Its take on probability as a way of modelling our brain is refreshing, for instance, the beginning of chapter 1:
Suppose some dark night a policeman walks down a street, apparently deserted; but suddenly he hears a burglar alarm, looks across the street, and sees a jewelry store with a broken window. Then a gentleman wearing a mask comes crawling out through the broken window, carrying a bag which turns out to be full of expensive jewelry. The policeman doesn't hesitate at all in deciding that this gentleman is dishonest. But by what reasoning process does he arrive at this conclusion? Let us first take a leisurely look at the general nature of such problems.
A moment's thought makes it clear that our policeman's conclusion was not a logical deduction from the evidence; for there may have been a perfectly innocent explanation for everything. It might be, for example, that this gentleman was the owner of the jewelry store and he was coming home from a masquerade party, and didn't have the key with him. But just as he walked by his store a passing truck threw a stone through the window; and he was only protecting his own property. Now while the policeman's reasoning process was not logical deduction, we will grant that it had a certain degree of validity. The evidence did not make the gentleman's dishonesty certain, but it did make it extremely plausible. This is an example of a kind of reasoning in which we haveall become more or less proficient, necessarily, long before studying mathematical theories. We are hardly able to get through one waking hour without facing some situation (e.g. will it rain or won't it?) where we do not have enough information to permit deductive reasoning; but still we must decide immediately what to do.
Ah, but policeman's actual thought process might have actually been purely deductive or purely reflexive, it might only be that the probabilities justify a series of a discreet calculations.
It's interesting that starting from the principles he states in section 1.7 the rules of probability can be derived. This formulation is from http://arxiv.org/abs/1507.06597 (where it is shown that the measure-theoretical approach is just an special case):
(Representation) Degrees of plausibility pl(A|B) can be identified as unique real numbers p ∈ R; and
(Qualitative Correspondence With Common Sense; Logical Internal Consistency) This has three sub-Principles:
– If there is more than one path to a correct conclusion, all such paths must lead to the same plausibility result;
– In assessing pl(A|B), You must always use all of the available information that You regard as relevant to the assessment; and
– Equivalent states of information about (A|B) always lead to the same pl(A|B).
I enjoy it, his papers and that books sit proudly on my desk. It's interesting that his collection of papers gives an entirely different approach to his approach to probability theory and hardly focuses on it being an extension of Aristotelean logic.
The fact that Shannon entropy is still relevant in numerous modern mathematics research is amazing.
Apart from minor typos that make him a "sloppy mathematician", he is a good educator based on my personal experience taking grad. courses from him. He's not the passionate high school STEM teacher type, but he offers great insights. I think which textbook he used is somewhat secondary. For courses he had taught before, he usually pick a standard text and teach based on the material he wrote in his blog post, including exercises.
I have a question that might be related to the topic. I have found that often, when solving problems related to probability theory, it is more convenient to think in terms of combinations than to think in terms of probabilities (perhaps because I'm a programmer). Would it be possible to devise a programming language that allows me to program a filter that selects the desired outcomes out of the complete set of combinations, and to have the compiler automatically deduce a closed-form formula for the probability (without enumeration)? If this is not possible in general, what would the restrictions on this programming language be, to make it work in practice?
So for example, given the question what the probability is that, when throwing 4 coins, 2 of which will be heads; I could write a function that generates all possible outcomes "TTTT", "TTTH", etc. And I could write a filter function that returns true for "TTHH", "THTH", "THHT", etc.
This is possible, and in fact probably implemented in some probabilistic programming languages, but I think you are looking at the wrong direction.
The point is that even for fairly simple real use cases, the computation complexity is so huge, that all computers in the world couldn't compute it in your lifetime if you don't employ some approximation or optimization and stick to naive algorithms.
So, that is what the whole field of machine learning is about: finding some clever ways to deal with random variables in a computationally feasible way...
All those probabilistic programming languages will become exponentially faster once we have feasible quantum computers, since BPP \in BQP. We currently use a weaker inclusion, BPP \in PSPACE, as the core execution model.
