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D-Lab
Spring
2010
Development through
Dialogue, Design and Dissemination
1
Today’s Class
• Logistics
• Charcoal Press Case Study
• Technology Case Studies
• Introduction to Mini Project
2 Logistics
• Project Selection (Mar 1)
– Design challenge descriptions due for
review by Wednesday, Feb 17
– Slides due by... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/4eb0375b0510c8988071071133302de5_MITEC_720JS10_lec03.pdf |
-Lab Instructor
17
“Everything should be made as
simple as possible, but not
simpler.”
- Albert Einstein
18
“Simplicity
is
the
ultimate
sophistication.”
- Leonardo Da Vinci
19 Now, Even Simpler
20 Design Mini Project
Affordable Rainwater
Harvesting
21 Background and
Inspiration
• ... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/4eb0375b0510c8988071071133302de5_MITEC_720JS10_lec03.pdf |
6.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems
2002
8. Generalizing Abstraction Functions
In this handout, we give a number of examples of specs and code for which simple abstraction
functions (of the kind we studied in handout 6 on abstraction functions) don’t exist, so that the
... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
way the spec is written. But this is wrong. A spec should be written to be as clear as possible
to the clients, not to make it easy to prove the correctness of code for. The reason for these
priorities is that we expect to have many more clients for the spec than implementers. The
examples below should make it clear... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
/Size()
FI >>
END StatDB
This spec is a very natural one that follows directly from the definitions of mean and variance.
The following code for the StatDB module does not retain db. Instead, it keeps track of the size,
sum, and sum of squares of the values in db. Simple algebra shows that this is enough to
compu... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
to the same future behavior. Then any two equivalent states yield the same future behavior
of the module. Each of these equivalence classes corresponds to a state of the code.
To get an abstraction function we must add history variables, as explained in the next section.
History variables
The problem in the StatDB ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
as the code StatDBImpl given earlier. Thus, if we can prove that StatDBImplH
implements StatDB, then it follows immediately that StatDBImpl implements StatDB.
However, we can prove that StatDBImplH implements StatDB using an abstraction function.
The abstraction function, AF, simply discards all components of the st... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
think of history variables as syntactic constructs in their own particular programming
languages; in this case, the restrictions on their use must be defined in terms of the language
syntax.
In the StatDB example, we have simply added a history variable that records the entire state of
the spec. This is not necessa... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
performed by both modules shows that the abstraction relation still holds for the new states.
2.
If t and s are reachable states of T and S respectively, with (t, s) ∈ R, and (t, π, t') is a step of
T, then there is a step of S from s to some s', having the same trace, and with (t', s') ∈ R.
An abstraction relation... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
described in Spec1 as:
TYPE T = [count: Int, sum: V, sumSquare: V]
S = [db: SEQ V]
FUNC AR(t, s) -> Bool =
% state of StatDBImpl
% state of StatDB
1 This is one of several ways to represent a relation, but it is the standard one in Spec. Earlier we described the
abstraction relation as a set of pairs (t, s). In ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
for MajReg. We consider cases, based on π. Again, the interesting cases are the
procedure bodies.
Abstraction relations vs. history variables
Notice that the invariant for the history variable db above bears an uncanny resemblance to the
abstraction relation AR. This is not an accident—the same ideas are used in bo... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
) and s' =
AF(t', h'). So AR(t', s') as required.
This correspondence makes it clear that any code that can be proved correct using history
variables can also be proved correct using an abstraction relation, and vice-versa. Some people
prefer using history variables because it allows them to use an abstraction func... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
abstraction function” or “abstraction relation”, we will
mean the generalized versions.
Some examples of the use of these generalized definitions appear in handout 7 on file systems,
where there are internal transitions of code that have no counterpart in the corresponding specs.
We will see examples later in the c... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
-
empty. RWMem’s pc for an active BigRead is the same as that for RWMemImpl. RWMem’s pc for
an active BigWrite is before the body if the pc in RWMemImpl is at the beginning of the body;
otherwise it is after the body.
In the proof that this is an abstraction function, all the atomic steps in a BigWrite of RWMemImpl ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
Premature choice
In all the examples we have done so far, whenever we have wanted to prove that one module
implements another (in the sense of trace inclusion), we have been able to do this using either an
abstraction function or else its slightly generalized version, an abstraction relation. Will this
always work?... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
in NonDet occurs before the output of 1,
whereas the choice in LateNonDet occurs later, after the output of 1. It is impossible to use an
abstraction relation to simulate an early choice with a later choice. If you think of constructing an
abstract execution to correspond to a concrete execution, this would mean tha... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
ivered messages outstanding (which can happen after a crash), the earlier one will be lost if
and only if the later one overtakes it. You don’t know until the overtaking happens whether the
first message will be lost. By this time the crash and subsequent recovery may be long since
over.
The following spec models t... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
nil, Allow(2), Outcome()=1
and Consensus would not.
Premature choice: Multi-word clock
Here is a third example of premature choice in a spec: reading a clock. The spec is simple:
MODULE Clock EXPORT Read =
Premature choice: Consensus
VAR t
: Int
% the current time
For another examples, consider the consensus pr... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
easily achieve this by
locking the clock representation, but then a slow process holding the lock (for instance, one that
gets pre-empted) could block other processes for a long time. A clever ‘wait-free’ code for Read
(which appears in handout 17 on formal concurrency) reads the various parts of the clock
represen... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
1
Handout 8. Generalizing Abstraction Functions
12
6.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems
2002
the VAR t2 makes the choice of tr, and it may choose a value of t some time ago. Any
abstraction relation from LateClock to Clock has to preserve t, because a thread that does ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
from the end of the trace instead of forward from the beginning.
A prophecy variable guesses in advance some non-deterministic choice that T is going to make
later. The guess gives enough information to construct an abstraction function to the spec that is
making a premature choice. When execution reaches the choice... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
For LateReliableMsgP we add a pDead flag to each marked message that forces Get to discard
it. Crash chooses which dead flags to set. The abstraction function just discards the marks and
the dead messages.
VAR q : SEQ [m, mark: Bool, pDead: Bool] := {}
% ABSTRACTION FUNCTION ReliableMsg.q = {x :IN LateReliableMsg.q... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
Handout 8. Generalizing Abstraction Functions
13
Handout 8. Generalizing Abstraction Functions
14
6.826—Principles of Computer Systems
2002
6.826—Principles of Computer Systems
2002
q := q.tail;
IF x.mark /\ (pQ = {} \/ x.m # pQ.head) => SKIP
[] pQ := pQ.tail; RET x.m
FI OD >>
APROC Crash() =
<< VAR q' | ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
body of Clock.Read is the Tick that gives t the
pre-chosen value. This seems odd, but since all these transitions are internal, they all have empty
external traces, so it is perfectly OK.
Just as we could use abstraction relations instead of adding history variables, we can use a
different kind of relation, satisfy... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
’s value that allows TP to later do what T
does.
Here is another way of looking at this. Condition (2) requires enough different values for the
prophecy variables pi to be carried forward from the points where they are set to the points where
they are used to ensure that as they are used, any set of choices that T ... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
. (3) is a new
condition ensuring that a backward run of T ending in an initial state simulates a backward run of
S ending in an initial state; since a forward simulation never ends, it has no analogous condition.
Theorem 3: If there exists a backward simulation from T to S then every finite trace of T is also
a tr... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
inclusion. The satisfying answer is that a combination of a forward and a backward simulation,
one after the other, will always work, at least to show finite trace inclusion. (Technicalities again
arise in the infinite case.) For proofs of this result and discussion of the technicalities, see the
papers by Abadi and... | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
Abstraction Functions
17
Handout 8. Generalizing Abstraction Functions
18 | https://ocw.mit.edu/courses/6-826-principles-of-computer-systems-spring-2002/4ec9d1bba9e2b44e72ecb1029ac5c590_8.pdf |
6.777J/2.372J: The MEMSclass
Introduction to MEMS and MEMS Design
Joel Voldman
(with ideas from SDS)
Massachusetts Institute of Technology
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massach... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
Lecture 1 - 3
Handouts
> Lecture notes
• Handed out at beginning of class
• Extra copies available at Carol’s office
> Library Orientation NEXT FRIDAY FEBRUARY 16
• Learn how to use online databases, journals, etc.
• This will be VERY useful for Problem Set 2
… and for life!
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unacceptable
» Use of prior years’ homework solutions
> Cooperation is essential in final design project
> No cooperation is allowed on take-home design problem
> Any breaches will be dealt severely, with no warnings
> Please consult us before doing anything questionable
> web.mit.edu/academicintegrity/handbook/handboo... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 1 - 8
What are MEMS?
> Micro-Electro-Mechanical Systems
> Microsystems
> Microfabrication
> Microtechnology
> Nanotechnology
> Etc.
Image removed due to copyright restrictions.
Image removed due to copyright restrictions.
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: Something moves and is sensed electrically
OR
» Actuators: An electrical signal moves something
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OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloade... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
372J/6.777J Spring 2007, Lecture 1 -
MEMS: Starting points
> Some starting points:
• 1961 first silicon pressure sensor (Kulite)
» Diffused Si piezoresistors mounted onto package
to form diaphragm
» Dr. Kurtz (founder) is MIT graduate, of course
• Mid 60’s: Westinghouse Resonant Gate Transistor
» H.C. Nathanson... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
of TI DMD
project
Figure 3 on page 1882 in: Terry, S. C., J. H. Jerman, and J. B.
Angell. "A Gas Chromatographic Air Analyzer Fabricated on a
Silicon Wafer." IEEE Transactions on Electron Devices 26,
no. 12 (1979): 1880-1886. © 1979 IEEE.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabricati... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
JV: 2.372J/6.777J Spring 2007, Lecture 1 -
MEMS: Important early work
> Electrostatic Micromotors
• Introduced in 1988-1990
• MIT and Berkeley
> Microchip capillary
electrophoresis and lab-on-a-
chip
• Introduced ~1990-1994
• A. Manz, D.J. Harrison,
others
Image removed due to copyright restrictions.
Fan et al., IED... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 1 -
MEMS: Some current hot topics
> RF MEMS
• Smaller, cheaper, better
way to manipulate RF
signals
• Re... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
permission.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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MEMS: Commercial su... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 1 -
Outline
> Class odds and ends
> Intro to MEMS
> The challenge of MEMS Design
> Course outline
> Design projects
JV: 2.372/6.777 Spring 2007,... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
romechanical Devices, Spring 2007. MIT
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Some solutions to this challenge
> Approach #1
• Make something easy or not useful, etc.
> Approach #2
• Do incorrect back-of-t... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 1 -
Outline
> Class odds and ends
> Intro to MEMS
> The challenge of MEMS Design
> Course outline
> Design projects
Cite as: Joel Voldman, course materia... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
)-3
Image removed due to copyright restrictions.
Photograph of Motorola MPX200 x-ducer.
> Material properties also matter greatly
• MEMS material properties are often poorly characterized
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JV: 2.372J/6.777J Spring 2007, Lecture 1 -
Course outline
> Split sessions
> Which session should I attend?
> Go for material you are less familiar with
• MEs go to Electronics
• EEs go to Elasticity/Structures
> Notes for both lectures will be available to all
> What if you don’t know either subject?
• We will hold m... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
the “resistance” of a microfluidic channel?
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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ends
> Intro to MEMS
> The challenge of MEMS Design
> Course outline
> Take-home and team design projects
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 1 -
Design projects
> Project timeline
• Short description March 9
• Your preferences March 23
• Teams assigned April ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
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JV: 2.372J/6.777J Spring 2007, Lecture 1 -
Images removed due copyright restrictions.
Student final presentation: Gerhardt, Antimony L., Saif A. Kahn, Adam D. Rosenthal,
Nicaulas A. Sabourin, and Keng-Hoong Wee. "A Piezoresis... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/4edf8c7168c8ef8d6dbde54cc08aec2d_07lecture01.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.438 Algorithms for Inference
Fall 2014
6 Gaussian Graphical Models
Today we describe how collections of jointly Gaussian random variables can be repre
sented as directed and undirected graphical models. Our knowled... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/4f312f9c99b48b35dae961d24f10c471_MIT6_438F14_Lec6.pdf |
such that
x = Au + b.
(ii) All linear combinations of elements of x are scalar Gaussian random variables:
y = aTx is Gaussian for all a.
(iii) Covariance form: The probability density function of x can be written as
px(x) =
1
(2π)N/2|Λ|1/2
�
1
exp −
2
(x − µ)TΛ−1(x − µ)
�
,
denoted as x ∼ N(µ, Λ) with mean µ... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/4f312f9c99b48b35dae961d24f10c471_MIT6_438F14_Lec6.pdf |
.
In contrast, computing marginals using the information form is more complicated:
x1 ∼ N(µ1, Λ11).
x1 ∼ N−1(h/, J/),
where h/ = h1 − J12J−1
the Schur complement.
22 h2
and J/ = J11 − J12J−1
22 J21.
The expression for J/ is called
Conditioning is easy when we have x represented in information form: We use the... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/4f312f9c99b48b35dae961d24f10c471_MIT6_438F14_Lec6.pdf |
x1
x2
(cid:27)
�
2 x2
(cid:27)
�
+ hT
1 x1 + hT
2 x2 −
1
T
x2 J22x2
2
1 T
2 x2 − x2 J22x2
2
(cid:27)
�
2
where the last step uses the fact that x2, which we are conditioning on, is treated as
a constant. In particular, we see that
x1|x2 ∼ N−1(h/
1, J11),
where h/
1 = h1 − J12x2. Whil... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/4f312f9c99b48b35dae961d24f10c471_MIT6_438F14_Lec6.pdf |
covariance and information forms are useful depending on
whether we are marginalizing or conditioning. Converting between the two requires
matrix inversion, e.g., solving linear equations. This involves Gaussian elimination
and use of the Schur complement, which we will say a little more about at the end of
today’s... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/4f312f9c99b48b35dae961d24f10c471_MIT6_438F14_Lec6.pdf |
process
An example of a directed Gaussian graphical model is the Gauss-Markov process,
shown in Figure 1. We can express the process in innovation form:
xi+1 = Axi + Bvi,
(1)
where x1 ∼ N(µ0, Λ0), vi ∼ i.i.d. N(0, Λv) are independent of x1, and Bvi is called the
innovation. This is a linear dynamical system becau... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/4f312f9c99b48b35dae961d24f10c471_MIT6_438F14_Lec6.pdf |
)
(cid:21)
0
G H −H−1G I
�
� �
G
G
� �
M
Z
G
X
M/H
��
⎡
⎤
⎦
=
⎣
E − FH−1G 0
,
H
�
G
0
�
G
W
where, M/H is the Schur complement. Noting that W−1 = Z−1M−1X−1, then M−1
is given by
M−1 = ZW−1X
(cid:20)
(cid:21) (cid:18)(cid:20)
=
=
I
0
−H−1G I
0
I
(cid:20)
I
−H−1G
(cid:21) (cid:20)
(M/H... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/4f312f9c99b48b35dae961d24f10c471_MIT6_438F14_Lec6.pdf |
fact that
(cid:20)
(cid:21)
A 0
=
C D
(cid:20)
(cid:21)
A B
= |A||D|
0 D
whenever A and D are square matrices. Rearranging terms in equation (3) gives
|M| = |M/H||H|, hence the notation for the Schur complement.
We could alternatively decompose M in terms of E and M/E = H − GE−1F to
get an expression for... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/4f312f9c99b48b35dae961d24f10c471_MIT6_438F14_Lec6.pdf |
Power Series Expansion of the Error Function
Several times in this course we’ve seen the error function:
2
erf(x) = √
π
�
x
0
2
e−t dt.
√
(The factor of 2 guarantees that limx→∞ erf(x) = 1.) This function is very
important in probability theory, but we don’t have a conventional algebraic
description of it.
π ... | https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/4f477c5893e496cd5bf18e777e95414c_MIT18_01SCF10_Ses100e.pdf |
.
1
MIT OpenCourseWare
http://ocw.mit.edu
18.01SC Single Variable Calculus�
Fall 2010 �
�
�
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/4f477c5893e496cd5bf18e777e95414c_MIT18_01SCF10_Ses100e.pdf |
Outline
• Problems with neural networks
• Support Vector Machines
• Controlling complexity in statistical models
Questions about neural networks
• Why do they have such a bad rap?
• To what extent are neural networks brain-
like?
• They take a long time to train. Is that a
good thing or a bad thing from the
standpoi... | https://ocw.mit.edu/courses/9-66j-computational-cognitive-science-fall-2004/4f68798f979c9662d232adf39772f476_nov_4_2004_final.pdf |
copyright considerations.
Questions about neural networks
• Why do they have such a bad rap?
• To what extent are neural networks brain-
like?
• They take a long time to train. Is that a
good thing or a bad thing from the
standpoint of cognitive modeling?
Image removed due to
copyright considerations.
(Kruschke)
Ou... | https://ocw.mit.edu/courses/9-66j-computational-cognitive-science-fall-2004/4f68798f979c9662d232adf39772f476_nov_4_2004_final.pdf |
z
1(
+=
x
⋅
x
)
i
)(
z
⋅
z
j
)(
=
exp(
−
||
x
i
)(
−
x
j
)(
2
||)
The original Perceptron idea
• Embed data in d-dimensional feature space
(d >> # data points, maybe infinite).
• Find optimal separating hyperplane in
feature space.
• Problems:
– Didn’t know the “kernel trick”, but inspired by
neural receptive fie... | https://ocw.mit.edu/courses/9-66j-computational-cognitive-science-fall-2004/4f68798f979c9662d232adf39772f476_nov_4_2004_final.pdf |
– Trading off complexity with fit to data
Outline
• Problems with neural networks
• Support Vector Machines
• Controlling complexity in statistical models
Overfitting in neural networks
Image removed due to copyright considerations.
Overfitting is a universal problem
• Concept learning as search: subset principle
• ... | https://ocw.mit.edu/courses/9-66j-computational-cognitive-science-fall-2004/4f68798f979c9662d232adf39772f476_nov_4_2004_final.pdf |
Lecture 12: Superdense Coding & Quantum Teleportation
Scribed by: Sebastian Bauer
Department of Mathematics, MIT
October 14, 2003
1 Superdense Coding & Quantum Teleportation
1.1 Repetition
1. EPR pair:
1
√
2
( 01
|
� − |
10 ) ,
�
which is in some sense (as we will soon see) equivalent to
2.
1
√2
( 00 + 11 ... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/4f97e39d2509d075121823e6b999455c_qc_lec12.pdf |
|
� − |
10
�
01
|
� − |
10
�
01
|
� − |
10
�
1
(
|
(
|
01
10
�
� − |
11
00
�
� − |
)
)
(
|
11
�
+
)
00
�
|
(
|
01
�
+
)
10
�
|
(1)
(2)
(3)
(4)
(5)
(6)
(7)
P. Shor – 18.435/2.111 Quantum Computation – Lecture 12
2
Any pair of the Bell basis states has inner product 0, e.g.
1
2 �
( 01
10
|
)
·
01 ... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/4f97e39d2509d075121823e6b999455c_qc_lec12.pdf |
1 qubit to Bob, can she only encode 1 bit of information?
EPR pair
Alice
� � � � � �
•
2 bits
••
� � � � � �
Bob
1 qubit
•
������
�
�
� � �
2 bits
t
Alice and Bob have shared an EPR pair
in the past, and now Alice sends a qubit.
This can transmit 2 bits of info.
Explanation:
•
1
Alice and Bob start wi... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/4f97e39d2509d075121823e6b999455c_qc_lec12.pdf |
her half of the EPR pair by just measuring
his half. He needs either Alice’s qubit, or the information about the operation Alice performed,
which would have to be transmitted to him classically, i.e. at maximum with the speed of light.
�
�
�
�
P. Shor – 18.435/2.111 Quantum Computation – Lecture 12
3
... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/4f97e39d2509d075121823e6b999455c_qc_lec12.pdf |
� = α 0 + β 1
| �
•
|
Alice and Bob share 2 qubits in state
EPR state, as the Algebra becomes a little bit easier that way.
+ 11
|
�
( 00
|
�
1
√
2
| �
•
�
). We use this state instead of the usual
•
•
=
1
2√2
1
2√2
1
The product state is √
2
(α 0 + β 1
|
�
| �
)
( 00
|
�
+ 11
|
�
)
⊗
Alice measures her qu... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/4f97e39d2509d075121823e6b999455c_qc_lec12.pdf |
1
|
�
β
0
|
�
�
) 0
|
�
) 0
|
�
) (α
11
0
|
�
� −
) (α
1
10
� −
|
|
�
)]
|
1
√
2
( 01
|
) 1
|
�
) 1
|
�
10
|
�
� ±
+ (α
01
|
+ (β
11
|
� −
� −
) and gets
�
α
10
|
�
β
00
|
�
)
1
|
�
) 1
|
�
]
(8)
•
When Bob receives the result of Alice’s measurement, he can apply the corresponding of the
4 Pauli matrices ... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/4f97e39d2509d075121823e6b999455c_qc_lec12.pdf |
3 Models of Quantum Computation
•
•
•
•
•
Quantum Turing Machine
Quantum Gate Array
Quantum Cellular Automata
Adiabatic Quantum Computer (see next lecture)
Linear Optics Quantum Computer (beam splitters, phase shifters, mirrors etc.; only
a few years ago the devices got sophisticated enough to produce and co... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/4f97e39d2509d075121823e6b999455c_qc_lec12.pdf |
−→
c
e
f
a
.
.
.
0
0
1
0
.
.
.
L
R
stay
stay
.
.
.
P. Shor – 18.435/2.111 Quantum Computation – Lecture 12
5
3.1.2 Classical Reversible Turing Machine
In order to be reversible, the table has to have the same number of columns on both sides.
head
state
tape
symbol
direction
(from)
head ... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/4f97e39d2509d075121823e6b999455c_qc_lec12.pdf |
�
i
i
� ⊗ |
−→ |
from R
�
⊗
� ⊗ | � −→ |
� ⊗ | � −→ |
from L
�
⊗
i
|
�
stay
⊗
�
1
�
i
−
|
i + 1
|
�
(10)
Transition function (corresponding to table in previous TM’s)1:
•
•
head state
|
”from” direction
i
�⊗|
�⊗| �⊗|
unitary map U on the vector space Vhead state
symbol i
head state
|
� −→
=
⇒
Vdirection
Vs... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/4f97e39d2509d075121823e6b999455c_qc_lec12.pdf |
start counting, so the tape looks like
answer
i
� ⊗ |
�
|
answer
−→ |
i
⊗ |
+ 1
�
�
(12)
After some time, the probability that the answer is in the first part of the tape hopefully
reaches 1.
k cells
��
�
�
rest of tape
��
�
�
Theorem 3. A quantum gate array with
for n steps.
