DAVID SONTAG: So today's lecture
is going to be about causality. Who's heard about
causality before? Raise your hand. What's the number one thing
that you hear about when thinking about causality? Yeah? AUDIENCE: Correlation
does not imply causation. DAVID SONTAG: Correlation
does not imply causation. Anything else come to mind? That's what came to my mind. Anything else come to mind? So up until now in
the semester, we've been talking about purely
predictive questions. And for purely
predictive questions, one could argue that
correlation is good enough. If we have some
signs in our data that are predictive of
some outcome of interest, we want to be able to
take advantage of that. Whether it's
upstream, downstream, the causal directionality is
irrelevant for that purpose. Although even that
isn't quite true, right, because Pete and I have been
hinting throughout the semester that there are times
when the data changes on you, for example, when you go
from one institution to another or when you have non-stationary. And in those situations,
having a deeper understanding about the data might
allow one to build an additional robustness to
that type of data set shift. But there are other
reasons as well why understanding something about
your underlying data generating processes can be
really important. It's because often,
the questions that we want to answer when
it comes to health care are not predictive questions,
their causal questions. And so what I'll do now is I'll
walk through a few examples of what I mean by this. Let's start out with what we saw
in Lecture 4 and in Problem Set 2, where we looked
at the question of how we can do early
detection of type 2 diabetes. You used Truven
MarketScan's data set to build a
risk stratification algorithm for
detecting who is going to be newly diagnosed
with diabetes one to three years from now. And if you think about
how one might then try to deploy that
algorithm, you might, for example, try to
get patients into the clinic to get them diagnosed. But the next set of
questions are usually about the so what question. What are you going to do
based on that prediction? Once diagnosed, how
will you intervene? And at the end of the
day, the interesting goal is not one of how do
you find them early, but how do you prevent them
from developing diabetes? Or how do you prevent the
patient from developing complications of diabetes? And those are questions
about causality. Now, when we built
a predictive model and we introspected
at the weight, we might have noticed
some interesting things. For example, if you looked at
the highest negative weights, which I'm not sure if we did
as part of the assignment but is something that I did
as part of my research study, you see that gastric
bypass surgery has the biggest negative weight. Does that mean that if you give
an obese person gastric bypass surgery, that will prevent
them from developing type 2 diabetes? That's an example of a causal
question which is raised by this predictive model. But just by looking
at the weight alone, as I'll
show you this week, you won't be able to
correctly infer that there is a causal relationship. And so part of what
we will be doing is coming up with a mathematical
language for thinking about how does one
answer, is there a causal relationship here? Here's a second example. Right before spring break
we had a series of lectures about diagnosis,
particularly diagnosis from imaging data of
a variety of kinds, whether it be
radiology or pathology. And often, questions
are of this sort. Here is a woman's breasts. She has breast cancer. Maybe you have an associated
pathology slide as well. And you want to know what is
the risk of this person dying in the next five years. So one can take a
deep learning model, learn to predict
what one observes. So in the patient in your
data set, you have the input and you have, let's
say, survival time. And you might use that
to predict something about how long it takes
from diagnosis to death. And based on those predictions,
you might take actions. For example, if you predict
that a patient is not risky, then you might
conclude that they don't need to get treatment. But that could be
really, really dangerous, and I'll just give
you one example of why that could be dangerous. These predictive models,
if you're learning them in this way, the outcome,
in this case let's say time to death, is
going to be affected by what's happened in between. So, for example,
this patient might have been receiving
treatment, and because of them receiving treatment in
between the time from diagnosis to death, it might have
prolonged their life. And so for this patient
in your data set, you might have observed that
they lived a very long time. But if you ignore what
happens in between and you simply learn to predict
y from X, X being the input, then a new patient comes
along and you predicted that new patient is going
to survive a long time, and it would be completely
the wrong conclusion to say that you don't need
to treat that patient. Because, in fact, the only
reason the patients like them in the training data
lived a long time is because they were treated. And so when it comes to this
field of machine learning and health care, we need
to think really carefully about these types of questions
because an error in the way that we formalize our
problem could kill people because of mistakes like this. Now, other questions
are ones about not how do we predict
outcomes but how do we guide treatment decisions. So, for example, as
data from pathology gets richer and
richer and richer, we might think that we
can now use computers to try to better predict
who is likely to benefit from a treatment than
humans could do alone. But the challenge
with using algorithms to do that is that people
respond differently to treatment, and the
data which is being used to guide treatment is
biased based on existing treatment guidelines. So, similarly, to the previous
question, we could ask, what would happen if we trained
to predict past treatment decisions? This would be the
most naive way to try to use data to guide
treatment decisions. So maybe you see David
gets treatment A, John gets treatment B,
Juana gets treatment A. And you might ask then,
OK, a new patient comes in, what should this new
patient be treated with? And if you've just
learned a model to predict from what you
know about the treatment that David is likely
to get, then the best that you could hope
to do is to do as well as existing clinical practice. So if we want to go beyond
current clinical practice, for example, to recognize
that there is heterogeneity in treatment response, then
we have to somehow change the question that we're asking. I'll give you one
last example, which is perhaps a more traditional
question of, does X cause y? For example, does
smoking cause lung cancer is a major question of
societal importance. Now, you might be familiar
with the traditional way of trying to answer questions
of this nature, which would be to do a randomized
controlled trial. Except this isn't
exactly the type of setting where you could do
randomized controlled trials. How would you feel if you were
a smoker and someone came up to you and said, you have
to stop smoking because I need to see what happens? Or how would you feel
if you were a non-smoker and someone came
up to you and said, you have to start smoking? That would be both not feasible
and completely unethical. And so if we want to
try to answer questions like this from data,
we need to start thinking about
how can we design, using observational
data, ways of answering questions like this. And the challenge
is that there's going to be bias in the data
because of who decides to smoke and who decides not to smoke. So, for example,
the most naive way you might try to
answer this question would be to look at the
conditional likelihood of getting lung
cancer among smokers and getting lung cancer
among non-smokers. But those numbers, as you'll
see in the next few slides, can be very misleading
because there might be confounding
factors, factors that would, for example, both
cause people to be a smoker and cause them to
receive lung cancer, which would differentiate
between these two numbers. And we'll have a
