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.
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.