"Random variables", "stochastic Processes" graduate course.
Lecture notes of Prof. H.Amindavar.
Professor of Electrical engineering at Amirkabir university of technology.
A brief discussion of Multivariate Gaussin, Rayleigh & Rician distributions
Prof. H.Amindavar complementary notes for the first session of "Advanced communications theory" course, Spring 2021
A brief discussion of cyclostationary processes.
Prof. H.Amindavar complementary notes for the first session of "Advanced communications theory" course, Spring 2021
"Random variables", "stochastic Processes" graduate course.
Lecture notes of Prof. H.Amindavar.
Professor of Electrical engineering at Amirkabir university of technology.
A brief discussion of Multivariate Gaussin, Rayleigh & Rician distributions
Prof. H.Amindavar complementary notes for the first session of "Advanced communications theory" course, Spring 2021
A brief discussion of cyclostationary processes.
Prof. H.Amindavar complementary notes for the first session of "Advanced communications theory" course, Spring 2021
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.
We use stochastic methods to present mathematically correct representation of the wave function. Informal construction was developed by R. Feynman. This approach were introduced first by H. Doss Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques. Communications in Mathematical Physics
October 1980, Volume 73, Issue 3, pp 247–264.
Primary intention is to discuss formal stochastic representation of the Schrodinger equation solution with its applications to the theory of demolition quantum measurements.
I will appreciate your comments.
Magnetohydrodynamic Rayleigh Problem with Hall Effect in a porous PlateIJERA Editor
This paper gives very significant analytical and numerical results to the magnetohydrodynamic flow version of
the classical Rayleigh problem including Hall Effect in a porous plate. An exact solution of the MHD flow of
incompressible, electrically conducting, viscous fluid past an uniformly accelerated and insulated infinite porous
plate has been presented. Numerical values for the effects of the Hall parameter N, the Hartmann number M and
the Porosity parameter P0 on the velocity components u and v are tabulated and their profiles are shown
graphically. The numerical results show that the velocity component u and v increases with the increase of N,
decreases with the increase of P0 and u decreases and v increases with the increase of M.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
This is an extended version of a previous talk. Some further progress has been made in the sense that there is computation of 6-d transport coefficients. Hopefully this allows us to generalize to higher dimensions.
Basic concepts and how to measure price volatility
Presented by Carlos Martins-Filho at the AGRODEP Workshop on Analytical Tools for Food Prices
and Price Volatility
June 6-7, 2011 • Dakar, Senegal
For more information on the workshop or to see the latest version of this presentation visit: http://www.agrodep.org/first-annual-workshop
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.
We use stochastic methods to present mathematically correct representation of the wave function. Informal construction was developed by R. Feynman. This approach were introduced first by H. Doss Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques. Communications in Mathematical Physics
October 1980, Volume 73, Issue 3, pp 247–264.
Primary intention is to discuss formal stochastic representation of the Schrodinger equation solution with its applications to the theory of demolition quantum measurements.
I will appreciate your comments.
Magnetohydrodynamic Rayleigh Problem with Hall Effect in a porous PlateIJERA Editor
This paper gives very significant analytical and numerical results to the magnetohydrodynamic flow version of
the classical Rayleigh problem including Hall Effect in a porous plate. An exact solution of the MHD flow of
incompressible, electrically conducting, viscous fluid past an uniformly accelerated and insulated infinite porous
plate has been presented. Numerical values for the effects of the Hall parameter N, the Hartmann number M and
the Porosity parameter P0 on the velocity components u and v are tabulated and their profiles are shown
graphically. The numerical results show that the velocity component u and v increases with the increase of N,
decreases with the increase of P0 and u decreases and v increases with the increase of M.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
This is an extended version of a previous talk. Some further progress has been made in the sense that there is computation of 6-d transport coefficients. Hopefully this allows us to generalize to higher dimensions.