very concrete example of this in just a few minutes. So to properly answer
all of these questions, one needs to be thinking
in terms of causal graphs. So rather than the
traditional setup in machine learning where you just
have inputs and outputs, now we need to have triplets. Rather than having
inputs and outputs, we need to be thinking
of inputs, interventions, and outcomes or outputs. So we now need be having
three quantities in mind. And we have to start
thinking about, well, what is the causal relationship
between these three? So for those of you who have
taken more graduate level machine learning
classes, you might be familiar with ideas
such as Bayesian networks. And when I went to
undergrad and grad school and I studied machine
learning, for the longest time I thought causal
inference had to do with learning causal graphs. So this is what I thought
causal inference was about. You have data of the
following nature-- 1, 0, 0, 1, dot, dot, dot. So here, there are
four random variables. I'm showing the realizations
of those four binary variables one per row, and you have
a data set like this. And I thought
causal inference had to do with taking data like
this and trying to figure out, is the underlying
Bayesian network that created that data, is it
X1 goes to X2 goes to X3 to X4? Or I'll say, this is X1,
that's X2, x3, and X4. Or maybe the causal graph
is X1, to X2, to X3, to x4. And trying to distinguish
between these different causal graphs from observational
data is one type of question that one can ask. And the one thing you learn
in traditional machine learning treatments of
this is that sometimes you can't distinguish between
these causal graphs from the data you have. For example, suppose you just
had two random variables. Because any distribution could
be represented by probability of X1 times probability
of X2 given X1, according to just rule of
conditional probability, and similarly, any
distribution can be represented as the opposite, probability
of X2 times probability of X1 given X2, which would
look like this, the statement that one would make
is that if you just had data involving X1 and
X2, you couldn't distinguish between these two causal graphs,
X1 causes X2 or X2 causes X1. And usually another
treatment would say, OK, but if you have a third variable
and you have a V structure or something like X1 goes
to x2, X1 goes to X3, this you could distinguish from,
let's say, a chain structure. And then the final
answer to what is causal inference
from this philosophy would be something like, OK, if
you're in a setting like this and you can't distinguish
between X1 causes X2 or X2 causes X1, then you
do some interventions, like you intervene on X1 and you
look to see what happens to X2, and that'll help you disentangle
these directions of causality. None of this is what we're
going to be talking about today. Today, we're going to be
talking about the simplest, simplest possible setting
you could imagine, that graph shown up there. You have three sets of
random variables, X, which is perhaps
a vector, so it's high dimensional, a
single random variable T, and a single
random variable Y. And we know the
causal graph here. We're going to
suppose that we know the directionality, that we
know that X might cause T and X and T might cause Y. And
the only thing we don't know is the strength of the edges. All right. And so now let's try to
think through this in context of the previous examples. Yeah, question? AUDIENCE: Just to make sure-- so
T does not affect X in any way? DAVID SONTAG: Correct,
that's the assumption we're going to make here. So let's try to
instantiate this. So we'll start
with this example. X might be what you know about
the patient at diagnosis. T, I'm going to assume for
the purposes of today's class, is a decision between two
different treatment plans. And I'm going to simplify
the state of the world. I'm going to say those
treatment plans only depend on what you know about
the patient at diagnosis. So at diagnosis, you
decide, I'm going to be giving them this
sequence of treatments at this three-month interval
or this other sequence of treatment at, maybe,
that four-month interval. And you make that decision
just based on diagnosis and you don't change it based
on anything you observe. Then the causal graph
of relevance there is, based on what you know about
the patient at diagnosis, which I'm going to
say X is a vector because maybe it's
based on images, your whole electronic
health record. There's a ton of data you have
on the patient at diagnosis. Based on that, you
make some decision about a treatment plan. I'm going to call that
T. T could be binary, a choice between two treatments,
it could be continuous, maybe you're deciding the
dosage of the treatment, or it could be
maybe even a vector. For today's lecture,
I'm going to suppose that T is just binary,
just involves two choices. But most of what
I'll tell you about will generalize to the setting
where T is non-binary as well. But critically,
I'm going to make the assumption for
today's lecture that you're not observing
new things in between. So, for example, in this
whole week's lecture, the following scenario
will not happen. Based on diagnosis, you make a
decision about treatment plan. Treatment plan starts,
you got new observations. Based on those new
observations, you realize that treatment
plan isn't working and change to another
treatment plan, and so on. So that scenario goes
by a different name, which is called
dynamic treatment regimes or off-policy
reinforcement learning, and that we'll learn
about next week. So for today's and
Thursday's lecture, we're going to suppose
you base on what you know about the patient at
this time, you make a decision, you execute the decision,
and you look at some outcome. So X causes T, not
the other way around. And that's pretty clear
because of our prior knowledge about this problem. It's not that the
treatment affects what their diagnosis was. And then there's the outcome
Y, and there, again, we suppose the outcome, what
happens to the patient, maybe survival time, for example, is
a function of what treatment they're getting and
aspects about that patient. So this is the causal graph. We know it. But we don't know,
does that treatment do anything to this patient? For whom does this
treatment help the most? And those are the types
of questions we're going to try to answer today. Is the setting clear? OK. Now, these questions
are not new questions. They've been studied
for decades in fields such as political science,
economics, statistics, biostatistics. And the reason why they're
studied in those other fields is because often you don't
have the ability to intervene, and one has to try to
answer these questions from observational data. For example, you might ask, what
will happen to the US economy if the Federal Reserve raises
US interest rates by 1%? When's the last time you
heard of the Federal Reserve doing a randomized
controlled trial? And even if they had done a
randomized controlled trial, for example, flipped a coin to
decide which way the interest rates would go, it wouldn't
be comparable had they done that experiment today to
if they had done that experiment two years from now because
the state of the world has changed in those years. Let's talk about
political science. I have close colleagues of mine
at NYU who look at Twitter, and they want to
ask questions like, how can we influence
elections, or how are elections influenced? So you might look at some
unnamed actors, possibly people supported by the
Russian government, who are posting to Twitter
or their social media. And you might ask the
question of, well, did that actually
influence the outcome of the previous
presidential election? Again, in that
scenario, it's one of, well, we have this
data, something happened in the
world, and we'd like to understand what was
the effect of that action, but we can't exactly go back
and replay to do something else. So these are fundamental
questions that appear all across the
sciences, and of course they're extremely relevant
in health care, but yet, we don't teach
them in our introduction to machine learning classes. We don't teach them in our
undergraduate computer science education. And I view this as a major
hole in our education, which is why we're