Most introductory books on probability I have seen open with the combinatorical approach you describe. When it is applicable, this approach is very powerfull. However there is essentially nothing novel in combinatorical probability relative to combinatorics, so research and advanced topics in this field are done under the title of combinatorics, instead of probability. Probability becomes a field in its own right once we move beyond the cases where a combinatorical approach will work.
let P = probability of 2 heads given 4 coins
let omega = (x + y)^n # probability space
let x = 50% # P(heads)
let y = 50% # P(tails)
let (n, k) = (4, 2) # 4 coins, 2 heads
P = (n, k)(x )^2(y )^2
(4, 2)(50%)^2(50%)^2
(6 )(25%) (25%) # (4, 2) = 6; see link
(6 )(6.25%)
(37.5%)
To each their own. Durrett's text is perfectly fine. The books you mention are older, and perhaps you originally learned from them, so you still hold a torch?
In particular, sending people to Loeve for their first course in measure theoretic probability would be really cruel.
Personally, I learned from Feller, and Billingsley, and finally Durrett.
> To each their own. Durrett's text is perfectly fine. The books you mention are older, and perhaps you originally learned from them, so you still hold a torch?
Durrett seems on most of the topics to have less than any of the four texts I mentioned.
> In particular, sending people to Loeve for their first course in measure theoretic probability would be really cruel.
I listed the authors in reverse order in which to read them! The easiest start is the last, Breiman. But some things in Loeve are good, e.g., sufficient statistics. Neveu is my favorite as the most elegant, but his exercises are the most difficult. Breiman and Chung are fine. IIRC, both Breiman and Neveu were Loeve students.
> Personally, I learned from Feller, and Billingsley, and finally Durrett.
I used Feller I and/or II for reference. Otherwise his writing seemed to lack an overall unification.
The only Billingsley text I used, and then only for a small topic, was Convergence of Probability Measures.
Out of curiosity, what's the background that lets you be so dismissive of a Fields Medal winner?
At this point I see a blog post written by a well respected mathematician whom I feel comfortable trusting and it's being brushed aside by I don't know who.
Well you can just read the reviews to see that the Durrett text isn't well regarded while others like Chung's are. And the criticism about a probability space not being a sample space is correct, but I think it's clear what Tao meant there, namely that the sample space would be a part of a probability space.
The Amazon UK reviews of Chung's book lead me to A Probability Path by Sidney Resnick which appears to be aimed at non-mathematicians. I have invested (speaking as a renegade physicist lacking a systematic exploration of measure theory).
Being a great mathematician doesn't necessarily make a great math educator. It is odd Tao chose Durret, but I assume it's due to the book being freely available online.
As the blog comments suggest there's a difference between a good self-study book and a good textbook for a class. The book will be accompanied by (at least) what look to be a good set of course notes.
Some googling around suggests his students are quite pleased with him. He has, according to Wikipedia's intro for him, the undisputed king of math blogs. Both of these point to him being at least a good or above average educator.
Background: From a famous,
world class research university,
Ph.D. from research
on stochastic optimal control.
Good to see that Durrett's book
touches on regular
conditional probabilities -- I needed
that topic!
I learned from a star student of
E. Cinlar, long at Princeton.
We used Royden, Rudin,
Neveu, and Chung, and there
were some nice topics in the
course not in any of those
texts, e.g., the Lindeberg-Feller
version of the central limit
theorem, a really nice, astounding,
result on an envelope for
Brownian motion, more on ergodic
theory, some on additive processes,
and more. Super nice course.
Of course. Appeal to authority was never a fallacy; the fallacy is "appeal to false authority". "Terry Tao knows much more about math in general and this in particular than I do, so I'll trust what he says here" is completely valid.
Appeal to authority is a dumb logical fallacy in reality geniuses are better at everything and we should prioritize their opinions by some decent weight over everyone else. Obviously.
And what he wrote, confusing a sample space and a probability space, just say that he had a bad day that day.
Could you expand on this? I'm reading his post now, but from the outside, it seems unlikely that Terry got the concepts completely wrong, and more likely that he's using slightly different definitions for the concepts than you are expecting.
That can be a real pain for
students trying to learn.
But, be warned: A lot of
people work with probability,
but only a tiny fraction ever
had a course in graduate
probability. So, eventually
have to learn to put up with,
and sometimes rewrite, some of
what is written that is not
very precise.
Here are the accepted definitions:
Take a non-empty set, usually
denoted by Omega, and call it the
sample space or the set of trials.
Take a collection of subsets of
Omega, usually denoted by script F,
and call it the set of events.