Basic concepts and how to measure price volatility
Presented by Carlos Martins-Filho at the AGRODEP Workshop on Analytical Tools for Food Prices
and Price Volatility
June 6-7, 2011 • Dakar, Senegal
For more information on the workshop or to see the latest version of this presentation visit: http://www.agrodep.org/first-annual-workshop
Expectation maximization (EM) algorithm is a popular and powerful mathematical method for parameter estimation in case that there exist both observed data and hidden data. Therefore, EM is appropriate to applications which aim to exploit latent aspects under heterogeneous data. This report focuses on probabilistic finite mixture model which is a popular and successful application of EM, which is fully explained in my book (Nguyen, Tutorial on EM algorithm, 2020, pp. 78-88). I also proposed a special regression model associated with mixture model in which missing values are acceptable.
A new Perron-Frobenius theorem for nonnegative tensorsFrancesco Tudisco
Based on the concept of dimensional partition we consider a general tensor spectral problem that includes all known tensor spectral problems as special cases. We formulate irreducibility and symmetry properties in terms of the dimensional partition and use the theory of multi-homogeneous order-preserving maps to derive a general and unifying Perron-Frobenius theorem for nonnegative tensors that either includes previous results of this kind or improves them by weakening the assumptions there considered.
Talk presented at SIAM Applied Linear Algebra conference Hong Kong 2018
Our experienced statistics help provider tutors offer support to complete your SAS homework help without any hassle. They are careful with your SAS homework and give 100% unique, well-researched, and authentic content. We have the best answers to do your SAS homework help. Because our statistics experts have plenty of years of experience in statistics, they also have excellent command over various statistics tools. They are having the best solution for your SAS homework .
We apply tensor train (TT) data format to solve an elliptic PDE with uncertain coefficients. We reduce complexity and storage from exponential to linear. Post-processing in TT format is also provided.
Handling missing data with expectation maximization algorithmLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating parameter of statistical models in case of incomplete data or hidden data. EM assumes that there is a relationship between hidden data and observed data, which can be a joint distribution or a mapping function. Therefore, this implies another implicit relationship between parameter estimation and data imputation. If missing data which contains missing values is considered as hidden data, it is very natural to handle missing data by EM algorithm. Handling missing data is not a new research but this report focuses on the theoretical base with detailed mathematical proofs for fulfilling missing values with EM. Besides, multinormal distribution and multinomial distribution are the two sample statistical models which are concerned to hold missing values.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
1. consider the problem of estimating an unknown
parameter of interest from a few of its noisy
observations. For example, determining the daily
temperature in a city, or the depth of a river at a
particular spot, are problems that fall into this
category. Observations (measurement) are made on
data that contain the desired nonrandom parameter θ
and undesired noise. Thus, for example,
Observation=signal(desired)+noise
or, the ith observation can be represented as
Xi = θ + ni, i = 1, 2, · · · , n.
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.1/27
2. Here θ represents the unknown nonrandom desired
parameter, and ni, i = 1, 2, · · · , n represent ran-
dom variables that may be dependent or independent
from observation to observation. Given n observations
{x1, x2, · · · , xn} the estimation problem is to obtain
the “best” estimator for the unknown parameter θ in
terms of these observations. Let us denote by θ̂(x) the
estimator for θ. Obviously θ̂(x) is a function of only
the observations. “Best estimator” in what sense? Var-
ious optimization strategies can be used to define the
term “best”. AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.2/27
3. Ideal solution would be when the estimate θ̂(x)
coincides with the unknown θ. This of course may not
be possible, and almost always any estimate will
result in an error given by
e = θ̂(x) − θ
One strategy would be to select the estimator θ̂(x) so
as to minimize some function of this error - such as
- minimization of the mean square error (MMSE), or
minimization of the absolute value of the error etc. A
more fundamental approach is that of the principle of
Maximum Likelihood (ML). AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.3/27
4. The underlying assumption in any estimation problem
is that the available data has something to do with the
unknown parameter θ. More precisely, we assume
that the joint PDF of given by fX(x1, x2, · · · , xn; θ)
depends on θ. The method of maximum likelihood
assumes that the given sample data set is
representative of the population fX(x1, x2, · · · , xn; θ)
and chooses that value for θ that most likely caused
the observed data to occur, i.e., once observations
{x1, x2, · · · , xn}
are given,
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.4/27
5. fX(x1, x2, · · · , xn; θ) is a function of θ alone, and the
value of θ that maximizes the above PDF is the most
likely value for θ, and it is chosen as the ML estimate
for θ̂ML(x) for θ (Figure 2).