spending two weeks on it in this course, which
is still not enough. But what has changed
between these fields, and what is relevant
in health care? Well, the traditional way
in which these questions were asked in
statistics were ones where you took a huge
amount of domain knowledge to, first of all, make sure
you're setting up the problem correctly, and that's always
going to be important. But then to think through what
are all of the factors that could influence the
treatment decisions called the confounding factors. And the traditional
approach is one would write down 10,
20 different things, and make sure that you do
some analysis, including the analysis I'll show you about
in today and Thursday's lecture using those 10 or 20 variables. But where this field
is going is one of now having high dimensional data. So I talked about how you
might have imaging data for X, you might have the whole entire
patient's electronic health record data facts. And the traditional approaches
that the statistics community used to work on no longer
work in this high dimensional setting. And so, in fact, it's actually
a really interesting area for research, one that my
lab is starting to work on and many other labs, where
we could ask, how can we bring machine learning
algorithms that are designed to work with high
dimensional data to answer these types of
causal inference questions? And in today's lecture, you'll
see one example of reduction from causal inference
to machine learning, where we'll be
able to use machine learning to answer one of those
causal inference questions. So the first thing we need
is some language in order to formalize these notions. So I will work within what's
known as the Rubin-Neyman Causal Model, where
we talk about what are called potential outcomes. What would have happened under
this world or that world? We'll call Y 0,
and often it will be denoted as Y underscore
0, sometimes it'll be denoted as Y parentheses
0, and sometimes it'll be denoted as Y given
X comma do Y equals 0. And all three of these
notations are equivalent. So Y is 0 corresponds
to what would have happened to this
individual if you gave them treatment to 0. And Y1 is the potential
outcome of what would have happened to this
individual had you gave them treatment one. So you could think about Y1
as being giving the blue pill and Y0 as being
given the red pill. Now, once you can talk about
these states of the world, then one could start
to ask questions of what's better, the red
pill or the blue pill? And one can formalize
that notion mathematically in terms of what's called the
conditional average treatment effect, and this
also goes by the name of individual treatment effect. So it's going to take
as input Xi, which I'm going to denote as
the data that you had at baseline for the individual. It's the covariance, the
features for the individual. And one wants to know, well,
for this individual with what we know about them, what's the
difference between giving them treatment one or giving
them treatment zero? So mathematically, that
corresponds to a difference in expectations. It's a difference in
expectation of Y1 from Y0. Now, the reason why I'm
calling this an expectation is because I'm not going to
assume that Y1 and Y0 are deterministic
because maybe there's some bad luck component. Like, maybe a medication usually
works for this type of person, but with a flip of a coin,
sometimes it doesn't work. And so that's the
randomness that I'm referring to when I talk about
probability over Y1 given Xi. And so the CATE looks
at the difference in those two expectations. And then one can now talk about
what the average treatment effect is, which is the
difference between those two. So the average treatment effect
is now the expectation of-- I'll say the expectation of
the CATE over the distribution of people, P of X.
Now, we're going to go through this in four
different ways in the next 10 minutes, and then you're
going to go over it five more ways doing your
homework assignment, and you'll go over it two more
ways on Friday in recitation. So if you don't get it
just yet, stay with me, you'll get it by the
end of this week. Now, in the data that you
observe for an individual, all you see is what happened
under one of the interventions. So, for example, if the i'th
individual in your data set received treatment Ti equals
1, then what you observe, Yi is the potential outcome Y1. On the other hand, if the
individual in your data set received treatment
Ti equals 0, then what you observed
for that individual is the potential outcome Y0. So that's the observed
factual outcome. But one could also talk
about the counterfactual of what would have
happened to this person had the opposite treatment
been done for them. Notice that I just swapped each
Ti for 1 minus Ti, and so on. Now, the key challenge in the
field is that in your data set, you only observe the
factual outcomes. And when you want to reason
about the counterfactual, that's where you have to impute
this unobserved counterfactual outcome. And that is known as
the fundamental problem of causal inference,
that we only observe one of the two outcomes
for any individual in the data set. So let's look at a
very simple example. Here, individuals
are characterized by just one feature, their age. And these two curves
that I'm showing you are the potential
outcomes of what would happen to this
individual's blood pressure if you gave them
treatment zero, which is the blue curve, versus
treatment one, which is the red curve. All right. So let's dig in a
little bit deeper. For the blue curve,
we see people who received the control, what
I'm calling treatment zero, their blood pressure
was pretty low for the individuals who
were low and for individuals whose age is high. But for middle age individuals,
their blood pressure on receiving treatment zero
is in the higher range. On the other hand,
for individuals who receive treatment
one, it's the red curve. So young people have much
higher, let's say, blood pressure under treatment one,
and, similarly, much older people. So then one could
ask, well, what about the difference between
these two potential outcomes? That is to say the CATE, the
Conditional Average Treatment Effect, is simply looking at the
distance between the blue curve and the red curve
for that individual. So for someone with
a specific age, let's say a young person
or a very old person, there's a very big difference
between giving treatment zero or giving treatment one. Whereas for a
middle aged person, there's very little difference. So, for example, if treatment
one was significantly cheaper than treatment zero,
then you might say, we'll give treatment one. Even though it's not quite
as good as treatment zero, but it's so much cheaper and
the difference between them is so small, we'll
give the other one. But in order to make that
type of policy decision, one, of course,
has to understand that conditional
average treatment effect for that individual,
and that's something that we're going to want
to predict using data. Now, we don't always
get the luxury of having personalized
treatment recommendations. Sometimes we have
to give a policy. Like, for example-- I took this example
out of my slides, but I'll give it to you anyway. The federal government
might come out with a guideline saying that
all men over the age of 50-- I'm making up that number-- need to get annual
prostate cancer screening. That's an example of a
very broad policy decision. You might ask, well, what is
the effect of that policy now applied over the
full population on, let's say, decreasing deaths
due to prostate cancer? And that would be
an example of asking about the average
treatment effect. So if you were to
average the red line, if you were to
average the blue line, you get those two dotted
lines I show there. And if you look at the
difference between them, that is the average
treatment effect between giving the
red intervention or giving the blue intervention. And if the average human
effect is very positive, you might say that, on
average, this intervention is a good intervention. If it's very negative, you
might say the opposite. Now, the challenge about
doing causal inference from observational data
is that, of course, we don't observe those
red and those blue curves, rather what we observe are
data points that might be distributed all over the place. Like, for example,
in this example, the blue treatment happens
to be given in the data more to young people, and
the red treatment happens to be given in the
data more to older people. And that can happen for
a variety of reasons. It can happen due to
access to medication. It can happen for
socioeconomic reasons. It could happen because existing
treatment guidelines say that old people should
receive treatment one and young people should
receive treatment zero. These are all reasons why
in your data who receives what treatment could
be biased in some way. And that's exactly what this
edge from X to T is modeling. But for each of
those people, you might want to know, well, what
would have happened if they had gotten the other treatment? And that's asking about
the counterfactual. So these dotted circles
are the counterfactuals for each of those observations. And by the way, you'll notice
that those dots are not on the curves, and the reason
they're not on the curve is because I'm
trying to point out that there could be some
stochasticity in the outcome. So the dotted lines are the
expected potential outcomes and the circles are the
realizations of them. All right. Everyone take out a calculator
or your computer or your phone, and I'll take out mine. This is not an opportunity to go
on Facebook, just to be clear. All you want is a calculator. My phone doesn't-- oh,
OK, it has a calculator. Good. All right. So we're going to do
a little exercise. Here's a data set on
the left-hand side. Each row is an individual. We're observing the
individual's age, gender, whether they exercise
regularly, which I'll say is a one or a zero,
and what treatment they got, which is A or B. On
the far right-hand side are their observed sugar
glucose sugar levels, let's say, at the end of the year. Now, what we'd like to
have, it looks like this. So we'd like to know what would
have happened to this person's sugar levels had they
received medication A or had they received
medication B. But if you look at
the previous slide, we observed for each individual
that they got either A or B. And so we're only
going to know one of these columns
for each individual. So the first row, for
example, this individual received treatment
A, and so you'll see that I've taken
the observed sugar level for that individual,
and since they received treatment A, that
observed level represents the potential outcome Ya, or Y0. And that's why I have a 6,
which is bolded under Y0. And we don't know what
would have happened to that individual
had they received treatment B. So in this
case, some magical creature came to me and told me
their sugar levels would have been 5.5, but we
don't actually know that. It wasn't in the data. Let's look at the
next line just to make sure we get what I'm saying. So the second
individual actually received treatment B. They're
observed sugar level is 6.5. OK. Let's do a little survey. That 6.5 number, should
it be in this column? Raise your hand. Or should it be in this column? Raise your hand. All right. About half of you
got that right. Indeed, it goes to
the second column. And again, what we would like
to know is the counterfactual. What would have been
their sugar levels had they received medication A? Which we don't actually
observe in our data, but I'm going to
hypothesize is-- suppose that someone
told me it was 7, then you would see that
value filled in there. That's the unobserved
counterfactual. All right. First of all, is
the setup clear? All right. Now here's when you
use your calculators. So we're going to
now demonstrate the difference between
a naive estimator of your average treatment effect
and the true average treatment effect. So what I want you
to do right now is to compute, first,
what is the average sugar level of the individuals who
got medication B. So for that, we're only going to
be using the red ones. So this is conditioning
on receiving medication B. And so this is equivalent
to going back to this one and saying, we're only going to
take the rows where individuals receive medication
B, and we're going to average their
observed sugar levels. And everyone should do that. What's the first number? 6.5 plus-- I'm getting 7.875. This is for the
average sugar, given that they received
medication B. Is that what other people are getting? AUDIENCE: Yeah. DAVID SONTAG: OK. What about for
the second number? Average sugar, given A? I want you to compute it. And I'm going to ask
everyone to say it out loud in literally one minute. And if you get it
wrong, of course you're going to be embarrassed. I'm going to try myself. OK. On the count of
three, I want everyone to read out what
that third number is. One, two, three. ALL: 7.125. DAVID SONTAG: All right. Good. We can all do arithmetic. All right. Good. So, again, we're just
looking at the red numbers here, just the red numbers. So we just computed
that difference, which is point what? AUDIENCE: 0.75. DAVID SONTAG: 0.75? Yeah, that looks about right. Good. All right. So that's a positive number. Now let's do
something different. Now let's compute the actual
average treatment effect, which is we're now going to average
every number in this column, and we're going to average
every number in this column. So this is the
average sugar level under the potential outcome
of had the individual received treatment B, and this is
the average sugar level under the potential outcome
that the individual received treatment A. All right. Who's doing it? AUDIENCE: 0.75. DAVID SONTAG: 0.75 is what? AUDIENCE: The difference. DAVID SONTAG: How do you know? AUDIENCE: [INAUDIBLE] DAVID SONTAG: Wow, you're fast. OK. Let's see if you're right. I actually don't know. OK. The first one is 0.75. Good, we got that right. I intentionally didn't post
the slides to today's lecture. And the second
one is minus 0.75. All right. So now let's put us in the
shoes of a policymaker. The policymaker has to
decide, is it a good idea to-- or let's say it's a
health insurance company. A health insurance
company is trying decide, should I reimburse for
treatment B or not? Or should I simply
say, no, I'm never going to reimburse for treatment
because it doesn't work well? So if they had done the
naive estimator, that would have been
the first example, then it would look
like medication B is-- we want lower
numbers here, so it would look like medication B
is worse than medication A. And if you properly
estimated what the actual average
treatment effect is, you get the absolute
opposite conclusion. You conclude that medication B
is much better than medication A. It's just a simple
example to really illustrate the difference
between conditioning and actually computing
that counterfactual. OK. So hopefully now you're
starting to get it. And again, you're going to have
many more opportunities to work through these things in your
homework assignment and so on. So by now you should be
starting to wonder, how the hell could I do anything in
this state of the world? Because you don't actually
observe those black numbers. These are all unobserved. And clearly there
is bias in what the values should
be because of what I've been saying all along. So what can we do? Well, the first thing
we have to realize is that typically, this is an
impossible problem to solve. So your instincts
aren't wrong, and we're going to have to make
a ton of assumptions in order to do anything here. So the first assumption
is called SUTVA. I'm not even going
to talk about it. You can read about
that in your readings. I'll tell you about
the two assumptions that are a little bit
easier to describe. The first critical assumption
is that there are no unobserved confounding factors. Mathematically
what that's saying is that your potential
outcomes, Y0 and Y1, are conditionally independent
of the treatment decision given what you observe on
the individual, X. Now, this could
be a bit hard to-- and that's called ignorability. And this can be a bit
hard to understand, so let me draw a picture. So X is your covariance, T
is your treatment decision. And now I've drawn for you
a slightly different graph. Over here I said X goes
to T, X and T go to Y. But now I don't have Y.
