Measuring Causality

We have an intuitive sense of causality, but how can define it mathematically? In Defining Causality we saw a definition which revolves around intervention. But in that post we assumed that 1. we could observe all the variables in our model and 2. we had complete access to the exact parameters of our underlying distribution. In reality 1. there are things we can’t observe and 2. the things we can observe are samples, not populations.

The Addictive Gene (recap)

Consider the causal diagram in Figure 1. In this example we have the binary variables indicating whether someone:

  1. Smokes
  2. Drinks
  3. Dies
  4. Has the (fictional) addictive gene, gene X, which makes an individual more likely to both smoke and drink

smoking.png

Figure 1: Causal diagram for smoking (observational distribution)

For some choice selection of parameters for this distribution, we arrive at the underlying distribution as visualised by the probability bars in Figure 2.

smoking-observational.png

Figure 2: Probability bars for smoking (observational distribution)

Our big causal question here is: does smoking cause death? By eyeballing Figure 2 we can see that under the smoking (orange) bars there is a higher rate of death (red) bars. Is our question as simple as that? Sadly no. The reason this is difficult is because there is also a higher rate of drinking (green) bars, due to the influence of gene X. How can we correctly adjust for the fact that the drinking variable also contributes to higher mortality?

The key to a correct answer is to examine a set of different distributions, namely the experimental distributions. The importance of these distributions was discussed in the last post, and they are visualised in the probability bars in Figure 3 and Figure 4.

Our task now is to estimate them.

smoking-everyone.png

Figure 3: Probability bars for the everyone experimental distribution.

smoking-noone.png

Figure 4: Probability bars for the no-one experimental distribution.

Recovering the experimental distribution

Suppose that we only have access to samples from the observational distribution, then it’s easy to estimate this observational distribution through sample averages. So given that we can (to some extent) recover the observational distribution, how do we use this to recover the experimental distributions? To answer this, let us begin with the following interpretation.

Imagine that we are Mother Nature, and it is our job to generate samples from the observational distribution. The causal diagram is important because it provides rules for the order in which we may proceed.

First, we draw the root nodes – which are our first causes – each drawn according to it’s own sacred distribution, not depending on any other node in the graph. Then we may draw the remaining nodes at any time so long as it’s parents have already been drawn; for it is the outcomes of the parents which determine the distribution from which we draw. In the smoking example, we can proceed with one of two different ordering strategies:

  1. gene X → drinking → smoking → death
  2. gene X → smoking → drinking → death

Okay, job done for Mother Nature generating observational samples, who we shall now refer to as Observational Mother Nature.

Now image that we are an Experimental Mother Nature, and we wish to generate samples from the experimental distribution. The obvious technique would be the intervention technique: we proceed as before but with one crucial change: at the moment where we are about to draw the experimental variable, instead we forcibly set this variable to our experimental value, then we continue on as normal drawing the remaining downstream variables. In our smoking example, just at the point where we would have randomly determined whether an individual was a smoker, instead we force them to be so.

But here is another technique: the restriction technique. The key insight here is that from the point of intervention, the life of a person who we force into being a smoker is exactly the same as the life of a person who was determined to be a smoker by chance. Therefore, given a large set of samples generated by Observational Mother Nature, Experimental Mother Nature can piggy back off these in the following way:

  1. Divide up Observational Mother Nature’s samples into groups depending on the history previous to that of the experimental value. That is, each separate combination of outcomes occurring in the variables drawn upstream of the experimental variable gets it’s own group.
  2. At that point, keep only the samples which agree with our experimental value.
  3. Stitch together these remaining samples, weighting each group by the history distribution, that is the original frequencies of each history group.

In our example, suppose Observational Mother Nature is following the first ordering strategy: gene X → drinking → smoking → death. Then there are four history groups before the smoking variable is drawn. The restricting and re-weighting which we describe has been visualised in Figure 5. This is also referred to as adjusting for gene X and drinking.

