What do I find conceptually puzzling?

There are lots of things that I don’t know, like, say, what the birth rate in Sweden is or what the effect of poverty on IQ is. There are also lots of things that I find really confusing and hard to understand, like quantum field theory and monetary policy. There’s also a special category of things that I find conceptually puzzling. These things aren’t difficult to grasp because the facts about them are difficult to understand or require learning complicated jargon. Instead, they’re difficult to grasp because I suspect that I’m confused about the concepts in use.

This is a much deeper level of confusion. It can’t be adjudicated by just reading lots of facts about the subject matter. It requires philosophical reflection on the nature of these concepts, which can sometimes leave me totally confused about everything and grasping for the solid ground of mere factual ignorance.

As such, it feels like a big deal when something I’ve been conceptually puzzled about becomes clear. I want to compile a list for future reference of things that I’m currently conceptually puzzled about and things that I’ve become un-puzzled about. (This is not a complete list, but I believe it touches on the major themes.)

Things I’m conceptually puzzled about

What is the relationship between consciousness and physics?

I’ve written about this here.

Essentially, at this point every available viewpoint on consciousness seems wrong to me.

Eliminativism amounts to a denial of pretty much the only thing that we can be sure can’t be denied – that we are having conscious experiences. Physicalism entails the claim that facts about conscious experience can be derived from laws of physics, which is wrong as a matter of logic.

Dualism entails that the laws of physics by themselves cannot account for the behavior of the matter in our brains, which is wrong. And epiphenomenalism entails that our beliefs about our own conscious experience are almost certainly wrong, and are no better representations of our actual conscious experiences than random chance.

How do we make sense of decision theory if we deny libertarian free will?

Written about this here and here.

Decision theory is ultimately about finding the decision D that maximizes expected utility EU(D). But to do this calculation, we have to decide what the set of possible decisions we are searching is.

EU confusion

Make this set too large, and you end up getting fantastical and impossible results (like that the optimal decision is to snap your fingers and make the world into a utopia). Make it too small, and you end up getting underwhelming results (in the extreme case, you just get that the optimal decision is to do exactly what you are going to do, since this is the only thing you can do in a strictly deterministic world).

We want to find a nice middle ground between these two – a boundary where we can say “inside here the things that are actually possible for us to do, and outside are those that are not.” But any principled distinction between what’s in the set and what’s not must be based on some conception of some actions being “truly possible” to us, and others being truly impossible. I don’t know how to make this distinction in the absence of a robust conception of libertarian free will.

Are there objectively right choices of priors?

I’ve written about this here.

If you say no, then there are no objectively right answers to questions like “What should I believe given the evidence I have?” And if you say yes, then you have to deal with thought experiments like the cube problem, where any choice of priors looks arbitrary and unjustifiable.

(If you are going to be handed a cube, and all you know is that it has a volume less than 1 cm3, then setting maximum entropy priors over volumes gives different answers than setting maximum entropy priors over side areas or side lengths. This means that what qualifies as “maximally uncertain” depends on whether we frame our reasoning in terms of side length, areas, or cube volume. Other approaches besides MaxEnt have similar problems of concept dependence.)

How should we deal with infinities in decision theory?

I wrote about this here, here, here, and here.

The basic problem is that expected utility theory does great at delivering reasonable answers when the rewards are finite, but becomes wacky when the rewards become infinite. There are a huge amount of examples of this. For instance, in the St. Petersburg paradox, you are given the option to play a game with an infinite expected payout, suggesting that you should buy in to the game no matter how high the cost. You end up making obviously irrational choices, such as spending $1,000,000 on the hope that a fair coin will land heads 20 times in a row. Variants of this involve the inability of EU theory to distinguish between obviously better and worse bets that have infinite expected value.

And Pascal’s mugging is an even worse case. Roughly speaking, a person comes up to you and threatens you with infinite torture if you don’t submit to them and give them 20 dollars. Now, the probability that this threat is credible is surely tiny. But it is non-zero! (as long as you don’t think it is literally logically impossible for this threat to come true)

An infinite penalty times a finite probability is still an infinite expected penalty. So we stand to gain an infinite expected utility by just handing over the 20 dollars. This seems ridiculous, but I don’t know any reasonable formalization of decision theory that allows me to refute it.

