Solutions Manual For Bayesian Statistical Methods 1st Edition
By Brian J. Reich; Sujit K. Ghosh 9781032093185 ALL Chapters .
in general, the goal of statistics is to - ANSWER: make an inference using observed data,x
to make an inference, the data is assumed to have originated from a probability model _________
and our inference is usually expressed in terms of an unobservable parameter ____ of the distribution
___ - ANSWER: f(x|θ) θ f
The Frequentist approach to statistics adopts the view that - ANSWER: only the data, x, is random
The fundamental principle of Bayesian statistics: everything is _______________ and can be treated
as ______________ and be associated with its own _________________________ provided we use a
_________________ interpretation of probability - ANSWER: uncertain, random, probability
distribution, subjective
in the Bayesian approach, the parameter θ is treated as a - ANSWER: random variable
the fundamental procedure of Bayesian statistics: use _______________________ probability with
f(θ) and f(x|θ) to find f(θ|x), which is the __________________ pdf of the parameter θ given the data
x - ANSWER: conditional
In the Bayesian approach, large samples aren't necessary for the methods to be valid but in some
cases with ________________________________ the results agree with the standard Frequentist
methods - ANSWER: large amounts of data
In the Frequentist approach, the parameter θ is considered as a - ANSWER: fixed but unknown value
In the Frequentist approach the sample size n has to be ____________ but in the Bayesian approach
it can be ______________________ - ANSWER: large, any size
The frequentist approach is based on ____________ whereas the Bayesian approach is based on
___________________________ - ANSWER: likelihood, likelihood combined with prior probabilities
What still apply to both methods? - ANSWER: the laws of probability
Bayesian: We use probability to describe all - ANSWER: uncertainty
Let X and Y be two continuous random variables with joint pdf f(x,y), then the marginal pdf of Y is
[NOT IN TERMS OF THE MULTIPLICATION RULE] - ANSWER: f(y) = ∫(X)f(x,y)dx (NOTE: (X) lies at the
bottom of the integral it is not included in the integral - see notes)
Let X and Y be two continuous random variables with joint pdf f(x,y), then the conditional pdf of Y
given X=x is - ANSWER: f(y|x) = f(x,y)/f(x)
f(x,y) = ..... = ...... = ....... [multiplication rule] - ANSWER: f(x|y)f(y) = f(y|x)f(x) = f(y,x)
f(x1,......,xn) = .........[generalised result of multiplication rule] - ANSWER: f(x1|x2,..,xn)f(x2|
x3,...,xn)...f(x(n-1)|xn)f(xn)
Let X and Y be two continuous random variables with joint pdf f(x,y), then the marginal pdf of Y is [IN
TERMS OF THE MULTIPLICATION RULE] - ANSWER: f(y) = ∫(X)f(y|x)f(x)dx (NOTE: (X) is the sample
space lies at the bottom of the integral it is not included in the integral - see notes) this is the
continuous version of the Partition Theorem
Let X and Y be two continuous random variables with joint pdf f(x,y), then the marginal pdf of Y
conditioned on a further variable Z is [IN TERMS OF THE MULTIPLICATION RULE] - ANSWER: f(y|z)=
∫(X)f(y|x,z)f(x|z)dx (NOTE: (X) is the sample space lies at the bottom of the integral it is not included in
the integral - see notes)
X and Y are independent if and only if f(x,y) = - ANSWER: f(x)f(y), so f(x)=f(x|y) and vv
Suppose X, Y and Z are random variables with f(y,z)>0, then X and Y are conditionally independent
given Z if and only if - ANSWER: f(x,y|z)=f(x|z)f(y|z), so f(x|y,z)=f(x|z) and vv
Suppose X and Y are continuous random variables and f(x)>0, then applying the partition theorem to
the denominator gives f(y|x) = - ANSWER: [f(x|y)f(y)]/[f(x)] = [f(x|y)f(y)]/[∫f(x|y)f(y)dy]
the prior distribution is represented by - ANSWER: f(θ)
After observing the data, X = x, you can formulate the likelihood, represented by_______ , of seeing
the data given the model or parameter value θ - ANSWER: f(x|θ)
the posterior distribution is represented by ____________ and expresses our uncertainty about the
value of the parameter value or statistical model θ given the information we have learned after we
have seen the data - ANSWER: f(θ|x)
Bayes theorem f(θ|x)= - ANSWER: (f(x|θ)f(θ))/f(x)
posterior = - ANSWER: (likelihood x prior)/data probability
f(θ|x)∝ - ANSWER: f(x|θ)f(θ)
By Brian J. Reich; Sujit K. Ghosh 9781032093185 ALL Chapters .
