameters control the shape of the Beta distribution.

The Gamma prior has parameters α and β. What do these parameters represent?

α: The shape parameter. β The rate parameter (inverse of scale). They control the skewness and scale of the Gamma distribution.