Is causality fundamental?

Causality has been nicely formalized by Pearl’s probabilistic graphical models. This is a simple extension of probability theory, out of which naturally falls causality and counterfactuals.

One can use this framework to represent the states of fundamental particles and how they change over time and interact with one another. What I’m confused about is that in some ways of looking at it, the causal relations appear to be useful but un-fundamental constructs for the sake of easing calculations. In other ways of looking at it, causal relations are necessarily built into the structure of the world, and we can go out and empirically discover them. I don’t know which is right. (Sorry for the vagueness in this one – it’s confusing enough to me that I have trouble even precisely phrasing the dilemma).

How should we deal with the apparent dependence of inductive reasoning upon our choices of concepts?

I’ve written about this here. Beyond just the problem of concept-dependence in our choices of priors, there’s also the problem presented by the grue/bleen thought experiment.

This thought experiment proposes two new concepts: grue (= the set of things that are either green before 2100 or blue after 2100) and bleen (the inverse of grue). It then shows that if we reasoned in terms of grue and bleen, standard induction would have us concluding that all emeralds will suddenly turn blue after 2100. (We repeatedly observed them being grue before 2100, so we should conclude that they will be grue after 2100.)

In other words, choose the wrong concepts and induction breaks down. This is really disturbing – choices of concepts should be merely pragmatic matters! They shouldn’t function as fatal epistemic handicaps. And given that they appear to, we need to develop some criterion we can use to determine what concepts are good and what concepts are bad.

The trouble with this is that the only proposals I’ve seen for such a criterion reference the idea of concepts that “carve reality at its joints”; in other words, the world is composed of green and blue things, not grue and bleen things, so we should use the former rather than the latter. But this relies on the outcome of our inductive process to draw conclusions about the starting step on which this outcome depends!

I don’t know how to cash out “good choices of concepts” without ultimately reasoning circularly. I also don’t even know how to make sense of the idea of concepts being better or worse for more than merely pragmatic reasons.

How should we reason about self defeating beliefs?

The classic self-defeating belief is “This statement is a lie.” If you believe it, then you are compelled to disbelieve it, eliminating the need to believe it in the first place. Broadly speaking, self-defeating beliefs are those that undermine the justifications for belief in them.

Here’s an example that might actually apply in the real world: Black holes glow. The process of emission is known as Hawking radiation. In principle, any configuration of particles with a mass less than the black hole can be emitted from it. Larger configurations are less likely to be emitted, but even configurations such as a human brain have a non-zero probability of being emitted. Henceforth, we will call such configurations black hole brains.

Now, imagine discovering some cosmological evidence that the era in which life can naturally arise on planets circling stars is finite, and that after this era there will be an infinite stretch of time during which all that exists are black holes and their radiation. In such a universe, the expected number of black hole brains produced is infinite (a tiny finite probability multiplied by an infinite stretch of time), while the expected number of “ordinary” brains produced is finite (assuming a finite spatial extent as well).

What this means is that discovering this cosmological evidence should give you an extremely strong boost in credence that you are a black hole brain. (Simply because most brains in your exact situation are black hole brains.) But most black hole brains have completely unreliable beliefs about their environment! They are produced by a stochastic process which cares nothing for producing brains with reliable beliefs. So if you believe that you are a black hole brain, then you should suddenly doubt all of your experiences and beliefs. In particular, you have no reason to think that the cosmological evidence you received was veridical at all!

I don’t know how to deal with this. It seems perfectly possible to find evidence for a scenario that suggests that we are black hole brains (I’d say that we have already found such evidence, multiple times). But then it seems we have no way to rationally respond to this evidence! In fact, if we do a naive application of Bayes’ theorem here, we find that the probability of receiving any evidence in support of black hole brains to be 0!