in general, the goal of statistics is to - ANSWER: make an inference using observed data,x
to make an inference, the data is assumed to have originated from a probability model _________
and our inference is usually expressed in terms of an unobservable parameter ____ of the distribution
___ - ANSWER: f(x|θ) θ f
The Frequentist approach to statistics adopts the view that - ANSWER: only the data, x, is random
The fundamental principle of Bayesian statistics: everything is _______________ and can be treated
as ______________ and be associated with its own _________________________ provided we use a
_________________ interpretation of probability - ANSWER: uncertain, random, probability
distribution, subjective
in the Bayesian approach, the parameter θ is treated as a - ANSWER: random variable
the fundamental procedure of Bayesian statistics: use _______________________ probability with
f(θ) and f(x|θ) to find f(θ|x), which is the __________________ pdf of the parameter θ given the data
x - ANSWER: conditional
In the Bayesian approach, large samples aren't necessary for the methods to be valid but in some
cases with ________________________________ the results agree with the standard Frequentist
methods - ANSWER: large amounts of data
In the Frequentist approach, the parameter θ is considered as a - ANSWER: fixed but unknown value
In the Frequentist approach the sample size n has to be ____________ but in the Bayesian approach
it can be ______________________ - ANSWER: large, any size
The frequentist approach is based on ____________ whereas the Bayesian approach is based on
___________________________ - ANSWER: likelihood, likelihood combined with prior probabilities
What still apply to both methods? - ANSWER: the laws of probability
Bayesian: We use probability to describe all - ANSWER: uncertainty
Let X and Y be two continuous random variables with joint pdf f(x,y), then the marginal pdf of Y is
[NOT IN TERMS OF THE MULTIPLICATION RULE] - ANSWER: f(y) = ∫(X)f(x,y)dx (NOTE: (X) lies at the
bottom of the integral it is not included in the integral - see notes)
Let X and Y be two continuous random variables with joint pdf f(x,y), then the conditional pdf of Y
given X=x is - ANSWER: f(y|x) = f(x,y)/f(x)
f(x,y) = ..... = ...... = ....... [multiplication rule] - ANSWER: f(x|y)f(y) = f(y|x)f(x) = f(y,x)
f(x1,......,xn) = .........[generalised result of multiplication rule] - ANSWER: f(x1|x2,..,xn)f(x2|
x3,...,xn)...f(x(n-1)|xn)f(xn)
Let X and Y be two continuous random variables with joint pdf f(x,y), then the marginal pdf of Y is [IN
TERMS OF THE MULTIPLICATION RULE] - ANSWER: f(y) = ∫(X)f(y|x)f(x)dx (NOTE: (X) is the sample
space lies at the bottom of the integral it is not included in the integral - see notes) this is the
continuous version of the Partition Theorem
Let X and Y be two continuous random variables with joint pdf f(x,y), then the marginal pdf of Y
conditioned on a further variable Z is [IN TERMS OF THE MULTIPLICATION RULE] - ANSWER: f(y|z)=
∫(X)f(y|x,z)f(x|z)dx (NOTE: (X) is the sample space lies at the bottom of the integral it is not included in
the integral - see notes)
X and Y are independent if and only if f(x,y) = - ANSWER: f(x)f(y), so f(x)=f(x|y) and vv
Suppose X, Y and Z are random variables with f(y,z)>0, then X and Y are conditionally independent
given Z if and only if - ANSWER: f(x,y|z)=f(x|z)f(y|z), so f(x|y,z)=f(x|z) and vv
Suppose X and Y are continuous random variables and f(x)>0, then applying the partition theorem to
the denominator gives f(y|x) = - ANSWER: [f(x|y)f(y)]/[f(x)] = [f(x|y)f(y)]/[∫f(x|y)f(y)dy]
the prior distribution is represented by - ANSWER: f(θ)
After observing the data, X = x, you can formulate the likelihood, represented by_______ , of seeing
the data given the model or parameter value θ - ANSWER: f(x|θ)
the posterior distribution is represented by ____________ and expresses our uncertainty about the
value of the parameter value or statistical model θ given the information we have learned after we
have seen the data - ANSWER: f(θ|x)
Bayes theorem f(θ|x)= - ANSWER: (f(x|θ)f(θ))/f(x)
posterior = - ANSWER: (likelihood x prior)/data probability
f(θ|x)∝ - ANSWER: f(x|θ)f(θ)