Have at least enough subsets of
Omega in the set of events script F
so that script F will be a sigma algebra.
So, script F has to have as an element
the empty set,
be closed under relative complements,
and be closed under countable unions.
We want all that for probability
theory. We want countably
infinite so that we can discuss,
say, the event that a coin never
or always
comes up heads. We don't want
uncountably infinite because
it would create a big mess
in the theory.
Then the ordered pair (Omega, script F)
is called a measurable space -- it
doesn't have a measure yet but
soon will.
On that measurable space, define
a positive measure P so that
P( Omega ) = 1.
Then P is a
probability measure on the
measurable space.
A probability space is
the triple ( Omega, script F, P ).
Intuitively,
a measure assigns to each
event A in script F a
non-negative real number P(A).
A signed measure permits
negative values. And
of course at times want
a measure that yields complex
numbers.
Measure theory is in, say,
P. Halmos, Measure Theory.
The back of Halmos has a
really nice introduction to
probability theory, the
Kolmogorov three series
theorem in stochastic processes,
etc. For more in stochastic
processes there is, of course,
the classic J. Doob, Stochastic
Processes. Halmos was a Doob
student and then served as
an assistant to von Neumann
at the Institute for Advanced
Study. From that he wrote
Halmos, Finite Dimensional
Vector Spaces, a finite
dimensional introduction to
Hilbert space. Later at
University of Chicago
Halmos worked in mathematical
statistics and made the
fundamental contribution to
sufficient statistics.
IIRC, it was at Chicago that
Halmos wrote Measure Theory.
Great details on measure theory
and much more
are in, say, H. Royden, Real
Analysis and W. Rudin, Real
and Complex Analysis.
To
narrow measure theory
to probability theory,
use the famous texts by the
authors I listed, Loeve, ....
Royden and Rudin are
terrific prerequisites to
Loeve, etc.
Commonly the sigma algebra of
events script F is the smallest
sigma algebra that contains
as a subset a topology, that is,
a collection of open sets,
on the
sample space Omega.
That's all standard, beginning
stuff in graduate probability --
ah, the only kind should bother
with anyway!
We want all that for probability theory. We want countably infinite so that we can discuss, say, the event that a coin never or always comes up heads. We don't want uncountably infinite because it would create a big mess in the theory.
Who should I read for an introduction to that "big mess"?
So many of the cases I'm actually interested in require real valued (continuous) inputs and outcomes. While one can quantize these to create an approximation of something countable, it seems like that a much simpler theory would be possible if it was built from the ground up to handle these common non-discrete real-world cases, rather than trying to shoe-horn them into standard probability theory. I was hoping this might be the direction that Terry was headed, with the emphasis on probabilistic methods.
I think you're misunderstanding. Standard basic (measure-theoretic) probability theory is designed to handle common non-discrete real-world cases: continuous random variables like height, temperature, etc. They're not approximated by something countable; instead theorems proving that they have the sort of behavior you'd want are established by proving them for a countable approximation, then taking limits. It is exactly like integration: prove things for step functions, then make the steps infinitely thin. The theory is clean and straightforward.
Here's where it gets less basic: say you want to look at temperature over time, but you don't want to model temperature as a variable that's measured daily, or hourly, or even every second (secondly?), but you want to model it as a process that evolves in continuous time. That's where the theory gets messy. Not necessarily at the level of a user of this theory, but definitely at the level of proving that the math you want to use is allowed.
If you actually need an introduction to that sort of probability, Lawrence Evans (Berkeley) has some old lecture notes aimed at undergrads [1] that he turned into a book [2]. If (more likely) you want standard measure-theoretic probability theory (as opposed to what's taught to undergraduates), David Pollard's book is pretty good [3].
I'm sure that @graycat will scoff at those recommendations, but his reading list would be considered excessively hardcore and time consuming even for a graduate student in math, which I'm assuming you're not.
ps: after looking at it again, the intro in Evans's notes is as gentle as it's going to get, so start there. And (as you'll find out, unless you're some sort of savant) this shit's hard. If you actually want to understand this stuff, graduate coursework is probably the only practical way to do it.
You're right, and I misunderstood. I'm a computer programmer trying to rapidly learn enough about probability theory to be able to communicate with some theoretical statisticians regarding causality, confounding, and longitudinal data analysis. I have a decent intuitive grasp of what's happening, but no ability to convey anything with proper terminology. I could certainly use a better grasp of the basics, and I'm trying to figure out where to start. Thanks for the links.
Okay, for that stuff probability theory is too abstract. For basic basics, Edward Tufte has a $2 ebook that's pretty good: Data Analysis For Politics And Policy[1] and for terminology in causality, Rubin has a short open access paper[2]. For a freshman-stats level treatment, OpenStax college's book looks legitimate but I haven't actually read it carefully[3].
Thanks for your references. I have always thought that many statistics/probability based explanations are adhoc. They are adhoc because they explain pre-selected facts; and their predictions are just a confirming instances (cf. positive vs confirming instance from Larry Laudan, a philosopher of science). Your point "definitely at the level of proving that the math you want to use is allowed" hints in that direction.
Probability's hard to teach. You can give informal statements and kind of wave your hands at the underlying theory, or you can give a rigorous well-founded treatment that's intellectually satisfying. But the rigorous foundation uses math that's a step or two beyond what undergraduate math majors learn. It's not necessarily harder than what math majors see, but it's a ton of extra material to teach, when the payoff is that you can now (after half a year) prove that the conditional probability is well-defined as
Pr(A | B) = Pr(A and B) / Pr(B)
instead of just telling it to students and drawing a few diagrams that drive the point home.
But I think it's more pragmatic than ad hoc. Any deep theory of probability that doesn't deliver
Pr(A | B) = Pr(A and B) / Pr(B)
is basically useless since that's how random phenomena seem to behave in real life. Having a deeper theory is useful because it allows you to derive other implications of that theory and makes certain calculations much easier. But if the theory disagrees with phenomena that we want to model, that can be a problem.
> I'm sure that @graycat will
scoff at those
recommendations, but his
reading list would be
considered excessively hardcore
and time consuming
even for a graduate student
in math, which I'm
assuming you're not.
Probability and stochastic
processes based on measure
theory are not very popular
in the US, even in graduate
math departments.
Uh, scoff, scoff. Okay?
The full measure theoretic details
of stochastic processes in
continuous time can be a bit
of a challenge. That topic can
be important, e.g., for Brownian
motion and stochastic differential
equations used in mathematical
finance. Of course, there is
Karatzas and Shreve,
Brownian Motion and Stochastic Calculus
and Chung and Williams,
Introduction to Stochastic Integration.
And there's much more, especially from
Russia and France.
But, otherwise, usually in practice,
what people are interested in is either
(1) second order stationary stochastic
processes, e.g., as in electronic or
acoustical signals and noise. There
are commonly interested in power spectral
estimation, digital filtering, maybe
Wiener filtering, the fast Fourier transform,
etc. or (2) what is in, say, Cinlar,
Introduction to Stochastic Processes.
In Cinlar,
for the continuous time case,
get a good introduction to
the Poisson process (the vanilla
arrival process, e.g.,
like clicks at a Geiger counter,
new sessions
at a Web site, and much more).
Also get
what else people are mostly interested
in in practice, Markov processes
in discrete time with a
discrete state space (that is, the
values are discrete).
The case of Markov processes in
continuous time and discrete state
space is not so tough if the
jumps are driven by just a Poisson
process. But there is still more
in Cinlar.
And there are other good texts on
stochastic processes.
For (1), look at some of the
texts used by EEs. The measure
theory approach is in
Doob, Stochastic Processes,
Loeve, Probability Theory,
and several more texts by quite
good authors.
E.g., without measure theory,
can just dive in via
Blackman and Tukey, The Measurement
of Power Spectra ....
With all these sources,
are able to get by without
measure theory.
Yes, without measure theory,
at some places will have to
not ask to understand too much
and just skip over some details
to get back to the applied stuff.
But for measure theory, the
Durrett text seems to get
a student to that unusually
quickly.
For more, at the MIT Web site,
there is an on-line course
in mathematical finance that
avoids measure theory.
They want to use the
Radon-Nikodym theorem
and Ito integration but
still avoid measure theory.
Uh, the Radon-Nikodym theorem
is a generalization of the
fundamental theorem of calculus.
Once see it, it's dirt simple,
but a good proof takes a bit
or follow von Neumann's proof
that knocks it all off in one
stroke (it's in
Rudin, Real and Complex Analysis).
All of this is fine for real-valued inputs and outcomes. It's the number of events (coin flips, measurements, etc) that we're restricting to be countable.
No, the number of events
is necessarily also finite or
uncountable.
Indeed, it is a nice exercise
that there are no countably
infinite sigma algebras
(extra credit for a solution!).
It's just can't take uncountably
many events, take their union,
and assume that the result is
also an event.
No problem. For the
foundations I outlined,
can work just fine with
continuous functions,
measurable functions,
stochastic processes,
random variables taking
values on the real line,
in the complex plane,
in finite dimensional
real or complex vector
spaces with, say,
the usual topology,
Hilbert and Banach spaces,
etc. Can do multi-dimensional
Markov processes, and much more.
And you can have each point on the
real line an event. Fine.
But you just can't take
the uncountable union of
any set of such events and assume
that the result is also an event.
As for the event a random variable
takes a value >= 0? Fine.
Or, let the Borel subsets of the
real line be the smallest sigma algebra
that contains all the open sets,
e.g., all the open intervals.
Then for Borel set A and
real valued random variable X,
can ask for the probability
X is in A.
I believe you will find that
you will have a solid foundation
for what you want.
To see all this stuff, need more
than just the sparse definitions
and, instead, need an actual
text and maybe a course. Recently
looked at the on-line materials
from MIT and didn't see such a course.
Graduate probability is not
all that popular in the US;
stochastic processes in continuous
time is still less popular.
To study graduate probability,
I'd recommend a good
undergraduate major in pure math
with good coverage of, say,
W. Rudin, Principles of Mathematical
Analysis. Then good coverage of
linear algebra from more than
one of the best known texts.
Likely also spend as much time
as you can in Halmos, Finite
Dimensional Vector Spaces.
E.g., at one time, Halmos,
Rudin, and Spivak, Calculus
on Manifolds were the three
main texts for Harvard's
famous Math 55.
Get good with proving the theorems.
I also recommend Fleming,
Functions of Several Variables.
Then, sure, Royden, Real Analysis.
Couldn't be prettier.
If not in a hurry, then
the real half of Rudin's
Real and Complex Analysis.
Especially if you like Fourier
theory!
Then of the probability books,
I believe that the nicest, first
book is L. Breiman, Probability.
He wrote that before he went
consulting and came back and
did CART and random forests.
Next, K. Chung,
A Course in Probability Theory.
Next, J. Neveu, Mathematical
Foundations of the Calculus of Probability.
Then, Loeve, Probability Theory.
Loeve is huge -- mostly just
use it for reference or browse.
E.g., it has sufficient statistics
and stationary stochastic processes
(the EEs love that) IIRC
not in the other
books.
IIRC, both Breiman and Neveu
were Loeve students at Berkeley.
If do well with Breiman, then
for graduate probability,
likely can stop there.
Else, Chung will then be
fast and easy reading and
reinforce what you learned in
Breiman. Neveu is elegant;
my favorite, but deserve extra
credit for each workable exercise
you can find, not actually work,
you understand, just find! Sure,
some of the exercises are terrific,
half a course in a few lines of
an exercise. E.g., he has one
of those on statistical decision
theory or some such. And see the
Tulcea material in the back.
Then there's more that you can
do on stochastic processes,
potential theory via Brownian
motion,
e.g., for mathematical finance,
stochastic optimal control,
and more.
I hope all this studying will pay off then, I have fremlin vol1/vol2 sitting underneath my text while I handle the per-requisite material hopefully the journey will pay off.
Measure theory can be used with a lot of stuff I guess.
In short measure theory is
ordinary freshman calculus done
in a way that is in some contexts
significantly more powerful.
Here measure is essentially
just a synonym for area, essentially
just ordinary area.
Measure theory addresses both
the
differentiation and integration
of freshman calculus,
but most of the focus is on
integration.
So, in freshman calculus, you
are given, say, a real valued function
of a real variable, say, function
f where, for real numbers x, we have,
e.g.,
f(x) = x^2 + 1
Then, say, we graph f and
want the area under the curve for
x in the interval [0,10]. Okay,
you saw that more than 10,000 times
in freshman calculus.
Well, in this case, measure theory
will give the same answer for the
area under the curve. The difference
is how that area is calculated.