So here we have it: miraculously we recover the entire experimental distribution without any experimental data!

smoking-adjustment.png

Figure 5: Adjusting for drinking and gene X, proof-by-picture. (I have switched the ordering of smoke and drink to make the comparison clearer)

Note on how to read these graphs

In this and all the pictures that follow, we have the observational distribution on the top, and the experimental distribution on the bottom (sometimes marginalised to a subset of variables). The blue boxes separate out each history group – i.e. each combination of our adjustment variables – but restricted to the experimental value.

The arrows indicate that after restricting and re-weighting, the distributions are the same.

Mother Nature: the second ordering strategy

What if Mother Nature decided to draw variables according to the other ordering strategy: gene X → smoking → drinking → death. This means only gene X is upstream of the smoking variable, and we only have two possible histories. And indeed adjusting only for gene X still works: see Figure 6.

smoking-x-adjustment.png

Figure 6: Adjusting for gene X only

Also, it is overkill to want to recover the entire experimental distribution: often we only care about the causal effect, which is the marginal distribution in the dependent variable. In this case mortality.

If this is the so, then according to this – the second ordering strategy – we don’t even need to see the outcome of the drinking variable in our observational sample; if it were hidden from us, we can still recover the marginal of the remaining three variables, and in particular, the causal effect. See Figure 7 for the visualisation.

smoking-adjustment-unobserved-drink.png

Figure 7: Adjusting for gene X, with the drinking variable being unobserved.

You will notice that this graph looks different because we can no longer group by whether the individual is a drinker or not. E.g. for those with gene X, all of the drinkers and non-drinkers are grouped together, but it still works.

Mother Nature: for two players

Let’s make things more complicated.

Imagine now that Observational Mother Nature is tired of drawing all the variables herself, so she divides the set of variables in two. Observational Mother Nature 1 (MN1) draws all the variables up to and including the experimental variable, then she passes her outcomes over to Observational Mother Nature 2 (MN2), who draws the remaining downstream variables. She only needs to pass along the relevant outcomes: those which have a child in the downstream variables.

Suppose we only care about the causal effect – i.e. the dependent variable – then MN1 doesn’t need to pass on all of her outcomes, only those which are relevant for drawing downstream variables: only those which have children among the downstream variables. For the smoking example in the case of the first ordering, this two-player method is visualised in Figure 8.

two-player.png

Figure 8: Two player generating of the observational distribution

Experimental Mother Nature (EMN) realises that to play the intervention technique, she doesn’t need to interfere with what MN1 is doing, she needs only to intervene with MN2. Her technique involves intervening just after receiving the outcomes from MN1, and her method is as follows:

  1. Take MN1’s outcomes (which include the experimental variable)
  2. Forcibly set the experimental variable to the experimental outcome
  3. Take over the task of MN2 in drawing all of the remaining variables.

For the smoking example, this method is depicted in Figure 9. Note that it is crucial that the experimental variable was the last thing to be drawn by MN1 before hand-off! So long as this is the case, we can be sure that tinkering with this variable would have no influence on the other variables handed-off by MN1.

two-player-experimental.png

Figure 9: Two player generating of the experimental distribution

And the restriction technique should still work. EMN might not know the outcomes of all the upstream variables drawn by MN1, but so long as EMN preserves the distribution of groups as given to her by MN1, then restricting and re-weighting ought to work just like before, for the same reason as before.

Returning then for the last time to our example, the restriction technique is the following:

  1. MN1 draws gene X, drinking, smoking
  2. MN1 passes only the drinking and smoking outcome to EMN
  3. EMN restricts only to the handed-off outcomes for which the individual smokes
  4. EMN re-weights according to the distribution of groups handed-off by MN1: i.e. preserving the original prevalence of drinkers and non-drinkers

This has been depicted in Figure 10. What is the corollary? We can recover the causal effect even when gene X is hidden from view!

smoking-switched-adjustment-unobserved-x.png

Figure 10: Adjusting for drinking, when gene X is unobserved.