Instead, I have Y0 and Y1, and I don't have any
edge from T to them. And that's because
now I'm actually using the potential
outcomes notation. Y0 is a potential
outcome of what would have happened to this
individual had they received treatment 0, and
Y1 is what would have happened to this individual
if they received treatment one. And because you already know
what treatment the individual has received, it
doesn't make sense to talk about an edge
from T to those values. That's why there's
no edge there. So then you might wonder,
how could you possibly have a violation of this
conditional independence assumption? Well, before I give
you that answer, let me put some names
to these things. So we might think about X as
being the age, gender, weight, diet, and so on
of the individual. T might be a medication, like
an anti-hypertensive medication to try to lower a
patient's blood pressure. And these would be
the potential outcomes after those two medications. So an example of a
violation of ignorability is if there is something else,
some hidden variable h, which is not observed
and which affects both the decision
of what treatment the individual in
your data set receives and the potential outcomes. Now it should be
really clear that this would be a violation of that
conditional independence assumption. In this graph, Y0 and
Y1 are not conditionally independent of T
given X. All right. So what are these
hidden confounders? Well, they might be things,
for example, which really affect treatment decisions. So maybe there's a
treatment guideline saying that for
diabetic patients, they should receive
treatment zero, that that's the right thing to do. And so a violation
of this would be if the fact that the
patient's diabetic were not recorded in the
electronic health record. So you don't know-- that's not up there. You don't know that,
in fact, the reason the patient received treatment
T was because of this h factor. And there's critically
another assumption, which is that h actually
affects the outcome, which is why you have these
edges from h to the Y's. If h were something
which might have affected treatment decision but not the
actual potential outcomes-- and that can happen, of course. Things like gender can often
affect treatment decisions, but maybe, for some diseases,
it might not affect outcomes. In that situation it wouldn't
be a confounding factor because it doesn't
violate this assumption. And, in fact, one would
be able to come up with consistent estimators
of average treatment effect under that assumption. Where things go to hell is when
you have both of those edges. All right. So there can't be
any of these h's. You have to observe
all things that affect both treatment and outcomes. The second big
assumption-- oh, yeah. Question? AUDIENCE: In practice, how
good of a model is this? DAVID SONTAG: Of what
I'm showing you here? AUDIENCE: Yeah. DAVID SONTAG: For hypertension? AUDIENCE: Sure. DAVID SONTAG: I have no idea. But I think what
you're really trying to get at here in asking your
question, how good of a model is this, is, well, oh,
my god, how do I know if I've observed everything? Right? All right. And that's where
you need to start talking to domain experts. So this is my
starting place where I said, no, I'm not
going to attempt to fit the causal graph. I'm going to assume I
know the causal graph and just try to
estimate the effects. That's where this starts to
become really irrelevant. Because if you notice, this
is another causal graph, not the one I drew on the board. And so that's something
where, really, talking with domain
experts would be relevant. So if you say, OK, I'm going
to be studying hypertension and this is the data I've
observed on patients, well, you can then go to a
clinician, maybe a primary care doctor who often treats
patients with hypertension, and you say, OK, what usually
affects your treatment decisions? And you get a set
of variables out, and then you check
to make sure, am I observing all of those
variables, at least the variables that would
also affect outcomes? So, often, there's
going to be a back and forth in that conversation
to make sure that you've set up your problem correctly. And again, this
is one area where you see a critical
difference between the way that we do causal
inference from the way that we do machine learning. Machine learning, if there's
some unobserved variables, so what? I mean, maybe your predictive
accuracy isn't quite as good as it could have
been, but whatever. Here, your conclusions
could be completely wrong if you don't get those
confounding factors right. Now, in some of the
optional readings for Thursday's lecture-- and we'll touch on it
very briefly on Thursday, but there's not much
time in this course-- I'll talk about ways and
you'll read about ways to try to assess
robustness to violations of these assumptions. And those go by the name
of sensitivity analysis. So, for example, the type
of question you might ask is, how would my
conclusions have changed if there were a
confounding factor which was blah strong? And that's something that one
could try to answer from data, but it's really starting
to get beyond the scope of this course. So I'll give you
some readings on it, but I won't be able to talk
about it in the lecture. Now, the second major
assumption that one needs is what's known
as common support. And by the way, pay
close attention here because at the end of today's
lecture-- and if I forget, someone must remind me-- I'm going to ask you where did
these two assumptions come up in the proof that I'm
about to give you. The first one I'm going to give
you will be a dead giveaway. So I'm going to answer to you
where ignorability comes up, but it's up to you
to figure out where does common support show up. So what is common support? Well, what common support
says is that there always must be some stochasticity
in the treatment decisions. For example, if in
your data patients only receive treatment A and no
patient receives treatment B, then you would never be able to
figure out the counterfactual, what would have happened if
patients receive treatment B. But what happens if it's
not quite that universal but maybe there is
classes of people? Some individual is X, let's
say, people with blue hair. People with blue hair always
receive treatment zero and they never
see treatment one. Well, for those people,
if for some reason something about them
having blue hair was also going to affect
how they would respond to the treatment,
then you wouldn't be able to answer anything
about the counterfactual for those individuals. This goes by the name of what's
called a propensity score. It's the probability of
receiving some treatment for each individual. And we're going to assume that
this propensity score is always bounded between 0 and 1. So it's between 1 minus
epsilon and epsilon for some small epsilon. And violations of
that assumption are going to completely
invalidate all conclusions that we could draw
from the data. All right. Now, in actual clinical
practice, you might wonder, can this ever hold? Because there are
clinical guidelines. Well, a couple of places where
you'll see this are as follows. First, often, there are settings
where we haven't the faintest idea how to treat patients,
like second line diabetes treatments. You know that the first thing
we start with is metformin. But if metformin doesn't help
control the patient's glucose values, there are several
second line diabetic treatments. And right now, we don't
really know which one to try. So a clinician might start
with treatments from one class. And if that's not working,
you try a different class, and so on. And it's a bit random
which class you start with for any one patient. In other settings, there might
be good clinical guidelines, but there is randomness
in other ways. For example, clinicians who
are trained on the west coast might be trained that this is
the right way to do things, and clinicians who are
trained in the east coast might be trained that this is
the right way to do things. And so even if any one
clinician's treatment decisions are deterministic
in some way, you'll see some stochasticity
now across clinicians. It's a bit subtle how to
use that in your analysis, but trust me, it can be done. So if you want to
do causal inference from observational
data, you're going to have to first start to
formalize things mathematically in terms of what is your X, what
is your T, what is your Y. You have to think through,
do these choices satisfy these assumptions
of ignorability and overlap? Some of these things you
can check in your data. Ignorability you can't
explicitly check in your data. But overlap, this thing,
you can test in your data. By the way, how? Any idea? Someone else who
hasn't spoken today. So just think back to
the previous example. You have this table of these X's
and treatment A or B and then sugar values. How would you test this? AUDIENCE: You could use
a frequentist approach and just count how
many things show up. And if there is zero, then you
could say that it's violated. DAVID SONTAG: Good. So you have this table. I'll just go back to that table. We have this table,
and these are your X's. Actually, we'll go back
to the previous slide where it's a bit easier to see. Here, we're going to ignore
the outcome, the sugar levels because,
remember, this only has to do with
probability of treatment given your covariance. The Y doesn't show
up here at all. So this thing on
the right-hand side, the observed sugar levels, is
irrelevant for this question. All we care about is
what goes on over here. So we look at this. These are your X's, and
this is your treatment. And you can look to
see, OK, here you have one 75-year-old
male who does exercise frequently and received
treatment A. Is there any one else in the data set who
is 75 years old and male, does exercise regularly
but received treatment B? Yes or no? No. Good. OK. So overlap is not satisfied
here, at least not empirically. Now, you might argue that I'm
being a bit too coarse here. Well, what happens if
the individual is 74 and received treatment B? Maybe that's close enough. So there starts to
become subtleties in assessing these things
when you have finite data. But it is something at
the fundamental level that you could start
to assess using data. As opposed to ignorability,
which you cannot test using data. All right. So you have to think about, are
these assumptions satisfied? And only once you start to think
through those questions can you start to do your analysis. And so that now brings me to
the next part of this lecture, which is how do we actually--
let's just now believe David, believe that these
assumptions hold. How do we do that
causal inference? Yeah? AUDIENCE: I just had a
question on [INAUDIBLE].. If you know that some patients,
for instance, healthy patients, are not tracking to
get any treatment, should we just remove
them, basically? DAVID SONTAG: So
the question is, what happens if you have
a violation of overlap? For example, you know that
healthy individuals never receive any treatment. Should you remove them
from your data set? Well, first of all, that has
to do with how do you formalize the question because not
receiving a treatment is a treatment. So that might be your control
arm, just to be clear. Now, if you're asking about
the difference between two treatments-- two different
classes of treatment for a condition, then often one
defines the relevant inclusion criteria in order to have
these conditions hold. For example, we could try to
redefine the set of individuals that we're asking about
so that overlap does hold. But then in that
situation, you have to just make sure that your
policy is also modified. You say, OK, I conclude that
the average treatment effect is blah for this type of people. OK? OK. So how could we possibly compute
the average treatment effect from data? Remember, average treatment
effect, mathematically, is the expectation between
potential outcome Y1 minus Y0. The key tool which we'll use
in order to estimate that is what's known as the
adjustment formula. This goes by many names in
the statistics community, such as the G-formula as well. Here, I'll give you
a derivation of it. We're first going to recognize
that this expectation is actually two
expectations in one. It's the expectation
over individuals X and it's the expectation over
potential outcomes Y given X. So I'm first just
going to write it out in terms of those
two expectations, and I'll write the expectations
related to X on the outside. That goes by name of law
of total expectation. This is trivial at this stage. And by the way, I'm just
writing out expectation of Y1. In a few minutes, I'll
show you expectation of Y0, but it's going to be
exactly analogous. Now, the next step is
where we use ignorability. I told you I was going
to give that one away. So remember, we said
that we're assuming that Y1 is conditionally
independent of the treatment T given X. What that
means is probability of Y1 given X is equal to
probability of Y1 given X comma T equals
whatever-- in this case I'll just say T equals 1. This is implied by Y1 being
conditionally independent of T given X. So I can just stick n
comma T equals 1 here, and that's explicitly because
of ignorability holding. But now we're in a really good
place because notice that-- and here I've just done
some short notation. I'm just going to
hide this expectation. And by the way, you could
do the same for Y0-- Y1, Y0. And now notice
that we can replace this average human effect
with now this expectation with respect to
all individuals X of the expectation of Y1 given
X comma T equals 1, and so on. And these are mostly
quantities that we can now observe from our data. So, for example, we can
look at the individuals who received treatment one,
and for those individuals we have realizations of Y1. We can look at individuals
who receive treatment zero, and for those individuals
we have realizations of Y0. And we could just average
those realizations to get estimates of the
corresponding expectations. So these we can easily
estimate from our data. And so we've made progress. We can now estimate some
part of this from our data. But notice, there
are some things that we can't yet directly
estimate from our data. In particular, we can't
estimate expectation of Y0 given X comma T equals 1
because we have no idea what would have happened to this
individual who actually got treatment one if they
had gotten treatment zero. So these we don't know. So these we don't know. Now, what is the trick
I'm planning on you? How does it help
that we can do this? Well, the key point is
that these quantities that we can estimate from
data show up in that term. In particular, if you
look at the individuals X that you've sampled from the
full set of individuals P of X, for that individual
X for which, in fact, we observed T equals 1, then we
can estimate expectation of Y1 given X comma T equals
1, and similarly for Y0. But what we need to be able
to do is to extrapolate. Because empirically, we only
have samples from P of X given T equals 1, P
of X given T equals 0 for those two potential
outcomes correspondingly. But we are going to also
get samples of X such that for those individuals
in your data set, you might have only
observed T equals 0. And to compute this formula,
you have to answer, for that X, what would it have been if
they got treatment equals one? So there are going to
be a set of individuals that we have to
extrapolate for in order to use this adjustment
formula for estimate. Yep? AUDIENCE: I thought because
common support is true, we have some patients that
received each treatment or a given type of X. DAVID SONTAG: Yes. But now-- so, yes, that's true. But that's a statement
about infinite data. And in reality, one
only has finite data. And so although common support
has to hold to some extent, you can't just build on
that to say that you always observe the counterfactual
for every individual, such as the pictures
I showed you earlier. So I'm going to leave this slide
up for just one more second to let it sink in and
see what it's saying. We started out from the goal of
computing the average treatment effect, expected
value of Y1 minus Y0. Using the adjustment
formula, we've gotten to now an equivalent
representation, which is now an expectation with
respect to all individuals sampling from P of X
of expected value of Y1 given X comma T equals
1, expected value of Y0 given X comma T equals 0. For some of the individuals,
you can observe this, and for some of them,
you have to extrapolate. So from here, there are
many ways that one can go. Hold your question
for a little while. So types of causal
inference methods that you will have
heard of include things like
covariance adjustment, propensity score re-weighting,