O
(n) qubits and
O
(n2) ga... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/4f97e39d2509d075121823e6b999455c_qc_lec12.pdf |
head and direction registers are changed. The two possible
head movements to the right and to the left correspond to the following operations:
�
head
|
to R
|
�
position i
��
�
0
0
�
�
|
|
position i
−
� �� �
1
�
0
0
|
�
�
|
position i+1
� �� �
to L
|
�
position i
��
�
�
�
head
|
−→
0
�
|
0
�
|
h... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/4f97e39d2509d075121823e6b999455c_qc_lec12.pdf |
�
0
�
|
|
0
�
0
�
|
|
0
�
0
|
�
|
head
|
�
to L
|
�
head
|
↓
from R
|
�
�
0
|
�
0
|
�
U2
→
head
|
to R
� |
�
0
�
0
�
|
|
0
0
� | �
|
↓
head from L
� |
|
�
0
�
0
|
�
|
0
�
0
|
�
|
head
|
�
to R
|
�
↓
head
|
to R
|
�
�
U1
←
head stay
|
�
|
�
↓
head stay
|
�
|
�
0
�
|
0
�
|
0
�
|
0
�
|
0
|
�
0
|
�
0
|
�
0
... | https://ocw.mit.edu/courses/18-435j-quantum-computation-fall-2003/4f97e39d2509d075121823e6b999455c_qc_lec12.pdf |
15.081J/6.251J Introduction to Mathematical
Programming
Lecture 5: The Simplex Method I
Slide 1
Slide 2
Slide 3
Slide 4
1 Outline
• Reduced Costs
• Optimality conditions
• Improving the cost
• Unboundness
• The Simplex algorithm
• The Simplex algorithm on degenerate problems
2 Matrix View
′
min c x
s.... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/4fdf1a06497288c24593f7eedef7576e_MIT6_251JF09_lec05.pdf |
, B−1 =
0
1 0 0
1 −1 0 0
1 1 0
1 0 1
−1
−1
d1
d3
d6
d7
′
−1
1
−1
−1
d5 = 1, d2 = d4 = 0,
= −B−1A5 =
Slide 13
′
y = x + θd ′ = (2 − θ, 0, 2 + θ, 0, θ, 1 − θ, 4 − θ)
What happens as θ increases?
xB
θ ∗ = min{i=1,...,m|dB(... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/4fdf1a06497288c24593f7eedef7576e_MIT6_251JF09_lec05.pdf |
,1,3)
x3
(0,0,3)
(1,0,3)
(2,0,2)
x1
(2,2,0)
(0,2,0)
B =
1 1 0 0
x21 0 1 0
0 1 0 0
0 1 0 1
−1
′
B
−1
, B =
1 0 −1 0
1 0
0 0
1 0
−1 1
0 0 −1 1
′
′
c = c − c B A = (0, 4, 0, −1, 0, 3, 0)
Need to continue, column A4 enters the basis.
6 Correctness
−
xB... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/4fdf1a06497288c24593f7eedef7576e_MIT6_251JF09_lec05.pdf |
Else select j : cj < 0.
4
Slide 16
Slide 17
Slide 18
Slide 19
Slide 20
Slide 21
3. Compute u = −d = B−1Aj .
• If u ≤ 0 ⇒ cost unbounded; stop
• Else
4. θ ∗ =
min
1≤i≤m,ui >0 ui
xB(i)
=
uB(l)
ul
5. Form a new basis by replacing AB(l) with Aj .
6. yj = θ ∗
yB(i) = xB(i) − θ ∗ ui
7.1 Finite Convergence... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/4fdf1a06497288c24593f7eedef7576e_MIT6_251JF09_lec05.pdf |
c
-
g
f
=0
x3
0
=
x
4
x
2
=
0
x
h
5
7.3 Pivot Selection
• Choices for the entering column:
(a) Choose a column Aj , with cj < 0, whose reduced cost is the most
negative.
(b) Choose a column with cj < 0 for which the corresponding cost de
crease θ ∗|cj | is largest.
• Choices for the exiting column:
smalle... | https://ocw.mit.edu/courses/6-251j-introduction-to-mathematical-programming-fall-2009/4fdf1a06497288c24593f7eedef7576e_MIT6_251JF09_lec05.pdf |
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
SLOAN SCHOOL OF MANAGEMENT
15.565 Integrating Information Systems:
Technology, Strategy, and Organizational Factors
15.578 Global Information Systems:
Communications & Connectivity Among Information Systems
Spring 2002
Lecture 8
WIDE-AREA NETWORKS ( WAN )
[ TELEPHONE & INTERNET... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/4fe0db520573eaedcc2c2c303ff44316_lecture08.pdf |
TELEPHONE SYSTEM (“POTS”)
• ALEXANDER GRAHAM-BELL -- 1876
• TWO PARTY
• PARTY LINE (RINGING)
• MANUAL “SWITCHED” PARTY LINE
• HOW DO YOU HANDLE VERY LARGE SCALE?