Here is the shortest description
of where measure theory is different:
As you recall, in freshman calculus
you found the area under the curve
by picking points on the X axis,
that is, partitioning the X axis
into little intervals, on each interval
building a rectangle that approximated
the area under the curve over that
little interval, inserting more in the
partition so that the longest of
the little intervals had its length
get as small as we pleased, adding up
the areas of the rectangles, and
taking the limit. That was it. That
was the definition of what the
integral was. Of course, to integrate
x^2 + 1 you learned about, for any
constant C, the anti-derivative
(1/3)x^3 + x + C
So, here's what measure theory does:
Yup, it also works with a partition
but, in this case, the partition is
on the Y axis instead of the X axis.
Then for each little interval on the
Y axis, we get a horizontal bar and
look at the parts that are under the
curve. Then as we add points to the
partition, we add up the ordinary
areas of the relevant parts of the
horizontal bars. The picture
is less nice than in freshman calculus
but, still, no biggie.
Now, how would one do, say, numerical
integration? Sure: The same
as in freshman calculus, say,
Simpson's rule where work with
little trapezoids. Nope, measure
theory is not suggesting that we
change that.
Here are four features of the
approach of measure theory:
(1) There are some goofy, pathological
functions that get the integration
theory (the Riemann integral) of
freshman calculus all confused.
E.g., consider the function 0 on
the rational numbers and 1 otherwise.
Then the upper and lower
Riemann sums of those little
rectangles never converge to what
we want. Bummer. Well, it follows
from some theorems that the integral
of measure theory rarely gets confused --
of course the theorems are very precise,
but the generality is mind blowing, so
much that it's darned tricky to
construct a case where the measure
theory integral fails.
(2) A big deal is what happens
when a sequence of functions
is used to approximate and converge
to another function. A leading
example is Fourier series. Well,
on taking limits during this
approximation, the Riemann integral
can get confused when the measure
theory integral (Lebesgue) does just fine.
H. Lebesgue was a student of
E. Borel in France and did
his work near 1900.
(3) Often we want to pass some
limits under the integral sign.
Again, Lebesgue does much better
here than Riemann. Indeed,
the Lebesgue integral has
a super nice theorem on
differentiation under the
integral sign (from the
TOC of Durrett, that theorem
may be the last topic in that
book -- it was a really fun
exercise when I was studying
that stuff).
(4) Notice a biggie: With
Lebesgue, actually we used
next to nothing about
the X axis, that is, about
the domain of the function
we are integrating. In this
way, right away, presto, bingo,
we get a theory of integration
that works on domains
with much, much less in assumptions
and properties than the real
numbers or the usual finite
dimensional real Euclidean
vector space. In particular,
we are now GO for doing
probability theory --
the Lebesgue integral is
used to define both probability
of events and expectation of
random variables. It was
A. Kolmogorov in 1933 who
noticed, and wrote a monster paper,
on how to use measure theory
and the Lebesgue integral
for a solid foundation for
probability theory. Since then
for essentially all serious
research in probability and
stochastic processes, much of
mathematical statistics,
nearly all of stochastic optimal
control, is based solidly
on the Kolmogorov,
i.e., measure theory, foundations.
So, from a mathematician not
much interested in probability,
probability theory is just
a special case of measure theory
where the total area (measure)
is just 1. That's not literally
or logically wrong
but does discard a fantastic
baby with any bathwater.
Some of the results in
probability are just astounding
and powerful -- both
beyond belief.
So, in measure theory, here
is what a measure (a definition
of area) is:
We start with a non empty set,
a space, say, X. From
X we have some subsets we
call measurable sets.
So, measurable set A is a
subset of X. In the special
case of probability,
the measurable sets are
the events, that is, e.g.,
all the trials where
our coin comes up heads.
We ask to have enough measurable
sets so that all of them
form a sigma algebra. Why?
Otherwise we don't have much.
A sigma algebra doesn't ask
for much. The sigma part is
supposed to suggest
finite or countably infinite
adding up, as in the usual use of
the capital Greek sigma for
summing.
Say our sigma algebra of
subsets of our measurable
space X is S (probability
theory usually uses script F).
Then we want the empty subset
of X to be an element of S.
For A in S we want X - A (the
relative complement) to be
an element of S. And for
B(i) in S for i = 1, 2, ...,
we want the union of
all the B(i) to be an element of
S. These conditions ensure that
we will have enough sets
in S to have a decently strong
theory.