doubly robust estimators, matching, and so on. And those are the tools of
the causal inference trade. And in this course,
we're only going to talk about the first two. And in today's
lecture, we're only going to talk about the first
one, covariate adjustment. And on Thursday, we'll
talk about the second one. So covariate adjustment
is a very natural way to try to do that extrapolation. It also goes by the name, by
the way, of response surface modeling. What we're going to
do is we're going to learn a function f, which
takes as an input X and T, and its goals is to predict
Y. So intuitively, you should think about f as
this conditional probability distribution. It's predicting Y
given X and T. So T is going to be an input
to the machine learning algorithm, which is going
to predict what would be the potential outcome Y for this
individual described by feature as X1 through Xd
under intervention T. So this is just from
the previous slide. And what we're going
to do now are-- this is now where we get the
reduction to machine learning-- is we're going to use empirical
risk minimization, or maybe some regularized empirical risk
minimization, to fit a function f which approximates the
expected value of YT given capital T equals little t. Got my X. And then once
you have that function, we're going to be able
to use that to estimate the average treatment effect
by just implementing now this formula here. So we're going to first take
an expectation with respect to the individuals
in the data set. So we're going to
approximate that with an empirical expectation
where we sum over the little n individuals in your data set. Then what we're
going to do is we're going to estimate the first
term, which is f of Xi comma 1 because that is approximating
the expected value of Y1 given T comma X-- T equals 1 comma
X. And we're going to approximate the second
term, which is just plugging now 0 for T instead of 1. And we're going to take the
difference between them, and that will be our estimator
of the average treatment effect. Here's a natural place
to ask a question. One thing you might wonder
is, in your data set, you actually did observe
something for that individual, right. Notice how your raw data
doesn't show up in this at all. Because I've done
machine learning, and then I've thrown
away the observed Y's, and I used this estimator. So what you could have done--
an alternative formula, which, by the way, is also a
consistent estimator, would have been to
use the observed Y for whatever the factual
is and the imputed Y for the counterfactual using f. That would have been
that would have also been a consistent estimator for
the average treatment effect. You could've done either. OK. Now, sometimes you're
not interested in just the average treatment
effect, but you're actually interested in understanding
the heterogeneity in the population. Well, this also now
gives you an opportunity to try to explore
that heterogeneity. So for each
individual Xi, you can look at just the
difference between what f predicts for
treatment one and what X predicts given treatment zero. And the difference
between those is your estimate of your
conditional average treatment effect. So, for example, if
you want to figure out for this individual, what
is the optimal policy, you might look to see is
CATE positive or negative, or is it greater than some
threshold, for example? So let's look at some pictures. Now what we're using is we're
using that function f in order to impute those counterfactuals. And now we have those
observed, and we can actually compute the CATE. And averaging over those,
you can estimate now the average treatment effect. Yep? AUDIENCE: How is f non-biased? DAVID SONTAG: Good. So where can this go wrong? So what do you mean
by biased, first? I'll ask that. AUDIENCE: For
instance, as we've seen in the paper like pneumonia
and people who have asthma, [INAUDIBLE] DAVID SONTAG: Oh, thank you so
much for bringing that back up. So you're referring
to one of the readings for the course
from several weeks ago, where we talked about using
just a pure machine learning algorithm to try to predict
outcomes in a hospital setting. In particular, what
happens for patients who have pneumonia in
the emergency department? And if you all remember,
there was this asthma example, where patients with
asthma were predicted to have better outcomes than
patients without asthma. And you're calling that bias. But you remember, when
I taught about this, I called it biased due
to a particular thing. What's the language I used? I said bias due to
intervention, maybe, is what I-- I can't remember
exactly what I said. [LAUGHTER] I don't know. Make it up. Now a textbook will be written
with bias by intervention. OK. So the problem
there is that they didn't formulize the
prediction problem correctly. The question that
they should have asked is, for asthma patients-- what you really want to ask is a
question of X and then T and Y, where T are the interventions
that are done for asthmatics. So the failure of that
paper is that it ignored the causal inference question
which was hidden in the data, and it just went to predict
Y given X marginalizing over T altogether. So T was never in
the predictive model. And said differently, they never
asked counterfactual questions of what would have happened
had you done a different T. And then they still used it
to try to guide some treatment decisions. Like, for example, should
you send this person home, or should you keep them for
careful monitoring or so on? So this is exactly
the same example as I gave in the
beginning of the lecture, where I said if you just
use a risk stratification model to make some decisions,
you run the risk that you're making the wrong decisions
because those predictions were biased by decisions
in your data. So that doesn't happen here
because we're explicitly accounting for T in
all of our analysis. Yep? AUDIENCE: In the data sets
that we've used, like MIMIC, how much treatment
information exists? DAVID SONTAG: So how much
treatment information is in MIMIC? A ton. In fact, one of the
readings for next week is going to be about trying to
understand how one could manage sepsis, which is a condition
caused by infection, which is managed by, for example,
giving broad spectrum antibiotics, giving
fluids, giving pressers and ventilators. And all of those
are interventions, and all those interventions
are recorded in the data so that one could then ask
counterfactual questions from the data, like
what would have happened if this patient
had they received a different set
of interventions? Would we have prolonged
their life, for example? And so in an intensive care unit
setting, most of the questions that we want to ask about,
not all, but many of them are about dynamic treatments
because it's not just a single treatment
but really about a service sequence of
treatments responding to the current
patient condition. And so that's where we'll really
start to get into that material next week, not in
today's lecture. Yep? AUDIENCE: How do you make sure
that your f function really learned from the relationship
between T and the outcome? DAVID SONTAG: That's
a phenomenal question. Where were you
this whole course? Thank you for asking it. So I'll repeat it. How do you know that
your function f actually learned something about the
relationship between the input X and the treatment
T and the outcome? And that really gets
to the question of, is my reduction actually valid? So I've taken this
problem and I've reduced it to this machine
learning problem, where I take my data, and
literally I just learn a function f to
try to predict well the observations in the data. And how do we know that
that function f actually does a good job at
estimating something like average treatment effect? In fact, it might not. And this is where
things start to get really tricky, particularly
with high dimensional data. Because it could happen, for
example, that your treatment decision is only one of a huge
number of factors that affect the outcome Y. And it
could be that a much more important factor is hidden in
X. And because you don't have much data, and because you have
to regularize your learning algorithm, let's say, with L1
or L2 regularization or maybe early stopping if you're
using deep neural network, your algorithm might never learn
the actual dependence on T. It might learn just to
throw away T and just use X to predict Y.