Back door paths

This is a very convoluted attempt to make intuitive what the literature refers to as blocking all the back door paths1. The main takeaway is that for any group of variables chosen in a way that is consistent with the above fairy-tale, the adjustment we describe here correctly transforms the observational distribution into the experimental distribution. These groups are exactly those which are said to block all the backdoor paths.

In fact there are other valid groups and different methods of adjustment, so the fun doesn’t stop here. If you have any ideas for crazy ways of explaining any of these other methods, then please get in touch.

Footnotes:

1

For a technical exposition of what it means to block all the back door paths, see Pearl, J. (2009). Causality. : Cambridge University Press.

Defining Causality

Causality is a confusing concept. It seems to be something that we understand intuitively, but in neither maths nor science do we have an agreed upon technical definition. Part of the problem, as usual, is that we are using one word to describe more than one thing. Here I will discuss forward and backward causality. And in the case of forward causality, I want to introduce interventions as a good candidate for an agreed upon definition.

Forward and Backward Causality

Example: House Fire

Consider the following collection of events: a short circuit in someone’s house creates a spark which sets on fire some nearby curtains. The local firefighters are nowhere to be found, and the house burns down.

In this example, what was the cause of the fire? Was it the short circuit, the curtain, or the absence of firefighters? How can the absence of something be a cause? If it was the short circuit then naturally we must ask, what was the cause of the short circuit? Was it dodgy manufacturing? Or misuse of the product? Essentially the question is this: who is to blame?

This is a question of backward causality. In general, the question goes “what were the inputs that lead to this output? And how much did each one contribute?” Answering this question is tricky business. In it’s truest interpretation, it feels like the answer is always “the big bang did it”. Indeed some philosophers give up on this type of question all together, and claim that our intuitive notion of causality in this sense is a fiction; our intuitions deceive us, just like how we are deceived by our intuitions for space and time. Free will, ethics, and the justice system all completely depend on this concept of backward causality.

A different question is that of forward causality. This is not the causality of credit assignment, but the causality of decision making. Here are some examples: What is the impact on my life if I take up smoking? If my dishwasher was built with a dodgy circuit, how does that affect the probability of my house burning down? These questions sound a bit more approachable, but still it’s not as easy as you might think. Two big problems are the inadequacy of correlation, and the complications which arise from confounding variables.

In summary, a backward question fixes a value of the outcome Y = y, and asks about the antecedents X. Whereas a forward question specifies or toggles a fixed input X=x, and asks about the consequences for some descendent Y. Backward questions seem very hard but, as far as I can tell, methods of intervention are a great way to deal with forward questions. Also known as the causal calculus, we will describe the method of interventions below. We will see that it goes beyond correlation, and gracefully handles confounders. Furthermore, we might hope that a deeper understanding of forward causality will give us an insight into the confusing world of backward causality.

The Addictive Gene

Let’s look at the simple example given in Figure 1. We suppose the existence of an “addictive gene”, gene X. People with gene X are more susceptible to addictive substances, such as smoking and drinking, both of which have an impact on that person’s mortality. Our aim is to interpret and answer the following question: does smoking cause death?

smoking.png

Figure 1: Causal diagram for smoking (observational distribution)

Without getting into too much detail, the causal diagram restricts the type of interactions we can have between these variables. In particular gene X can influence death only via the means of making someone more likely to smoke or drink. Correspondingly, if we know whether someone drinks and/or smokes, then whether they have gene X or not is no longer relevant in determining their mortality.

There are many distributions with this causal diagram, and we shall choose the one with the following parameters.