So we have a few options. First, we could rule out any possible skeptical scenarios like black hole brains, as well as anything that could provide any amount of evidence for them (no matter how tiny). Or we could accept the possibility of such scenarios but face paralysis upon actually encountering evidence for them! Both of these seem clearly wrong, but I don’t know what else to do.

How should we reason about our own existence and indexical statements in general?

This is called anthropic reasoning. I haven’t written about it on this blog, but expect future posts on it.

A thought experiment: imagine a murderous psychopath who has decided to go on an unusual rampage. He will start by abducting one random person. He rolls a pair of dice, and kills the person if they land snake eyes (1, 1). If not, he lets them free and hunts down ten new people. Once again, he rolls his pair of die. If he gets snake eyes he kills all ten. Otherwise he frees them and kidnaps 100 new people. On and on until he eventually gets snake eyes, at which point his murder spree ends.

Now, you wake up and find that you have been abducted. You don’t know how many others have been abducted alongside you. The murderer is about to roll the dice. What is your chance of survival?

Your first thought might be that your chance of death is just the chance of both dice landing 1: 1/36. But think instead about the proportion of all people that are ever abducted by him that end up dying. This value ends up being roughly 90%! So once you condition upon the information that you have been captured, you end up being much more worried about your survival chance.

But at the same time, it seems really wrong to be watching the two dice tumble and internally thinking that there is a 90% chance that they land snake eyes. It’s as if you’re imagining that there’s some weird anthropic “force” pushing the dice towards snake eyes. There’s way more to say about this, but I’ll leave it for future posts.

Things I’ve become un-puzzled about

Newcomb’s problem – one box or two box?

To almost everyone, it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly.

– Nozick, 1969

I’ve spent months and months being hopelessly puzzled about Newcomb’s problem. I now am convinced that there’s an unambiguous right answer, which is to take the one box. I wrote up a dialogue here explaining the justification for this choice.

In a few words, you should one-box because one-boxing makes it nearly certain that the simulation of you run by the predictor also one-boxed, thus making it nearly certain that you will get 1 million dollars. The dependence between your action and the simulation is not an ordinary causal dependence, nor even a spurious correlation – it is a logical dependence arising from the shared input-output structure. It is the same type of dependence that exists in the clone prisoner dilemma, where you can defect or cooperate with an individual you are assured is identical to you in every single way. When you take into account this logical dependence (also called subjunctive dependence), the answer is unambiguous: one-boxing is the way to go.

Summing up:

Things I remain conceptually confused about:

  • Consciousness
  • Decision theory & free will
  • Objective priors
  • Infinities in decision theory
  • Fundamentality of causality
  • Dependence of induction on concept choice
  • Self-defeating beliefs
  • Anthropic reasoning

On existence

Epistemic status: This is a line of thought that I’m not fully on board with, but have been taking more seriously recently. I wouldn’t be surprised if I object to all of this down the line.

The question of whether or not a given thing exists is not an empty question or a question of mere semantics. It is a question which you can get empirical evidence for, and a question whose answer affects what you expect to observe in the world.

Before explaining this further, I want to draw an analogy between ontology and causation (and my attitudes towards them).

Early in my philosophical education, my attitude towards causality was sympathetic to the Humean-style eliminativism, in which causality is a useful construct that isn’t reflected in the fundamental structure of the world. That is, I quickly ‘tossed out’ the notion of causality, comfortable to just talk about the empirical regularities governed by our laws of physics.

Later, upon encountering some statisticians that exposed me to the way that causality is actually calculated in the real world, I began to feel that I had been overly hasty. In fact, it turns out that there is a perfectly rigorous and epistemically accessible formalization of causality, and I now feel that there is no need to toss it out after all.

Here’s an easy way of thinking about this: While the slogan “Correlation does not imply causality” is certainly true, the reverse (“Causality does not imply correlation”) is trickier. In fact, whenever you have a causal relationship between variables, you do end up expecting some observable correlations. So while you cannot deductively conclude a causal relationship from a merely correlational one, you can certainly get evidence for some causal models.

