Consider the Bayesian linear regression model with K regressors where v) Now suppose that we have an uninformative prio...
Bayesian regression Consider the Bayesian linear regression model with K regressors where (v) Now suppose that we have an uninformative prior such that Show that the posterior verifies 2a2 where VĮß-σ2 (XX)-1. (vi) Now suppose that there is only one regressor li (ie. K = 1). Show that o2 N2 vii) Comment on how the result in part (vi) relates to the choice of prior and standard frequentist (i.e. non-Bayesian) estimators. Bayesian regression Consider the Bayesian linear regression model with...
Bayesian regression Consider the Bayesian linear regression model with K regressors where (v) Now suppose that we have an uninformative prior such that Show that the posterior verifies 2a2 where VĮß-σ2 (XX)-1. (vi) Now suppose that there is only one regressor li (ie. K = 1). Show that o2 N2 vii) Comment on how the result in part (vi) relates to the choice of prior and standard frequentist (i.e. non-Bayesian) estimators. Bayesian regression Consider the Bayesian linear regression model with...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (iii) Suppose that we have a prior μ ~ N(a, b-1) where b > 0, Show that the prior distribution π(A) verifies r(11) x exp (iv) Show that the posterior π(μ|y) verifies (v) which distribution is π(μ|y)? Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (vi) Suppose that ( of y with a -ab1. Suppose that you observe a realization Compute the posterior distribution value of 1. π(μ|1) and explain how it relates to π(μ). vii) Suppose now that you observe a second realization of y with a value of -1. Update the posterior π(p11) to incorporate this new information. Bayesian...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (vi) Suppose that ( of y with a -ab1. Suppose that you observe a realization Compute the posterior distribution value of 1. π(μ|1) and explain how it relates to π(μ). vii) Suppose now that you observe a second realization of y with a value of -1. Update the posterior π(p11) to incorporate this new information. Bayesian...
(al This question asks you to consider a Bayesian approach to inference about λ, the mortality rate in an exponential model for survival time. In order to take a Bayesian . Show that the gamma distribution is a conjugate prior distribution for the distribution is also Gamma, with parameters that depend on a, P, n,y. approach, we specify a prior distribution for A which is gamma distribution exponential model, ie. if we specify that λ~Gamma (α, β) a priori, then...
in a Bayesian view. Consider the prior π(a)-1 for all a e R Consider a Gaussian linear model Y = aX+ E Determine whether each of the following statements is true or false. π(a) a uniform prior. (1) (a) True (b) False L(Y=y14=a,X=x) (2) π(a) is a jeffreys prior when we consider the likelihood (where we assume xis known) (a) True (b)False Y-XB+ σε where ε E R" is a random vector with Consider a linear regression model E[ε1-0, E[eErJ-1....
Consider the following simple regression model: a. Suppose that OLS assumptions 1 to 4 hold true. We know that homoskedasticity assumption is statedas: Var[UjIx] = σ2 for all i Now, suppose that homoskedasticity does not hold. Mathematically, this is expressed as In other words, the subscript i in σ12 means that the conditional variance of errors for each individual i is different. Under heteroskedasticity, we can derive the expression for the variance of Var(B) as SST Where SSTx is the...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (i) Write down the conditional probability density function of y given μ (ii) Show that rw1p) amp(剖-rr) Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (i) Write down the conditional probability density function of y given μ (ii) Show that...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (i) Write down the conditional probability density function of y given μ (ii) Show that rw1p) amp(剖-rr) Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (i) Write down the conditional probability density function of y given μ (ii) Show that...