Exercise 22.12. [3, p.310] Maximum likelihood fitting of an exponential-family model. Assume that a variable x comes from a probability distribution of the form 1 P(x|w) = Z(w) exp (anf(x)). fk where the functions f(x) are given, and the parameters w = {w} are not known. A data set {x(n)} of N points is supplied. (22.31) Show by differentiating the log likelihood that the maximum-likelihood parameters WML satisfy ΣP(X|WML) fk (X) = = Σ fk(x(n)), n (fk) Data (22.32) where the left-hand sum is over all x, and the right-hand sum is over the data points. A shorthand for this result is that each function-average under the fitted model must equal the function-average found in the data: (fk) P(x|WML) (22.33)
Exercise 22.12. [3, p.310] Maximum likelihood fitting of an exponential-family model. Assume that a variable x comes from a probability distribution of the form 1 P(x|w) = Z(w) exp (anf(x)). fk where the functions f(x) are given, and the parameters w = {w} are not known. A data set {x(n)} of N points is supplied. (22.31) Show by differentiating the log likelihood that the maximum-likelihood parameters WML satisfy ΣP(X|WML) fk (X) = = Σ fk(x(n)), n (fk) Data (22.32) where the left-hand sum is over all x, and the right-hand sum is over the data points. A shorthand for this result is that each function-average under the fitted model must equal the function-average found in the data: (fk) P(x|WML) (22.33)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![▷ Exercise 22.12. [3, p.310] Maximum likelihood fitting of an exponential-family
model.
Assume that a variable x comes from a probability distribution of the
form
P(x|w) =
=
1
Z(w)
where the functions f(x) are given, and the parameters w =
not known. A data set {x)} of N points is supplied.
X
-exp (Σ wk.fk (x)
k
ΣP(X|WML.) fk (X) = Σ fix(x(")),
n
(fk) p
Show by differentiating the log likelihood that the maximum-likelihood
parameters WML satisfy
+(x)).
P(x|WML)
=
where the left-hand sum is over all x, and the right-hand sum is over the
data points. A shorthand for this result is that each function-average
under the fitted model must equal the function-average found in the
data:
(fk) Data
(22.31)
▪
= {wk} are
(22.32)
(22.33)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F36632e20-53e6-4c4b-83e3-efdc24c19507%2Fbacc7316-b021-4486-830b-3764a47f4e7e%2Fqwrtudi_processed.png&w=3840&q=75)
Transcribed Image Text:▷ Exercise 22.12. [3, p.310] Maximum likelihood fitting of an exponential-family
model.
Assume that a variable x comes from a probability distribution of the
form
P(x|w) =
=
1
Z(w)
where the functions f(x) are given, and the parameters w =
not known. A data set {x)} of N points is supplied.
X
-exp (Σ wk.fk (x)
k
ΣP(X|WML.) fk (X) = Σ fix(x(")),
n
(fk) p
Show by differentiating the log likelihood that the maximum-likelihood
parameters WML satisfy
+(x)).
P(x|WML)
=
where the left-hand sum is over all x, and the right-hand sum is over the
data points. A shorthand for this result is that each function-average
under the fitted model must equal the function-average found in the
data:
(fk) Data
(22.31)
▪
= {wk} are
(22.32)
(22.33)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 24 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman