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MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: It involves real
phenomena out there. So we have real stuff
that happens. So it might be an arrival
process to a bank that we're trying to model. This is a reality, but
this is what we have been doing so far. We have been playing
with models of probabilistic phenomena. And somehow we need to
tie the two together. The way these are tied is that
we observe the real world and this gives us data. And then based on these data, we
try to come up with a model of what exactly is going on. For example, for an arrival
process, you might ask the model in question, is my arrival
process Poisson or is it something different? If it is Poisson, what is the
rate of the arrival process? Once you come up with your model
and you come up with the parameters of the model, then
you can use it to make predictions about reality or to
figure out certain hidden things, certain hidden aspects
of reality, that you do not observe directly, but you try
to infer what they are. So that's where the usefulness
of the model comes in. Now this field is of course
tremendously useful. And it shows up pretty
much everywhere. So we talked about the polling
examples in the last couple of lectures. This is, of course, a
real application. You sample and on the basis of
the sample that you have, you try to make some inferences
about, let's say, the preferences in a given
population. Let's say in the medical field,
you want to try whether a certain drug makes a
difference or not. So people would do medical
trials, get some results, and then from the data somehow you
need to make sense of them and make a decision. Is the new drug useful
or is it not? How do we go systematically
about the question of this type? A sexier, more recent topic,
there's this famous Netflix competition where Netflix gives
you a huge table of movies and people. And people have rated the
movies, but not everyone has watched all of the
movies in there. You have some of the ratings. For example, this person gave a
4 to that particular movie. So you get the table that's
partially filled. And the Netflix asks
you to make recommendations to people. So this means trying to guess. This person here, how much
would they like this particular movie? And you can start thinking,
well, maybe this person has given somewhat similar ratings
with another person. And if that other person has
also seen that movie, maybe the rating of that other
person is relevant. But of course it's a lot more
complicated than that. And this has been a serious
competition where people have been using every heavy, wet
machinery that there is in statistics, trying to
come up with good recommendation systems. Then the other people, of
course, are trying to analyze financial data. Somebody gives you the sequence
of the values, let's say of the SMP index. You look at something like this and you can ask questions. How do I model these data using
any of the models that we have in our bag of tools? How can I make predictions about
what's going to happen afterwards, and so on? On the engineering side,
anywhere where you have noise inference comes in. Signal processing, in
some sense, is just an inference problem. You observe signals that are
noisy and you try to figure out exactly what's happening
out there or what kind of signal has been sent. Maybe the beginning of the field
could be traced a few hundred years ago where people
would observe, make astronomical observations
of the position of the planets in the sky. They would have some beliefs
that perhaps the orbits of planets is an ellipse. Or if it's a comet, maybe it's
a parabola, hyperbola, don't know what it is. But they would have
a model of that. But, of course, astronomical
measurements would not be perfectly exact. And they would try to find the
curve that fits these data. How do you go about choosing
this particular curve on the base of noisy data and
try to do it in a somewhat principled way? OK, so questions of this
type-- clearly the applications are all
over the place. But how is this related
conceptually with what we have been doing so far? What's the relation between the
field of inference and the field of probability
as we have been practicing until now? Well, mathematically speaking,
what's going to happen in the next few lectures could be just
exercises or homework problems in the class in based
on what we have done so far. That means you're not going
to get any new facts about probability theory. Everything we're going to do
will be simple applications of things that you already
do know. So in some sense, statistics
and inference is just an applied exercise
in probability. But actually, things are
not that simple in the following sense. If you get a probability
problem, there's a correct answer. There's a correct solution. And that correct solution
is unique. There's no ambiguity. The theory of probability has
clearly defined rules. These are the axioms. You're given some information
about probability distributions. You're asked to calculate
certain other things. There's no ambiguity. Answers are always unique. In statistical questions, it's
no longer the case that the question has a unique answer. If I give you data and I ask
you what's the best way of estimating the motion of that
planet, reasonable people can come up with different
methods. And reasonable people will try
to argue that's my method has these desirable properties but
somebody else may say, here's another method that has certain
desirable properties. And it's not clear what
the best method is. So it's good to have some
understanding of what the issues are and to know at least
what is the general class of methods that one tries
to consider, how does one go about such problems. So we're going to see
lots and lots of different inference methods. We're not going to tell
you that one is better than the other. But it's important to understand
what are the concepts between those
different methods. And finally, statistics can
be misused really badly. That is, one can come up with
methods that you think are sound, but in fact they're
not quite that. I will bring some examples next
time and talk a little more about this. So, they want to say, you have
some data, you want to make some inference from them, what
many people will do is to go to Wikipedia, find a statistical
test that they think it applies to that
situation, plug in numbers, and present results. Are the conclusions that they
get really justified or are they misusing statistical
methods? Well, too many people actually
do misuse statistics and conclusions that people
get are often false. So it's important to, besides
just being able to copy statistical tests and use them,
to understand what are the assumptions between the
different methods and what kind of guarantees they
have, if any. All right, so we'll try to do a
quick tour through the field of inference in this lecture and
the next few lectures that we have left this semester and
try to highlight at the very high level the main concept
skills, and techniques that come in. Let's start with some
generalities and some general statements. One first statement is that
statistics or inference problems come up in very
different guises. And they may look as if they are
of very different forms. Although, at some fundamental
level, the basic issues turn out to be always pretty
much the same. So let's look at this example. There's an unknown signal
that's being sent. It's sent through some medium,
and that medium just takes the signal and amplifies it
by a certain number. So you can think of
somebody shouting. There's the air out there. What you shouted will be
attenuated through the air until it gets to a receiver. And that receiver then observes
this, but together with some random noise. Here I meant S. S is the signal
that's being sent. And what you observe is an X. You observe X, so what kind
of inference problems could we have here? In some cases, you want to build
a model of the physical phenomenon that you're
dealing with. So for example, you don't know
the attenuation of your signal and you try to find out what
this number is based on the observations that you have. So the way this is done in
engineering systems is that you design a certain signal, you
know what it is, you shout a particular word, and then
the receiver listens. And based on the intensity of
the signal that they get, they try to make a guess about A. So
you don't know A, but you know S. And by observing X,
you get some information about what A is. So in this case, you're trying
to build a model of the medium through which your signal
is propagating. So sometimes one would call
problems of this kind, let's say, system identification. In a different version of an
inference problem that comes with this picture, you've
done your modeling. You know your A. You know the
medium through which the signal is going, but it's
a communication system. This person is trying
to communicate something to that person. So you send the signal S, but
that person receives a noisy version of S. So that person
tries to reconstruct S based on X. So in both cases, we have a
linear relation between X and the unknown quantity. In one version, A is the unknown
and we know S. In the other version, A is known,
and so we try to infer S. Mathematically, you can see that
this is essentially the same kind of problem
in both cases. Although, the kind of practical
problem that you're trying to solve is a
little different. So we will not be making any
distinctions between problems of the model building type as
opposed to models where you try to estimate some unknown
signal and so on. Because conceptually, the tools
that one uses for both types of problems are
essentially the same. OK, next a very useful
classification of inference problems-- the unknown quantity that you're
trying to estimate could be either a discrete
one that takes a small number of values. So this could be discrete
problems, such as the airplane radar problem we encountered
back a long time ago in this class. So there's two possibilities-- an airplane is out there or an
airplane is not out there. And you're trying to
make a decision between these two options. Or you can have other problems
would you have, let's say, four possible options. You don't know which one is
true, but you get data and you try to figure out which
one is true. In problems of these kind,
usually you want to make a decision based on your data. And you're interested in the
probability of making a correct decision. You would like that
probability to be as high as possible. Estimation problems are
a little different. Here you have some continuous
quantity that's not known. And you try to make a good
guess of that quantity. And you would like your guess to
be as close as possible to the true quantity. So the polling problem
was of this type. There was an unknown fraction
f of the population that had some property. And you try to estimate f as
accurately as you can. So the distinction here is that
usually here the unknown quantity takes on discrete
set of values. Here the unknown quantity
takes a continuous set of values. Here we're interested in the
probability of error. Here we're interested in
the size of the error. Broadly speaking, most inference
problems fall either in this category or
in that category. Although, if you want to
complicate life, you can also think or construct problems
where both of these aspects are simultaneously present. OK, finally since we're in
classification mode, there is a very big, important dichotomy
into how one goes about inference problems. And here there's two
fundamentally different philosophical points of view,
which is how do we model the quantity that is unknown? In one approach, you say there's
a certain quantity that has a definite value. It just happens that
they don't know it. But it's a number. There's nothing random
about it. So think of trying to estimate
some physical quantity. You're making measurements, you
try to estimate the mass of an electron, which
is a sort of universal physical constant. There's nothing random
about it. It's a fixed number. You get data, because you have
some measuring apparatus. And that measuring apparatus,
depending on what that results that you get are affected by the
true mass of the electron, but there's also some noise. You take the data out of your
measuring apparatus and you try to come up with
some estimate of that quantity theta. So this is definitely a
legitimate picture, but the important thing in this picture
is that this theta is written as lowercase. And that's to make the point
that it's a real number, not a random variable. There's a different
philosophical approach which says, well, anything that I
don't know I should model it as a random variable. Yes, I know. The mass of the electron
is not really random. It's a constant. But I don't know what it is. I have some vague sense,
perhaps, what it is perhaps because of the experiments
that some other people carried out. So perhaps I have a prior
distribution on the possible values of Theta. And that prior distribution
doesn't mean that the nature is random, but it's more of a
subjective description of my subjective beliefs of where do
I think this constant number happens to be. So even though it's not truly
random, I model my initial beliefs before the experiment
starts. In terms of a prior
distribution, I view it as a random variable. Then I observe another related
random variable through some measuring apparatus. And then I use this again
to create an estimate. So these two pictures
philosophically are very different from each other. Here we treat the unknown
quantities as unknown numbers. Here we treat them as
random variables. When we treat them as a random
variables, then we know pretty much already what we
should be doing. We should just use
the Bayes rule. Based on X, find
the conditional distribution of Theta. And that's what we will be doing
mostly over this lecture and the next lecture. Now in both cases, what you end
up getting at the end is an estimate. But actually, that estimate is
what kind of object is it? It's a random variable
in both cases. Why? Even in this case where
theta was a constant, my data are random. I do my data processing. So I calculate a function
of the data, the data are random variables. So out here we output something
which is a function of a random variable. So this quantity here
will be also random. It's affected by the noise and
the experiment that I have been doing. That's why these estimators
will be denoted by uppercase Thetas. And we will be using hats. Hat, usually in estimation,
means an estimate of something. All right, so this is
the big picture. We're going to start with
the Bayesian version. And then the last few lectures
we're going to talk about the non-Bayesian version or
the classical one. By the way, I should say that
statisticians have been debating fiercely for 100 years
whether the right way to approach statistics is to go
the classical way or the Bayesian way. And there have been tides going
back and forth between the two sides. These days, Bayesian methods
tend to become a little more popular for various reasons. We're going to come back
to this later. All right, so in Bayesian
estimation, what we got in our hands is Bayes rule. And if you have Bayes rule,
there's not a lot that's left to do. We have different forms of the
Bayes rule, depending on whether we're dealing with
discrete data, And discrete quantities to estimate, or
continuous data, and so on. In the hypothesis testing
problem, the unknown quantity Theta is discrete. So in both cases here,
we have a P of Theta. We obtain data, the X's. And on the basis of the X that
we observe, we can calculate the posterior distribution
of Theta, given the data. So to use Bayesian inference,
what do we start with? We start with some priors. These are our initial
beliefs about what Theta that might be. That's before we do
the experiment. We have a model of the
experimental aparatus. And the model of the
experimental apparatus tells us if this Theta is true, I'm
going to see X's of that kind. If that other Theta is true, I'm
going to see X's that they are somewhere else. That models my apparatus. And based on that knowledge,
once I observe I have these two functions in my hands, we
have already seen that if you know those two functions, you
can also calculate the denominator here. So all of these functions are
available, so you can compute, you can find a formula for
this function as well. And as soon as you observe the
data, that X's, you plug in here the numerical value
of those X's. And you get a function
of Theta. And this is the posterior
distribution of Theta, given the data that you have seen. So you've already done
a fair number of exercises of these kind. So we not say more about this. And there's a similar formula as
you know for the case where we have continuous data. If the X's are continuous random
variable, then the formula is the same, except
that X's are described by densities instead of being
described by a probability mass functions. OK, now if Theta is continuous,
then we're dealing with estimation problems. But the story is once
more the same. You're going to use the Bayes
rule to come up with the posterior density of Theta,
given the data that you have observed. Now just for the sake of the
example, let's come back to this picture here. Suppose that something is flying
in the air, and maybe this is just an object in the
air close to the Earth. So because of gravity, the
trajectory that it's going to follow it's going to
be a parabola. So this is the general equation
of a parabola. Zt is the position of my
objects at time t. But I don't know exactly
which parabola it is. So the parameters of the
parabola are unknown quantities. What I can do is to go and
measure the position of my objects at different times. But unfortunately, my
measurements are noisy. What I want to do is to model
the motion of my object. So I guess in the picture, the
axis would be t going this way and Z going this way. And on the basis of the
data that they get, these are my X's. I want to figure
out the Thetas. That is, I want to figure
out the exact equation of this parabola. Now if somebody gives you
probability distributions for Theta, these would
be your priors. So this is given. We need the conditional
distribution of the X's given the Thetas. Well, we have the conditional
distribution of Z, given the Thetas from this equation. And then by playing with this
equation, you can also find how is X distributed if Theta
takes a particular value. So you do have all of the
densities that you might need. And you can apply
the Bayes rule. And at the end, your end result
would be a formula for the distribution of Theta,
given to the X that you have observed-- except for one sort of
computation, or to make things more interesting. Instead of these X's and Theta's
being single random variables that we have here,
typically those X's and Theta's will be
multi-dimensional random variables or will correspond
to multiple ones. So this little Theta here
actually stands for a triplet of Theta0, Theta1, and Theta2. And that X here stands here for
the entire sequence of X's that we have observed. So in reality, the object that
you're going to get at to the end after inference is done is
a function that you plug in the values of the data and you
get the function of the Theta's that tells you the
relative likelihoods of different Theta triplets. So what I'm saying is that this
is no harder than the problems that you have dealt
with so far, except perhaps for the complication that's
usually in interesting inference problems. Your Theta's and X's are often
the vectors of random variables instead of individual
random variables. Now if you are to do estimation
in a case where you have discrete data, again the
situation is no different. We still have a Bayes rule of
the same kind, except that densities gets replaced
by PMF's. If X is discrete, you put a P
here instead of putting an f. So an example of an estimation
problem with discrete data is similar to the polling
problem. You have a coin. It has an unknown
parameter Theta. This is the probability
of obtaining heads. You flip the coin many times. What can you tell me about
the true value of Theta? A classical statistician, at
this point, would say, OK, I'm going to use an estimator,
the most reasonable one, which is this. How many heads did they
obtain in n trials? Divide by the total
number of trials. This is my estimate of
the bias of my coin. And then the classical
statistician would continue from here and try to prove some
properties and argue that this estimate is a good one. For example, we have the weak
law of large numbers that tells us that this particular
estimate converges in probability to the
true parameter. This is a kind of guarantee
that's useful to have. And the classical statistician
would pretty much close the subject in this way. What would the Bayesian
person do differently? The Bayesian person would start
by assuming a prior distribution of Theta. Instead of treating Theta as
an unknown constant, they would say that Theta would speak
randomly or pretend that it would speak randomly
and assume a distribution on Theta. So for example, if you don't
know they need anything more, you might assume that any value
for the bias of the coin is as likely as any other value
of the bias of the coin. And this way so the probability
distribution that's uniform. Or if you have a little more
faith in the manufacturing processes that's created that
coin, you might choose your prior to be a distribution
that's centered around 1/2 and sits fairly narrowly centered
around 1/2. That would be a prior
distribution in which you say, well, I believe that the
manufacturer tried to make my coin to be fair. But they often makes some
mistakes, so it's going to be, I believe, it's approximately
1/2 but not quite. So depending on your beliefs,
you would choose an appropriate prior for the
distribution of Theta. And then you would use the
Bayes rule to find the probabilities of different
values of Theta, based on the data that you have observed. So no matter which version of
the Bayes rule that you use, the end product of the Bayes
rule is going to be either a plot of this kind or a
plot of that kind. So what am I plotting here? This axis is the Theta axis. These are the possible values
of the unknown quantity that we're trying to estimate. In the continuous
case, theta is a continuous random variable. I obtain my data. And I plot for the posterior
probability distribution after observing my data. And I'm plotting here the
probability density for Theta. So this is a plot
of that density. In the discrete case, theta can
take finitely many values or a discrete set of values. And for each one of those
values, I'm telling you how likely is that the value to be
the correct one, given the data that I have observed. And in general, what you would
go back to your boss and report after you've done all
your inference work would be either a plot of this kinds
or of that kind. So you go to your boss
who asks you, what is the value of Theta? And you say, well, I only
have limited data. That I don't know what it is. It could be this, with
so much probability. There's probability. OK, let's throw in some
numbers here. There's probability 0.3 that
Theta is this value. There's probability 0.2 that
Theta is this value, 0.1 that it's this one, 0.1 that it's
this one, 0.2 that it's that one, and so on. OK, now bosses often want
simple answers. They say, OK, you're
talking too much. What do you think Theta is? And now you're forced
to make a decision. If that was the situation and
you have to make a decision, how would you make it? Well, I'm going to make a
decision that's most likely to be correct. If I make this decision,
what's going to happen? Theta is this value with
probability 0.2, which means there's probably 0.8 that
they make an error if I make that guess. If I make that decision, this
decision has probably 0.3 of being the correct one. So I have probably
of error 0.7. So if you want to just maximize
the probability of giving the correct decision, or
if you want to minimize the probability of making an
incorrect decision, what you're going to choose to report
is that value of Theta for which the probability
is highest. So in this case, I would
choose to report this particular value, the most
likely value of Theta, given what I have observed. And that value is called them
maximum a posteriori probability estimate. It's going to be this
one in our case. So picking the point in the
posterior PMF that has the highest probability. That's the reasonable
thing to do. This is the optimal thing to do
if you want to minimize the probability of an incorrect
inference. And that's what people do
usually if they need to report a single answer, if they need
to report a single decision. How about in the estimation
context? If that's what you know about
Theta, Theta could be around here, but there's also some
sharp probability that it is around here. What's the single answer that
you would give to your boss? One option is to use the same
philosophy and say, OK, I'm going to find the Theta at which
this posterior density is highest. So I would pick this point
here and report this particular Theta. So this would be my Theta,
again, Theta MAP, the Theta that has the highest a
posteriori probability, just because it corresponds to
the peak of the density. But in this context, the
maximum a posteriori probability theta was the
one that was most likely to be true. In the continuous case, you
cannot really say that this is the most likely value
of Theta. In a continuous setting, any
value of Theta has zero probability, so when we
talk about densities. So it's not the most likely. It's the one for which the
density, so the probabilities of that neighborhoods,
are highest. So the rationale for picking
this particular estimate in the continuous case is much
less compelling than the rationale that we had in here. So in this case, reasonable
people might choose different quantities to report. And the very popular one would
be to report instead the conditional expectation. So I don't know quite
what Theta is. Given the data that I have,
Theta has this distribution. Let me just report the average
over that distribution. Let me report to the center
of gravity of this figure. And in this figure, the center
of gravity would probably be somewhere around here. And that would be a different
estimate that you might choose to report. So center of gravity is
something around here. And this is a conditional
expectation of Theta, given the data that you have. So these are two, in some sense,
fairly reasonable ways of choosing what to report
to your boss. Some people might choose
to report this. Some people might choose
to report that. And a priori, if there's no
compelling reason why one would be preferable than other
one, unless you set some rules for the game and you describe
a little more precisely what your objectives are. But no matter which one you
report, a single answer, a point estimate, doesn't really
tell you the whole story. There's a lot more information
conveyed by this posterior distribution plot than
any single number that you might report. So in general, you may wish to
convince your boss that's it's worth their time to look at the
entire plot, because that plot sort of covers all
the possibilities. It tells your boss most likely
we're in that range, but there's also a distinct change
that our Theta happens to lie in that range. All right, now let us try to
perhaps differentiate between these two and see under what
circumstances this one might be the better estimate
to perform. Better with respect to what? We need some rules. So we're going to throw
in some rules. As a warm up, we're going to
deal with the problem of making an estimation if you
had no information at all, except for a prior
distribution. So this is a warm up for what's
coming next, which would be estimation that takes
into account some information. So we have a Theta. And because of your subjective
beliefs or models by others, you believe that Theta is
uniformly distributed between, let's say, 4 and 10. You want to come up with
a point estimate. Let's try to look
for an estimate. Call it c, in this case. I want to pick a number
with which to estimate the value of Theta. I will be interested in the size
of the error that I make. And I really dislike large
errors, so I'm going to focus on the square of the error
that they make. So I pick c. Theta that has a random value
that I don't know. But whatever it is, once it
becomes known, it results into a squared error between
what it is and what I guessed that it was. And I'm interested in making
a small air on the average, where the average is taken
with respect to all the possible and unknown
values of Theta. So the problem, this is a least
squares formulation of the problem, where we
try to minimize the least squares errors. How do you find the optimal c? Well, we take that expression
and expand it. And it is, using linearity
of expectations-- square minus 2c expected
Theta plus c squared-- that's the quantity that
we want to minimize, with respect to c. To do the minimization, take the
derivative with respect to c and set it to 0. So that differentiation gives us
from here minus 2 expected value of Theta plus
2c is equal to 0. And the answer that you get by
solving this equation is that c is the expected
value of Theta. So when you do this
optimization, you find that the optimal estimate, the
things you should be reporting, is the expected
value of Theta. So in this particular example,
you would choose your estimate c to be just the middle
of these values, which would be 7. OK, and in case your
boss asks you, how good is your estimate? How big is your error
going to be? What you could report is the
average size of the estimation error that you are making. We picked our estimates to be
the expected value of Theta. So for this particular way that
I'm choosing to do my estimation, this is the mean
squared error that I get. And this is a familiar
quantity. It's just the variance
of the distribution. So the expectation is that
best way to estimate a quantity, if you're interested
in the mean squared error. And the resulting mean squared
error is the variance itself. How will this story change if
we now have data as well? Now having data means that
we can compute posterior distributions or conditional
distributions. So we get transported into a new
universe where instead the working with the original
distribution of Theta, the prior distribution, now we work
with the condition of distribution of Theta,
given the data that we have observed. Now remember our old slogan that
conditional models and conditional probabilities are
no different than ordinary probabilities, except that we
live now in a new universe where the new information has
been taken into account. So if you use that philosophy
and you're asked to minimize the squared error but now that
you live in a new universe where X has been fixed to
something, what would the optimal solution be? It would again be the
expectation of theta, but which expectation? It's the expectation which
applies in the new conditional universe in which we
live right now. So because of what we did
before, by the same calculation, we would find that
the optimal estimates is the expected value of X of
Theta, but the optimal estimate that takes
into account the information that we have. So the conclusion, once you get
your data, if you want to minimize the mean squared error,
you should just report the conditional estimation of
this unknown quantity based on the data that you have. So the picture here is that
Theta is unknown. You have your apparatus that
creates measurements. So this creates an X. You take
an X, and here you have a box that does calculations. It does calculations and it
spits out the conditional expectation of Theta, given the
particular data that you have observed. And what we have done in this
class so far is, to some extent, developing the
computational tools and skills to do with this particular
calculation-- how to calculate the posterior
density for Theta and how to calculate expectations,
conditional expectations. So in principle, we know
how to do this. In principle, we can program a
computer to take the data and to spit out condition
expectations. Somebody who doesn't think like
us might instead design a calculating machine that does
something differently and produces some other estimate. So we went through this argument
and we decided to program our computer to
calculate conditional expectations. Somebody else came up with some
other crazy idea for how to estimate the random
variable. They came up with some function
g and the programmed it, and they designed a machine
that estimates Theta's by outputting a certain
g of X. That could be an alternative
estimator. Which one is better? Well, we convinced ourselves
that this is the optimal one in a universe where we have
fixed the particular value of the data. So what we have proved so far
is a relation of this kind. In this conditional universe,
the mean squared error that I get-- I'm the one who's using
this estimator-- is less than or equal than the
mean squared error that this person will get, the person
who uses that estimator. For any particular value of
the data, I'm going to do better than the other person. Now the data themselves
are random. If I average over all possible
values of the data, I should still be better off. If I'm better off for any
possible value X, then I should be better off on the
average over all possible values of X. So let us average both sides of
this quantity with respect to the probability distribution
of X. If you want to do it formally, you can write
this inequality between numbers as an inequality between
random variables. And it tells that no matter
what that random variable turns out to be, this quantity
is better than that quantity. Take expectations of both
sides, and you get this inequality between expectations
overall. And this last inequality tells
me that the person who's using this estimator who produces
estimates according to this machine will have a mean squared
estimation error that's less than or equal to
the estimation error that's produced by the other person. In a few words, the conditional
expectation estimator is the optimal
estimator. It's the ultimate estimating
machine. That's how you should solve
estimation problems and report a single value. If you're forced to report a
single value and if you're interested in estimation
errors. OK, while we could have told you
that story, of course, a month or two ago, this is really
about interpretation -- about realizing that conditional
expectations have a very nice property. But other than that, any
probabilistic skills that come into this business are just the
probabilistic skills of being able to calculate
conditional expectations, which you already
know how to do. So conclusion, all of optimal
Bayesian estimation just means calculating and reporting
conditional expectations. Well, if the world were that
simple, then statisticians wouldn't be able to find jobs
if life is that simple. So real life is not
that simple. There are complications. And that perhaps makes their
life a little more interesting. OK, one complication is that we
would deal with the vectors instead of just single
random variables. I use the notation here
as if X was a single random variable. In real life, you get
several data. Does our story change? Not really, same argument-- given all the data that you
have observed, you should still report the conditional
expectation of Theta. But what kind of work does it
take in order to report this conditional expectation? One issue is that you need to
cook up a plausible prior distribution for Theta. How do you do that? In a given application , this
is a bit of a judgment call, what prior would you
be working with. And there's a certain
skill there of not making silly choices. A more pragmatic, practical
issue is that this is a formula that's extremely nice
and compact and simple that you can write with
minimal ink. But the behind it there could
be hidden a huge amount of calculation. So doing any sort of
calculations that involve multiple random variables really
involves calculating multi-dimensional integrals. And the multi-dimensional
integrals are hard to compute. So implementing actually this
calculating machine here may not be easy, might be
complicated computationally. It's also complicated in terms
of not being able to derive intuition about it. So perhaps you might want to
have a simpler version, a simpler alternative to this
formula that's easier to work with and easier to calculate. We will be talking about
one such simpler alternative next time. So again, to conclude, at
the high level, Bayesian estimation is very, very simple,
given that you have mastered everything that
has happened in this course so far. There are certain practical
issues and it's also good to be familiar with the concepts
and the issues that in general, you would prefer to
report that complete posterior distribution. But if you're forced to report a
point estimate, then there's a number of reasonable
ways to do it. And perhaps the most reasonable
one is to just the report the conditional
expectation itself.