And if that's the case, you will never be able to
infer these average treatment effects accurately. You'll have huge errors. And that gets back
to one of the slides that I skipped, where I
started out from this picture. This is the machine learning
picture saying, OK, a reduction to machine learning is-- now you add an
additional feature, which is your
treatment decision, and you learn that
black box function f. But this is where machine
learning causal inference starts to differ because
we don't actually care about the quality
of predicting Y. We can measure your
root mean squared error in predicting Y given your
X's and T's, and that error might be low. But you can run into
these failure modes where it just completely
ignores T, for example. So T is special here. So really, the picture
we want to have in mind is that T is some
parameter of interest. We want to learn a model f
such that if we twiddle T, we can see how there is a
differential effect on Y based on twiddling T.
That's what we truly care about when we're
using machine learning for causal inference. And so that's really
the gap, that's the gap in our
understanding today. And it's really an
active area of research to figure out how do you change
the whole machine learning paradigm to recognize that when
you're using machine learning for causal inference,
you're actually interested in something
a little bit different. And by the way, that's a major
area of my lab's research, and we just published
a series of papers trying to answer that question. Beyond the scope of
this course, but I'm happy to send you those
papers if anyone's interested. So that type of question
is extremely important. It doesn't show up quite as much
when your X's aren't very high dimensional and where
things like regularization don't become important. But once your X becomes
high dimensional and once you want to start to
consider more and more complex f's during your
fitting, like you want to use deep neural
networks, for example, these differences in goals
become extremely important. So there are other ways
in which things can fail. So I want to give you
here an example where-- shoot, I'm answering
my question. OK. No one saw that slide. Question-- where did
the overlap assumptions show up in our approach for
estimating average treatment effect using
covariate adjustment? Let me go back to the formula. Someone who hasn't
spoken today, hopefully. You can be wrong, it's fine. Yeah, in the back? AUDIENCE: Is it the
version with the same age in receiving treatment
B and treatment B? DAVID SONTAG: So maybe you have
an individual with some age-- we're going to want
to be able to look at the difference between what
f predicts for that individual if they got treatment
A versus treatment B, or one versus zero. And let me try to lead
this a little bit. And it might happen
in your data set that for individuals
like them, you only ever observe treatment one and
there's no one even remotely like them who you
observe treatment zero. So what's this function
going to output then when you input zero for
that second argument? Everyone say out loud. Garbage? Right? If in your data set you never
observed anyone even remotely similar to Xi who
received treatment zero, then this function is basically
undefined for that individual. I mean, yeah, your function
will output something because you fit it, but it's not
going to be the right answer. And so that's where this
assumption starts to show up. When one talks about the
sample complexity of learning these functions f to do
covariate adjustment, and when one talks
about the consistency of these arguments--
for example, you'd like to be
able to make claims that as the amount of
data grows to, let's say, infinity, that this is
the right answer-- gives you the right estimate. So that's the type
of proof which is often given in the
causal inference literature. Well, if you have overlap,
then as the amount of data goes to infinity, you
will observe someone, like the person who
received treatment one, you'll observe someone who
also received treatment zero. It might have taken you a huge
amount of data to get there because treatment zero
might have been much less likely than treatment one. But because the probability
of treatment zero is not zero, eventually you'll see
someone like that. And so eventually
you'll get enough data in order to learn a function
which can extrapolate correctly for that individual. And so that's where
overlap comes in in giving that type of
consistency argument. Of course, in reality, you
never have infinite data. And so these questions
about trade-offs between the amount
of data you have and the fact that
you never truly have empirical overlap with
a small amount of data, and answering when can
you extrapolate correctly despite that is the
critical question that one needs to answer,
but is, by the way, not studied very well
in the literature because people don't usually
think in terms of sample complexity in that field. That's where computer
scientists can start really to contribute to this
literature and bringing things that we often think
about in machine learning to this new topic. So I've got a couple
of minutes left. Are there any other
questions, or should I introduce some new
material in one minute? Yeah? AUDIENCE: So you said that
the average treatment effect estimator here is consistent. But does it matter if
we choose the wrong-- do we have to choose some
functional form of the features to the effect? DAVID SONTAG: Great question. AUDIENCE: Is it consistent even
if we choose a completely wrong function or formula? DAVID SONTAG: No. AUDIENCE: That's
a different thing? DAVID SONTAG: No, no. You're asking all
the right questions. Good job today, everyone. So, no. If you walk through
that argument I made, I assume two things. First, that you observe
enough data such that you can have any chance
of extrapolating correctly. But then implicit
in that statement is that you're
choosing a function family which is
powerful enough that it can extrapolate correctly. So if your true function is-- if you think back to this
figure I showed you here, if the true potential
outcome functions are these quadratic functions
and you're fitting them with a linear function,
then no matter how much data you
have you're always going to get wrong estimates
because this argument really requires that you're considering
more and more complex non-linearity as your
amount of data grows. So now here's a visual
depiction of what can go wrong if you don't have overlap. So now I've taken out-- previously, I had one or two
red points over here and one or two blue points over here,
but I've taken those out. So in your data all you
have are these blue points and those red points. So all you have are
the points, and now one can learn as good functions,
as you can imagine, to try to, let's say, minimize the mean
squared error of predicting these blue points and minimize
the mean squared error of predicting those red points. And what you might get
out is something-- maybe you'll decide on
a linear function. That's as good as you
could do if all you have are those red points. And so even if you were
willing to consider more and more complex
hypothesis classes, here, if you tried to consider
a more complex hypothesis class than this line, you'd
probably just over-fitting to the data you have. And so you decide
on that line, which, because you had
no data over here, you don't even know that it's
not a good fit to the data. And then you notice
that you're getting completely wrong estimates. For example, if you asked about
the CATE for a young person, it would have the wrong sign
over here because they flipped, the two lines. So that's an example of how
one can start to get errors. And when we begin on
Thursday's lecture, we're going to pick up right
where we left off today, and I'll talk about this issue
a little bit more in detail. I'll talk about how, if one were
to learn a linear function, how one could actually,
under the assumption that the true potential
outcomes are linear, how one could actually
interpret the coefficients of that linear function
in a causal way under the very strong
assumption that the two potential outcomes are linear. So that's what we'll
return to on Thursday.
