We say a parameter is identifiable if for any two members of the family of distributions...
We say a parameter is identifiable if for any two members of the family of distributions that are equal (as functions in their arguments), then the corresponding parameters are equal.
(1) Can you verify that the Gamma model is identifiable?
(2) Can you verify that the bivariate normal model is identifiable?
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We say a parameter is identifiable if for any two members of the
family of distributions...
He second form for one-parameter exponential family distributions, introduced during lecture 09.1...
he second form for one-parameter exponential family distributions, introduced during lecture 09.1, was Jy (y | θ) = b(y)ec(0)t(y)-d(0) Let η = c(0). If c is an invertible function, we can rewrite (1) as where η is called the natural, or canonical, parameter and K(n) = d(C-1(n)). Expression (2) is referred to as the canonical representation of the exponential family distribution (a) Function κ(η) is called the log-normalizer: it ensures that the distribution fy(y n) integrates to one. Show that,...
Show that the following distributions belong to the exponential family. Find the natural parameter θ, scale...
Show that the following distributions belong to the exponential family. Find the natural parameter θ, scale parameter p and convex function b(9). Also find the E(Y) and Var(Y) as functions of the natural parameter. Specify the canonical link functions 1. Exponential distribution Bxp ), f(y:λ) λe-Ag. Binomial distribution known; f(y: π- C)π"(1-π)n-y, where n is 2. Bin(n,π). 3. Poisson distribution Pois(A), f(y:A)-e
Question 1. Take a moment to convince yourself that the exponential and gamma distributions are exponential...
Question 1. Take a moment to convince yourself that the exponential and gamma distributions are exponential family models. Show that, if the data is exponentially distributed as above with a gamma prior q(0) Gamma(00,%) the posterior is again a gamma, and find the formula for the posterior parameters. (In other words, adapt the computation we performed in class for general exponential families to the specific case of the exponential/gamma model.) In detail: Ignore multiplicative constants and normalization terms, such as...
11. Chi-Squared Test for a Family of Discrete Distributions A Bookmark this page In the problems...
11. Chi-Squared Test for a Family of Discrete Distributions A Bookmark this page In the problems on this page you will apply the goodness of fit test to determine whether or not a sample has a binomial distribution So far, we have used the x test to determine if our data had a categorical distribution with specific parameters (e.s uniform on an set). element For the problems on this page, we extend the discussion on x tests beyond what was...
Gelman's (2006) paper on noninformative prior distributions for variance parameters has attracted over 3,100 citations (Google scholar). The Half-t distribution introduced in that paper is now co...
Gelman's (2006) paper on noninformative prior distributions for variance parameters has attracted over 3,100 citations (Google scholar). The Half-t distribution introduced in that paper is now considered the "default" prior to use for any standard deviation parameter a) Show that if σ2 is generated from the following hierarchical model, a ~ Inv-Gamma(α-1/2, λ-1/A2), then the (positive) square root σ has a marginal Half-t(v, A) distribution with density Here, the "scale" A > 0 and "degrees-of-freedom" v > 0 are fixed...
would this not be a two parameter exponential family? if not why not im struggling to...
would this not be a two parameter exponential family?
if not why not im struggling to understand
(a) We can write the density as fe(y) v2rez exp{-282 (y – 0)2} exp{-} log(20) – į log(02) – 2 +% – }} = We are not able to identify c(O), T(y), d(0), S(y) as this form exp{c(0)T(y) + d(0) + S(y)} This shows that this distribution does NOT belong to the exponential family.
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by θ, . We choose to employ a linear model connecting the time points t with the canonical parameter of the Poisson distribution above, i.e., n other words, we choose a generalized linear model with Poisson distribution and its canonical link function. That also means that conditioned on t,...
Section B - Answer any two questions. 2. (a) Consider the one-parameter family of nonlinear ordin...
Section B - Answer any two questions. 2. (a) Consider the one-parameter family of nonlinear ordinary differential equations dr where a is a real parameter. i. Find all equilibrium points. ii. Sketch the bifurcation diagram, and indicate the behaviour of non- equilibrium solutions using appropriate arrows. ii. Find all bifurcation points and classify them. 10 Marks (b) Consider the second order differential equation i. Show that (1) can be written as the system of ordinary differential equations (y R for...
The members of a family of four (two parents and two children) are trying to be...