In what we commonly
do in applied probability,
we wouldn't settle for less.
E.g., the empty set is the
event it never happens.
If H is the event of heads,
then T = X - H, the relative
complement, is the event tails.
If H(i) is the event that
flip i comes up heads, then
the union of all the H(i)
is the event the coin comes
up heads at least once or
doesn't always come up tails.
In probability, those are all
simple situations, and
just from those we need
a sigma algebra of events.
And it turns out, that's enough,
and has been since 1933.
So, for a measure, say,
m, to measurable set A
there is real number m(A),
the measure (think area
or, in probability theory, the
probability) of A. Of course
in probability we call the
measure P instead of m
and write P(A) for the probability
of event A.
You can begin to see that we
are essentially forced into
how Kolmogorov applied measure
theory whether we like it or not.
Sorry 'bout that!
Well, for a measure
m, we want countable
additivity. So,
for disjoint measurable
sets B(i), i = 1, 2, ...,
we want
m(union B(i)) = sum m(B(i))
for some sloppy math notation
since I can't type TeX here!
Usually m(A) is real with
m(A) >= 0, and commonly we
need to have m(X) = infinity
so that m can take on value
infinity.
We can also extend
to m(A) any real number or
any complex number.
Measure theory is the
total cat's meow for
Fourier theory!
To get a sigma algebra
of measurable sets we want,
commonly we start with
a topology, that is,
its collection of open
sets, and ask for the
unique smallest sigma
algebra for which each open
set is also a measurable
set in the sigma algebra.
When we do this on the real
line and assign
the measure of intervals
their ordinary length
and extend that to
as many subsets of the
reals as we can, we get
Lebesgue measure for the
real line. We get a lot
of sets! It's a tricky
exercise, that uses the
axiom of choice, even to
construct a subset of the
reals that is not Lebesgue
measurable. Powerful theory!
Suppose we have spaces X
and Y, each with a sigma
algebra of subsets and
a function
f: X --> Y
Then f is measurable
if for each measurable
subset B of Y
f^(-1)(B) is also
a measurable subset
of X. In measure
theory, when we integrate
a function, we ask that
it be measurable.
In the usual cases, it's
even tough to construct
a function that is not
measurable. Darned
near any limit of
measurable functions is
also measurable -- super
nice theory.
In probability theory,
a random variable
is just a measurable
function where its
domain is
a probability space,
that is, a sample space
Omega with a sigma
algebra of subsets
script F and
a probability measure P.
Of course, there's much
more, stacks, shelves,
racks of books
as difficult as you wish,
but the above is
a simple, intuitive view
from 10,000 feet up.
Or, measure theory is
a nicer theory of
area and area under a curve
that in all the simple cases
gives you just what
you have been used to!
I'm saving your responses to view later as I really appreciate them. Nassim also highly recommends Leo Breiman's text, I'll actually be working through the path you recommend , or something very similar.
> In the standard foundations of probability theory, as laid out by Kolmogorov, we can then model these events and random variables by introducing a sample space (which will be a probability space) to capture all the ambient sources of randomness; events are then modeled as measurable subsets of this sample space, and random variables are modeled as measurable functions on this sample space.
This matches the definitions of Wikipedia pretty closely: "A probability space consists of three parts: A sample space [...], A set of events [...], The assignment of probabilities to the events".
So either you misunderstood that sentence or both Wikipedia and Terrence Tao are wrong.
The sample space is not the probability space. The probability space has a sample space though, which is what the Wikipedia definition says, but not what Tao literally said. I think it is pretty clear what Tao meant though.
Nonsense. Tao is using perfectly idiomatic language here - "The sample space will be a probability space (once we have endowed it with some additional structure)".
That's not good: We don't
want to have to use words
such as endowed and structure
that are not precisely
defined in the context and, thus,
are conceptually fuzzy.
Some intuitive overviews, clearly
labeled as such, are fine and can
be helpful, but "idiomatic language"
just is not. Won't find such in
the writings of W. Rudin, P. Halmos,
J. Neveu, or any of a long list
of authors of some of the best
math books. In a good math or
computer science
journal, a reviewer or the editor
would likely reject "idiomatic
language".
https://www.google.com/#q=probability+the+logic+of+science