This is just a peek into the world of statistical discovery of causal relationships – going further requires a lot more work. But that’s not necessary for my aim here. I just want to express the following parallel:

Rather than trying to set up a perfect set of necessary and sufficient conditions for application of the term ’cause’, we can just take a basic axiom that any account of causation must adhere to. Namely: Where there’s causation, there’s correlation.

And rather than trying to set up a perfect set of necessary and sufficient conditions for the term ‘existence’, we can just take a basic axiom that any account of existence must adhere to. Namely: If something affects the world, it exists.

This should seem trivially obvious. While there could conceivably be entities that exist without affecting anything, clearly any entity that has a causal impact on the world must exist.

The contrapositive of this axiom is that if something doesn’t exist, it does not affect the world.

Again, this is not a controversial statement. And importantly, it makes ontology amenable to scientific inquiry! Why? Because two worlds with different ontologies will have different repertoires of causes and effects. A world in which nothing exists is a world in which nothing affects anything – a dead, static region of nothingness. We can rule out this world on the basis of our basic empirical observation that stuff is happening.

This short argument attempts to show that ontology is a scientifically respectable concept, and not merely a matter of linguistic game-playing. Scientific theories implicitly assume particular ontologies by relying upon laws of nature which reference objects with causal powers. Fundamentally, evidence that reveals the impotence of these supposed causal powers serves as evidence against the ontological framework of such theories.

I think the temptation to wave off ontological questions as somehow disreputable and unscientific actually springs from the fundamentality of this concept. Ontology isn’t a minor add-on to our scientific theories done to appease the philosophers. Instead, it is built in from the ground floor. We can’t do science without implicitly making ontological assumptions. I think it’s better to make these assumptions explicit and debate about the fundamental principles by which we justify them, then it is to do it invisibly, without further analysis.

Concepts we keep and concepts we toss out

Often when we think about philosophical concepts like identity, existence, possibility, and so on, we find ourselves confronted with numerous apparent paradoxes that require us to revise our initial intuitive conception. Sometimes, however, the revisions necessary to render the concept coherent end up looking so extreme as to make us prefer to just throw out the concept altogether.

An example: claims about identity are implicit in much of our reasoning (“I was thinking about this in the morning” implicitly presumes an identity between myself now and the person resembling me in my apartment this morning). But when we analyze our intuitive conception of identity, we find numerous incoherencies (e.g. through Sorites-style paradoxes in which objects retain their identity through arbitrarily small transformations, but then end up changing their identity upon the conjunction of these transformations anyway).

When faced with these incoherencies, we have a few options: first of all, we can decide to “toss out” the concept of identity (i.e. determine that the concept is too fundamentally paradoxical to be saved), or we can decide to keep it. If we keep it, then we are forced to bite some bullets (e.g. by revising the concept away from our intuitions to a more coherent neighboring concept, or by accepting the incoherencies).

In addition, keeping the concept does not mean thinking that the concept actually successfully applies to anything. For instance, one might keep the concept of free will (in that they have a well-defined personal conception of it), while denying that free will exists. This is the difference between saying “People don’t have free will, and that has consequences X, Y, and Z” and saying “I think that contradictions are so deeply embedded in the concept of free will that it’s fundamentally unsavable, and henceforth I’m not going to reason in terms of it.” I often hop back and forth between these positions, but I think they are really quite different.

One final way to describe this distinction: When faced with a statement like “X exists,” we have three choices: We can say that the statement is true, that it is false, or that it is not a proposition. This third category is what we would say about statements like “Arghleschmargle” or “Colorless green ideas sleep furiously”. While they are sentences that we can speak, they just aren’t the types of things that could be true or false. To throw out the concept of existence is to say that a statement like “X exists” is neither true nor false, and to keep it is to treat it as having a truth value.

I have a clear sense for any given concept whether or not I think it’s better to keep or toss out, and I imagine that others can do the same.  Here’s a table of some common philosophical concepts and my personal response to each:

Keep
Causality
Existence
Justification
Free will
Time
Consciousness
Randomness
Meaning (of life)
Should (ethical)
Essences
Representation / Intentionality

Toss Out
Knowledge
Identity
Possibility
Objects
Forms
Purposes (in the teleological sense)
Beauty

Many of these I’m not sure about, and I imagine I could have my mind easily changed (e.g. identity, possibility, intentionality). Some I’ve even recently changed my mind about (causality, existence). And others I feel quite confident about (e.g. knowledge, randomness, justification).

I’m curious about how others’ would respond… What philosophical concepts do you lean towards keeping, and which concepts do you lean towards tossing out?

Explanation is asymmetric

We all regularly reason in terms of the concept of explanation, but rarely think hard about what exactly we mean by this explanation. What constitutes a scientific explanation? In this post, I’ll point out some features of explanation that may not be immediately obvious.

Let’s start with one account of explanation that should seem intuitively plausible. This is the idea that to explain X to a person is to give that person some information I that would have allowed them to predict X.

For instance, suppose that Janae wants an explanation of why Ari is not pregnant. Once we tell Janae that Ari is a biological male, she is satisfied and feels that the lack of pregnancy has been explained. Why? Well, because had Janae known that Ari was a male, she would have been able to predict that Ari would not get pregnant.

Let’s call this the “predictive theory of explanation.” On this view, explanation and prediction go hand-in-hand. When somebody learns a fact that explains a phenomenon, they have also learned a fact that allows them to predict that phenomenon.

 To spell this out very explicitly, suppose that Janae’s state of knowledge at some initial time is expressed by

K1 = “Males cannot get pregnant.”

At this point, Janae clearly cannot conclude anything about whether Ari is pregnant. But now Janae learns a new piece of information, and her state of knowledge is updated to

K2 = “Ari is a male & males cannot get pregnant.”

Now Janae is warranted in adding the deduction

K’ = “Ari cannot get pregnant”

This suggests that added information explains Ari’s non-pregnancy for the same reason that it allows the deduction of Ari’s non-pregnancy.

Now, let’s consider a problem with this view: the problem of relevance.

Suppose a man named John is not pregnant, and somebody explains this with the following two premises:

  1. People who take birth control pills almost certainly don’t get pregnant.
  2. John takes birth control pills regularly.

Now, these two premises do successfully predict that John will not get pregnant. But the fact that John takes birth control pills regularly gives no explanation at all of his lack of pregnancy. Naively applying the predictive theory of explanation gives the wrong answer here.

You might have also been suspicious of the predictive theory of explanation on the grounds that it relied on purely logical deduction and a binary conception of knowledge, not allowing us to accommodate the uncertainty inherent in scientific reasoning. We can fix this by saying something like the following:

What it is to explain X to somebody that knows K is to give them information I such that

(1) P(X | K) is small, and
(2) P(X | K, I) is large.

“Small” and “large’ here are intentionally vague; it wouldn’t make sense to draw a precise line in the probabilities.

The idea here is that explanations are good insofar as they (1) make their explanandum sufficiently likely, where (2) it would be insufficiently likely without them.

We can think of this as a correlational account of explanation. It attempts to root explanations in sufficiently strong correlations.

First of all, we can notice that this doesn’t suffer from a problem with irrelevant information. We can find relevance relationships by looking for independencies between variables. So maybe this is a good definition of scientific explanation?

Unfortunately, this “correlational account of explanation” has its own problems.

Take the following example.

uploaded image

This flagpole casts a shadow of length L because of the angle of elevation of the sun and the height of the flagpole (H). In other words, we can explain the length of the shadow with the following pieces of information:

I1 =  “The angle of elevation of the sun is θ”
I2 = “The height of the lamp post is H”
I3 = Details involving the rectilinear propagation of light and the formation of shadows

Both the predictive and correlational theory of explanation work fine here. If somebody wanted an explanation for why the shadow’s length is L, then telling them I1, I2, and I3 would suffice. Why? Because I1, I2, and Ijointly allow us to predict the shadow’s length! Easy.