θ̂ML(x)
fX(x1,··· ,xn;θ)
θ
Figure 1: ML principle
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.5/27
6. Given {x1, x2, · · · , xn} the JPDF fX(x1, · · · , xn; θ)
represents the likelihood function, and the ML
estimate can be determined either from the likelihood
equation
sup
θ̂ML
fX(x1, · · · , xn; θ)
or using the log-likelihood function (sup represents
the supremum operation)
L(x1, x2, · · · , xn; θ) = log fX(x1, x2, · · · , xn; θ).
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.6/27
7. If L(x1, x2, · · · , xn; θ) is differentiable and a
supremum θ̂ML exists, then that must satisfy the
equation
∂ log fX(x1, x2, · · · , xn; θ)
∂θ
12. Example: Let Xi = θ + wi, i = 1, 2, · · · , n
represent n observations where θ is the unknown
parameter of interest, and wi are zero mean
independent normal RVs with common variance.
Determine the ML estimate for θ.
Solution:
Since wi are independent RVs and θ is an unknown
constant, we have Xis are independent normal random
variables.
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.8/27
13. Thus the likelihood function takes the form
fX(x1, x2, · · · , xn; θ) =
n
Y
i=1
fXi
(xi; θ).
fXi
(xi; θ) =
1
√
2πσ2
e−(xi−θ)2
/2σ2
.
fX(x1, x2, · · · , xn; θ) =
1
(2πσ2)n/2
e
−
n
P
i=1
(xi−θ)2
/2σ2
.
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.9/27
24. E[θ̂ML(x)] =
1
n
n
X
i=1
E(Xi) = θ,
the expected value of the estimator does not differ
from the desired parameter, and hence there is no bias
between the two. Such estimators are known as
unbiased estimators. Moreover the variance of the
estimator is given by
V ar(θ̂ML) = E[(θ̂ML − θ)2
] = 1
n2 E
( n
P
i=1
Xi − θ
2
)
= 1
n2
(
n
P
i=1
E(Xi − θ)2
+
n
P
i=1
n
P
j=1,i6=j
E(Xi − θ)(Xj − θ)
)
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.11/27
25. V ar(θ̂ML) =
1
n2
n
X
i=1
V ar(Xi) =
nσ2
n2
=
σ2
n
.
V ar(θ̂ML) → 0, as n → ∞,
another desired property. We say such estimators are
consistent estimators. Next two examples show that
ML estimator can be highly nonlinear.
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.12/27
26. Example: Let
X1, X2, · · · , Xn
be independent, identically distributed uniform
random variables in the interval (0, θ) with common
PDF
fXi
(xi; θ) =
1
θ
, 0 xi θ,
where θ is an unknown parameter. Find the ML esti-
mate for θ.
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.13/27
27. Solution:
fX(x1, x2, · · · , xn; θ) = 1
θn , 0 xi 6 θ, i = 1, · · · , n
= 1
θn , 0 6 max(x1, x2, · · · , xn) 6 θ.
the likelihood function in this case is maximized by
the minimum value of θ, and since
θ max(X1, X2, · · · , Xn),
we get
θ̂ML(X) = max(X1, X2, · · · , Xn) (3)
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.14/27
28. represents a nonlinear function of the observations. To
determine whether (3) represents an unbiased estimate
for θ, we need to evaluate its mean. To accomplish
that in this case, it is easier to determine its PDF and
proceed directly. Z = max(X1, X2, · · · , Xn)
FZ(z) = P[max(X1, X2, · · · , Xn) 6 z] =
P(X1 6 z, X2 6 z, · · · , Xn 6 z)
=
n
Y
i=1
P(Xi 6 z) =
n
Y
i=1
FXi
(z) =
z
θ
n
, 0 z θ,
fZ(z) =
nzn−1
θn , 0 z θ,
0, AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.15/27
29. E[θ̂ML(X)] = E(Z) =
Z θ
0
zfZ(z)dz =
θ
(1 + 1/n)
.