\mathbb{P}\left(\text{gene X}\right) =50\%
\mathbb{P}\left(\text{smoke}\mid\text{gene X}\right), \mathbb{P}\left(\text{drink}\mid\text{gene X}\right) =75\%
\mathbb{P}\left(\text{smoke}\mid{\neg\text{{gene X}}}\right), \mathbb{P}\left(\text{drink}\mid\neg\text{{gene X}}\right) =25\%
\mathbb{P}\left(\text{death}\mid\text{smoke}\right), \mathbb{P}\left(\text{death}\mid\text{drink}\right) =50\%
\mathbb{P}\left(\text{death}\mid \text{{smoke}} \,\&\, \text{{drink}} \right) =75\%
\mathbb{P}\left(\text{death}\mid \neg\text{{smoke}} \,\&\, \neg\text{{drink}} \right) =20\%

An intuitive picture of the resulting distribution is given in Figure 2. I call these “probability bars”; they show that the population is divided into sixteen groups, one for each combination of these four binary variables, and the width of each group represents their probability within the distribution.

smoking-observational.png

Figure 2: Probability bars for smoking (observational distribution)

Correlation

Here is an bad definition of causality:

Bad definition

X is a cause of Y if P(Y|X) > P(Y). That is, observing X increases the likelihood of observing Y relative to the base-rate.

This is a tempting definition because most of the causal relationships we like to imagine do indeed satisfy this relationship. For that reason, I would even argue that it serves as a good proxy for causality. In the above setup, for example, our intuition tells us that smoking is a cause of death, and this definition agrees with that. We can calculate the required probabilities exactly to be:

\mathbb{P}(\text{death}) \approx 55.47\%
\mathbb{P}(\text{death} \mid \text{smoke}) = 66.25\%

And we can sanity check these numbers by eyeballing the probability bars and e.g. seeing that around a third of the orange bars are also red.

Thus according to the definition above, smoking is indeed a cause of death. So why is this a bad definition? It might already be clear to you, especially if you are familiar with the phrase correlation does not imply correlation, but what we have here is an artefact of correlation. With a sillier example we can see that it is clearly insufficient.

Rain and raincoats

rain.png

Figure 3: Graphical model for rain and raincoats

Suppose when it rains sometimes we see raincoats and independently sometimes we see umbrellas. We give the graphical model in Figure 3, the probabilities below, and the probability bars in 4.

\mathbb{P}(\text{rain}) =50\%
\mathbb{P}(\text{raincoats} \mid \text{rain}) =90\%
\mathbb{P}(\text{umbrellas} \mid \text{rain}) =90\%

rain-observational.png

Figure 4: Probability bars for rain and raincoats

Therefore we have that \mathbb{P}(\text{umbrellas} | \text{raincoats}) = 90\%, an increase on the base rate for umbrellas \mathbb{P}(\text{umbrellas}) which stands at 45%. Therefore according to our definition, raincoats cause umbrellas. Suspicious.

Experiments

It’s clear in the raincoats example that we have a confounding variable, i.e. both raincoats and umbrellas have the shared cause of rain. What a nuisance. In fact, in some contexts such a variable is also known as a nuisance variable. How can we account for this? How can we decide whether raincoats cause umbrellas?

Solution: perform an experiment. First we force raincoats into existence, then we measure the impact of this intervention on our output variable umbrellas. If the intervention gives an increase in the rate of umbrellas then we shall decree that raincoats are a cause of rain.

This type of experiment is called an RCT and is considered to be the gold standard for measuring causal inference. To carry this out correctly over the course of say a month we would randomly choose some subset of days in which to do nothing, and on the remaining days we would perform our intervention.

We can see that this experiment would reveal that raincoats actually have no impact on umbrellas. But how can we formalise this conclusion?

Formalisation of interventions

The subtlety here is that while what we see in the natural world comes from one distribution, questions of forward causality pertain to a different distribution, what’s called the experimental distribution. This is the distribution we get from performing an experiment, such as the one described above with the umbrellas.

We must introduce a retronym for what we have left behind: the observational distribution. This was our first distribution, with probability bars in Figure 2, and it is to be thought of as the natural distribution governing the business-as-usual relationships of these variables. If we were to sample from the real world, then the data would behave as if it were drawn from this distribution.