5
EARLY “STEP-BY-STEP” AUTOMATIC CIRCUIT SWITCH
A
B
C
D
E
AUTOMATED
SWITCH
A
B
C
D
E
• ALMON B. STOWGER DEVELOPER – 1889
• NO MAJOR IMPACT UNTI... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/4fe0db520573eaedcc2c2c303ff44316_lecture08.pdf |
ESS [1965], #5 ESS [MIT], #4 ESS)
• ROUTING
-- COULD TAKE 9 CONNECTIONS
-- TIME-OF-DAY IMPACTS (NYC TO MIAMI)
7
TRADITIONAL AT&T LONG DISTANCE NETWORK
8
SIGNALING ISSUES
• SIGNALING (INCREASED INTELLIGENCE WITHIN NETWORK)
--
--
“IN-BAND” VS. “OUT-OF-BAND”
COMMON CHANNEL INTER-OFFICE SIGNALING (CCIS)
--
--
197... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/4fe0db520573eaedcc2c2c303ff44316_lecture08.pdf |
-2-3-4-5
ETC.
ROUTING BASED UPON
- SPEED OF COMMUNICATION LINKS
- RELIABILITY & AVAILABILITY OF LINKS AND NODES
- NETWORK TRAFFIC LOAD
11
PACKET-BASED COMMUNICATION
FROM
TO
i
OF
PACKET
n
Processing
• Disassembly
• Routing
• Re-assembly
•
Applications exchange packets
– Message divided into packets
–... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/4fe0db520573eaedcc2c2c303ff44316_lecture08.pdf |
[SprintLink, E-Net,
J-Net]
AOL
gateway
RN#2
Connections:
DS1 = 64K, T1 = 1.5M
T3/DS3 = 45M
OC-1 = 155M, OC-12 = 622M
RN#3
Local Area Network #n
UCLA
15
INTERNET GROWTH
Source <www.nw.com>
16
WIDE AREA NETWORKS (WAN):
Network convergence business Drivers
• Market Volume: Internet traffic is doubling every... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/4fe0db520573eaedcc2c2c303ff44316_lecture08.pdf |
Announcements
• No class monday.
• Metric embedding seminar.
Review
• expectation
• notion of high probability.
•
Markov.
Today: Book 4.1, 3.3, 4.2
Chebyshev.
• Remind variance, standard deviation. σ2 = E[(X − µX )2]
X
• E[XY ] = E[X]E[Y ] if independent
• variance of independent variables: sum of variances
... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/50050d7f144bc798f479c2a411b18eae_n5.pdf |
=
E[exp(tXi)] = pie t + (1 − pi)
= 1 + pi(e t − 1)
≤
exp(pi(e t − 1))
exp(pi(et − 1)) = exp(µ(e − 1))
t
�
So overall bound is
•
exp((et − 1)µ)
exp(t(1 + �)µ)
True for any t. To minimize, plug in t = ln(1 + �).
2
• Simpler bounds:
– less than e−µ�2/3 for � < 1
– less than e−µ�2/4 for � < 2e − 1.
– L... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/50050d7f144bc798f479c2a411b18eae_n5.pdf |
take fences before and after sample median
– keep items between fences. sort.
• Analysis
– claim (i) median within fences and (ii) few items between fences.
– Without loss of generality, L contains 1, . . . , n. (ok for comparison
based algorithm)
– Samples s1, . . . , sm in sorted order.
– lemma: Sr near rn/s. ... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/50050d7f144bc798f479c2a411b18eae_n5.pdf |
ernoff: analysis of load balancing.
• Already saw balls in bins example
• synchronous message passing
• bidirectional links, one message per step
• queues on links
• permutation routing
• oblivious algorithms only consider self packet.
• Theorem Any deterministic oblivious permutation routing requires
Ω( N/d) step... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/50050d7f144bc798f479c2a411b18eae_n5.pdf |
– Adversary doesn’t know our routing so cannot plan worst permu
tation
• What if don’t wait for next phase?
– FIFO queuing
– total time is length plus delay
– Expected delay ≤ E[ T (el)] = n/2.
– Chernoff bound? no. dependence of T (ei).
�
• High prob. bound:
– consider paths sharing i’s fixed route (e0, . . . , ... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/50050d7f144bc798f479c2a411b18eae_n5.pdf |
l at ej +1 next time)
�
∗
∗ charge one delay to w.
Summary:
• 2 key roles for chernoff
• sampling
• load balancing
• “high probability” results at log n means.