E[θ̂ML(X)] 6= θ,
hence the ML estimator is not an unbiased estimator
for θ. However, as n → ∞
lim
n→∞
E[θ̂ML(X)] = lim
n→∞
θ
(1 + 1/n)
= θ,
ML is an asymptotically unbiased estimator.
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.16/27
30. E(Z2
) =
Z θ
0
z2
fZ(z)dz =
n
θn
Z θ
0
zn+1
dz =
nθ2
n + 2
V ar[θ̂ML(X)] = E(Z2
) − [E(Z)]2
=
nθ2
n + 2
−
n2
θ2
(n + 1)2
=
nθ2
(n + 1)2(n + 2)
.
Hence, we have a consistent estimator.
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.17/27
31. Example 12.3: Let {X1, X2, · · · , Xn} be IID Gamma
random variables with unknown parameters α and β.
Determine the ML estimator for α and β.
Solution: xi 0
fX(x1, x2, · · · , xn; α, β) =
βnα
(Γ(α))n
n
Y
i=1
xα−1
i e
−β
n
P
i=1
xi
.
L(x1, x2, · · · , xn; α, β) = log fX(x1, x2, · · · , xn; α, β)
= nα log β−n log Γ(α)+(α−1)
n
X
i=1
log xi
!
−β
n
X
i=1
xi.
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.18/27
43. log α̂ML −
Γ′
(α̂ML)
Γ(α̂ML)
= log
1
n
n
X
i=1
xi
!
−
1
n
n
X
i=1
xi.
This is a highly nonlinear equation in α̂ML.
In general the (log)-likelihood function can have more
than one solution, or no solutions at all. Further, the
(log)-likelihood function may not be even differen-
tiable, or it can be extremely complicated to solve ex-
plicitly
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.20/27
44. Cramer - Rao Bound: Variance of any unbiased
estimator based on observations {x1, x2, · · · , xn} for
θ must satisfy the lower bound
V ar(θ̂)
1
E
∂ ln fX(x1,x2,··· ,xn;θ)
∂θ
2 (4)
=
−1
E
∂2 ln fX(x1,x2,··· ,xn;θ)
∂θ2
. (5)
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.21/27
45. This important result states that the right side of above
equation acts as a lower bound on the variance of all
unbiased estimator for θ, provided their JPDF satis-
fies certain regularity restrictions. Naturally any un-
biased estimator whose variance coincides with that in
(4), must be the best. There are no better solutions!
Such estimates are known as efficient estimators.
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.22/27
46. For this purpose, let’s consider (1,2).
∂ ln fX(x1, x2, · · · , xn; θ)
∂θ
2
=
1
σ4
n
X
i=1
(Xi − θ)
!2
;
E
∂ ln fX(x1, x2, · · · , xn; θ)
∂θ
2
=
1
σ4
n
X
i=1
E[(Xi − θ)2
] +
n
X
i=1
n
X
j=1,i6=j
E[(Xi − θ)(Xj − θ)]
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.23/27
48. So far, we discussed nonrandom parameters that are
unknown. What if the parameter of interest is a
RV with a-priori PDF f(θ). How does one ob-
tain a good estimate for θ based on the observations
{x1, x2, · · · , xn}?
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.25/27
49. One technique is to use the observations to compute
its a-posteriori probability density function Of course,
we can use the Bayes’ theorem to obtain this
a-posteriori PDF. This gives
fθ|X(θ|x1, · · · , xn) =
fX|θ(x1, · · · , xn|θ)fθ(θ)
fX(x1, · · · , xn)
. (6)
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.26/27
50. θ is the variable in (6), given {x1, x2, · · · , xn} the
most probable value of θ suggested by the above
a-posteriori PDF Naturally, the most likely value for θ
is that corresponding to the maximum of the
a-posteriori PDF. This estimator - maximum of the
a-posteriori PDF is known as the MAP estimator for
θ. It is possible to use other optimality criteria as well.
Of course, that should be the subject matter of another
course!
θ̂MAP (x)
fX(θ|x1,··· ,xn)
θ
Figure 2: MAP principle
AKU-EE/Parameter/HA, 1st Semester, 84-85 – p.27/27