The members of a family of four (two parents and two children) are trying to be more conscious of their water usage. To find opportunities where they might make the most impact, they researched the average household water consumption for various uses. They already have a new house with efficient appliances, which will decrease water usage for the dishwasher and clothes washer. Help them determine where they can make the most impact with their water usage. Type of Use U.S....
C++ 2. (a) Write a function, printdixisors, that takes a single integer parameter and prints all...
C++
2. (a) Write a function, printdixisors, that takes a single integer parameter and prints all the numbers less that the parameter that are divisors of the parameter (i.e. divides it without a remainder) including 1. So printdivisors(6) will print 1,2,3. Note you may use a wrapper function or default parameters. (b) Write a function, sumdixisors, that takes a single integer parameter and returns the sum of all the divisors of the parameter (including 1). So sumdivisors(6) will return 6...
he second form for one-parameter exponential family distributions, introduced during lecture 09.1, was Jy (y | θ) = b(y)ec(0)t(y)-d(0) Let η = c(0). If c is an invertible function, we can rewrite (1) as where η is called the natural, or canonical, parameter and K(n) = d(C-1(n)). Expression (2) is referred to as the canonical representation of the exponential family distribution (a) Function κ(η) is called the log-normalizer: it ensures that the distribution fy(y n) integrates to one. Show that,...
Show that the following distributions belong to the exponential family. Find the natural parameter θ, scale parameter p and convex function b(9). Also find the E(Y) and Var(Y) as functions of the natural parameter. Specify the canonical link functions 1. Exponential distribution Bxp ), f(y:λ) λe-Ag. Binomial distribution known; f(y: π- C)π"(1-π)n-y, where n is 2. Bin(n,π). 3. Poisson distribution Pois(A), f(y:A)-e
Question 1. Take a moment to convince yourself that the exponential and gamma distributions are exponential family models. Show that, if the data is exponentially distributed as above with a gamma prior q(0) Gamma(00,%) the posterior is again a gamma, and find the formula for the posterior parameters. (In other words, adapt the computation we performed in class for general exponential families to the specific case of the exponential/gamma model.) In detail: Ignore multiplicative constants and normalization terms, such as...
11. Chi-Squared Test for a Family of Discrete Distributions A Bookmark this page In the problems on this page you will apply the goodness of fit test to determine whether or not a sample has a binomial distribution So far, we have used the x test to determine if our data had a categorical distribution with specific parameters (e.s uniform on an set). element For the problems on this page, we extend the discussion on x tests beyond what was...
Gelman's (2006) paper on noninformative prior distributions for variance parameters has attracted over 3,100 citations (Google scholar). The Half-t distribution introduced in that paper is now considered the "default" prior to use for any standard deviation parameter a) Show that if σ2 is generated from the following hierarchical model, a ~ Inv-Gamma(α-1/2, λ-1/A2), then the (positive) square root σ has a marginal Half-t(v, A) distribution with density Here, the "scale" A > 0 and "degrees-of-freedom" v > 0 are fixed...
would this not be a two parameter exponential family?
if not why not im struggling to understand
(a) We can write the density as fe(y) v2rez exp{-282 (y – 0)2} exp{-} log(20) – į log(02) – 2 +% – }} = We are not able to identify c(O), T(y), d(0), S(y) as this form exp{c(0)T(y) + d(0) + S(y)} This shows that this distribution does NOT belong to the exponential family.
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by θ, . We choose to employ a linear model connecting the time points t with the canonical parameter of the Poisson distribution above, i.e., n other words, we choose a generalized linear model with Poisson distribution and its canonical link function. That also means that conditioned on t,...
Section B - Answer any two questions. 2. (a) Consider the one-parameter family of nonlinear ordinary differential equations dr where a is a real parameter. i. Find all equilibrium points. ii. Sketch the bifurcation diagram, and indicate the behaviour of non- equilibrium solutions using appropriate arrows. ii. Find all bifurcation points and classify them. 10 Marks (b) Consider the second order differential equation i. Show that (1) can be written as the system of ordinary differential equations (y R for...
C++
2. (a) Write a function, printdixisors, that takes a single integer parameter and prints all the numbers less that the parameter that are divisors of the parameter (i.e. divides it without a remainder) including 1. So printdivisors(6) will print 1,2,3. Note you may use a wrapper function or default parameters. (b) Write a function, sumdixisors, that takes a single integer parameter and returns the sum of all the divisors of the parameter (including 1). So sumdivisors(6) will return 6...
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