8 | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/50050d7f144bc798f479c2a411b18eae_n5.pdf |
r
e
t
p
a
h
5
C 5
Minimax Lower Bounds
In the previous chapters, we have proved several upper bounds and the goal of
this chapter is to assess their optimality. Specifically, our goal is to answer the
following questions:
1. Can our analysis be improved? In other words: do the estimators that
we have studied actually sa... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
we observe Y =
(Y1, . . . , Yd)⊤, defined by
Yi = θi∗ + εi ,
i = 1, . . . , d ,
(5.1)
(0, σ2
n
1
d
∼ N
Θ is the parameter
where ε = (ε , . . . , ε )⊤
d
IRd is a given set of parameters. We will need a more precise
of interest and Θ
notation for probabilities and expectations throughout this chapter. Denote by
IPθ∗ and I... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
prove lower bounds in terms of the Fisher information. Neverthe-
less, this notion of optimality is too stringent and often leads to nonexistence
of optimal estimators. Rather, we prefer here the notion of minimax optimality
that characterizes how fast θ∗ can be estimated uniformly over Θ.
φ(Θ)?
∈
5.2. Reduction to fin... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
θ |
ˆθ θ
Θ
(cid:3)
(cid:2)
for some positive constants A an
equality also implies a lower
d C”. The above in
bound with high probability. We can therefore employ the following alternate
definition for minimax optimality.
2
2 > Aφ(Θ)
C”
≥
−
θ
∈
|
Definition 5.2.
satisfies either (5.2) or (5.3) and there exists C′ > 0 such ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
set Θ.
θj −
4φ(Θ) .
2
2 ≥
(5.7)
Then we can use the following trivial observations:
ˆ
inf sup IPθ θ
|
ˆθ θ
Θ
2
2 > φ(Θ)
θ
|
−
inf max IP
j
ˆ 1
θ ≤
≤
M
θ
j
≥
∈
(cid:2)
(cid:3)
θj 2 > φ(Θ) .
ˆ
2
θ
| − |
(cid:2)
(cid:3)
2. Reduction to a testing problem. In this second step, the necessity
of the constraint (5.7) becomes a... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
≥
inf max IPθ
ˆ 1
j M
θ
≤ ≤
inf max IPθ
j
ψ 1
≤
M
≤
ˆ
j ψ θ) = j
(
(cid:2)
j ψ = j
(cid:3)
(cid:2)
where the infimum is taken over all tests based on Y and
in
1, . . . , M
.
(cid:3)
ake values
that t
{
}
Conclusion: it is sufficient for proving lower bounds to find θ1, . . . , θM
such that
4φ(Θ) and
2
θj −
|
θk|2 ≥
Θ
∈
inf... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
Lemma indicates that any test ψ is bound to commit one
of these two types of error with positive probability unless IP0 and IP1 have
essentially disjoint support.
IP0) or an error of type II (ψ = 0 whereas Z
∼
∼
Let ν be a sigma finite measure satisfying IP0 ≪
ν. For example
we can take ν = IP0 + IP1. It follows from th... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
(R)
−
p
p0
R − 1
p0 −
R R⋆
∩
=
1 +
= 1 +
Z
Z
= 1
−
R
Z
∩
p0
R⋆ | − |
p1 +
p0
R (R⋆)c −
Z
p1 +
p0
∩
p1
p1
− |
|
(R⋆)c
ZR
∩
(R⋆)c)
1I(R
∩
−
R⋆)
(cid:3)
∩
The above quantity is clearly minimized for R = R⋆.
Z
(cid:2)
|
= 1 +
p0 −
p1|
1I(R
The lower bound in the Neyman-Pearson lemma is related to a well known
quantity: the... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
the other expressions. Hereafter, we show that (iii) = (iv). To that end,
observe th
at
p0 −
|
p1|
=
p1 −
p0 +
p0
p1<p0 −
Z
p1
p1≥
Z
p0
Z
=
p1 +
p0
p1≥
Z
= 1
−
p1<p0
Z
p1 + 1
p1<p0
Z
2
= 2
−
min(p0, p1)
Z
min(p0, p1)
p0 −
Z
−
p1≥
Z
p0
p0 −
Z
min(p0, p1)
5.3. Lower bounds based on two hypotheses
107
In view of the Neym... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
when IP1 ≪
IP0 (though it can be equal to
even in this case). Unlike the total variation
distance, the Kullback-Leibler divergence is not a distance. Actually, it is not
even symmetric. Nevertheless, it enjoys properties that are very useful for our
purposes.
∞
Proposition 5.6. Let IP and Q be two probability measures.... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
5.7. For any θ
(θ, σ2Id). Then
N
∈
IRd,
let Pθ denote the distribution of Y
∼
K (P
L θ, Pθ′) =
d
(θ
i
i=1
X
2
i′)
θ
−
2σ2
2
θ′ 2
θ
= | − |
2σ2
.
The proof is left as an exercise (see Problem 5.1).
The Kullback-Leibler divergence is easier to manipulate than the total vari-
ation distance but only the latter is related ... | https://ocw.mit.edu/courses/18-s997-high-dimensional-statistics-spring-2015/501374d1714bfd55ff6345189b9c2e26_MIT18_S997S15_Chapter5.pdf |
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