Returning to the experimental distribution, first we must decide on an experimental variable, and an experimental value. Then we proceed as if we have intervened on our experimental variable, setting it to our experimental value. I.e., we take our causal diagram for the observational distribution and remove all of the arrows going into our experimental variable. Then for the children of this variable, we proceed as if we had observed a value equal to our experimental value.

For the gene X example, our experimental variable is the smoking variable, and our experimental value is “true”. What we get is the result of hypothetically forcing everyone to smoke, regardless of gene X, regardless of whether they drink, regardless of anything! The causal diagram for the resulting experimental distribution is given in Figure 5. Again we can visualise this distribution by looking at the corresponding probability bars, these are given in 6.

smoking-experimental.png

Figure 5: Causal diagram for smoking (experimental distribution)

smoking-everyone.png

Figure 6: Probability bars for the everyone experimental distribution.

There are in fact two experimental distributions in this case: the first is when we intervene telling everyone that they must smoke, the second is when we tell everyone that they must not smoke. We shall call the case when everyone smokes the “everyone experimental distribution” — probability bars in Figure 6 — and the case when no-one smokes we will call the “no-one experimental distribution” — probability bars in Figure 7.

smoking-noone.png

Figure 7: Probability bars for the no-one experimental distribution.

A better definition of Causality

These new distributions give rise to new probability measures. We could write these as \mathbb{P}_\text{everyone} and \mathbb{P}_\text{no-one}. Then we can calculate for example \mathbb{P}_\text{everyone}(\text{death}) = 62.5\%. But we will opt for some slightly more suggestive notation, making use of the do-operator1. The do-operator way to write this expression is as follows: \mathbb{P}(\text{death} \mid \text{\textit{do}(smoke)}) = 62.5\%. The idea being that we are considering the distribution in which we make everyone smoke, or – perhaps more confusingly put – we make everyone do smoking.

So what does this mean for our question: does smoking cause death? We now have the tools to give the intervention definition of causality.

Definition: Causality

We say that X is a cause of Y if \mathbb{P}(Y \mid \text{\textit{do}}(X)) > \mathbb{P}(Y).

Applied to our example of smoking and death, we see that \mathbb{P}(\text{death} \mid \text{\textit{do}}(\text{smoke})) = 62.5\% > 55.47\% \approx \mathbb{P}(\text{death}). So we still conclude that smoking is a cause of death. What about the example with the raincoats and the umbrellas? Let’s take a look at the probability bars for the experimental distribution in which we force people to wear raincoats, Figure 8.

rain-everyone.png

Figure 8: Probability bars for the everyone wears raincoats experimental distribution.

We can calculate that \mathbb{P}(\text{umbrellas} \mid \text{\textit{do}}(\text{raincoats})) = 45\% which is identical to the base-rate that we have in the observational. I.e. raincoats do not cause umbrellas.

Conclusion

The consideration of experimental distributions gives us a working definition of causality. We have shown that this is a clear improvement over correlation, but the many other merits of this definition remain to be shown.

Next time I would like to discuss the case where we have unknown parameters. In the examples so far given we have assumed complete knowledge of the underlying distribution. But in reality we don’t know \mathbb{P}(\text{death} \mid \text{smoke}). These parameters are hidden from view, and they must be estimated from the data. But data is usually drawn from an observational distribution, so how do we estimate the experimental distributions? How do we estimate \mathbb{P}(\text{death} \mid \text{\textit{do}(smoking)})? One method is to perform a randomised control trial, where we literally go out there and tell some people that they must smoke, and others that they must not. But surprisingly this is not always necessary, and in some cases it is possible to estimate the experimental distributions from observational data alone.

Footnotes:

1

The ideas behind the do-operator were developed in the mid 90s by Judea Pearl. His textbook on causality covers this material in great detail.