X = “The length of the shadow is L.”
(I1 & I2 & I3) ⇒ X
So I1 & I2 & I3 explain X.

And similarly, P(X | I1 & I2 & I3) is large, and P(X) is small. So on the correlational account, the information given explains X.

But now, consider the following argument:

(I1 & I3 & X) ⇒ I2
So I1 & I3 & X explain I2.

The predictive theory of explanation applies here. If we know the length of the shadow and the angle of elevation of the sun, we can deduce the height of the flagpole. And the correlational account tells us the same thing.

But it’s clearly wrong to say that the explanation for the height of the flagpole is the length of the shadow!

What this reveals is an asymmetry in our notion of explanation. If somebody already knows how light propagates and also knows θ, then telling them H explains L. But telling them L does not explain H!

In other words, the correlational theory of explanation fails, because correlation possesses symmetry properties that explanation does not.

This thought experiment also points the way to a more complete account of explanation. Namely, the relevant asymmetry between the length of the shadow and the height of the flagpole is one of causality. The reason why the height of the flagpole explains the shadow length but not vice versa, is that the flagpole is the cause of the shadow and not the reverse.

In other words, what this reveals to us is that scientific explanation is fundamentally about finding causes, not merely prediction or statistical correlation. This causal theory of explanation can be summarized in the following:

An explanation of A is a description of its causes that renders it intelligible.

More explicitly, an explanation of A (relative to background knowledge K) is a set of causes of A that render X intelligible to a rational agent that knows K.

What is integrated information?

Integrated information theory relates consciousness to degrees of integrated information within a physical system. I recently became interested in IIT and found it surprisingly hard to locate a good simple explanation of the actual mathematics of integrated information online.

Having eventually just read through all of the original papers introducing IIT, I discovered that integrated information is closely related to some of my favorite bits of mathematics, involving information theory and causal modeling.  This was exciting enough to me that I decided to write a guide to understanding integrated information. My goal in this post is to introduce a beginner to integrated information in a rigorous and (hopefully!) intuitive way.

I’ll describe it increasing levels of complexity, so that even if you eventually get lost somewhere in the post, you’ll be able to walk away having learned something. If you get to the end of this post, you should be able to sit down with a pencil and paper and calculate the amount of integrated information in simple systems, as well as how to calculate it in principle for any system.

Level 1

So first, integrated information is a measure of the degree to which the components of a system are working together to produce outputs.

A system composed of many individual parts that are not interacting with each other in any way is completely un-integrated – it has an integrated information ɸ = 0. On the other hand, a system composed entirely of parts that are tightly entangled with one another will have a high amount of integrated information, ɸ >> 0.

For example, consider a simple model of a camera sensor.

tut_sensors_grid2

This sensor is composed of many independent parts functioning completely separately. Each pixel stores a unit of information about the outside world, regardless of what its neighboring pixels are doing. If we were to somehow sever the causal connections between the two halves of the sensor, each half would still capture and store information in exactly the same way.

Now compare this to a human brain.

FLARE-Technique-Offers-Snapshots-of-Neuron-Activity

The nervous system is a highly entangled mesh of neurons, each interacting with many many neighbors in functionally important ways. If we tried to cut the brain in half, severing all the causal connections between the two sides, we would get an enormous change in brain functioning.

Makes sense? Okay, on to level 2.

Level 2

So, integrated information has to do with the degree to which the components of a system are working together to produce outputs. Let’s delve a little deeper.

We just said that we can tell that the brain is integrating lots of information, because the functioning would be drastically disrupted if you cut it in half. A keen reader might have realized that the degree to which the functioning is disrupted will depend a lot on how you cut it in half.

For instance, cut off the front half of somebody’s brain, and you will end up with total dysfunction. But you can entirely remove somebody’s cerebellum (~50% of the brain’s neurons), and end up with a person that has difficulty with coordination and is a slow learner, but is otherwise a pretty ordinary person.

Human head, MRI and 3D CT scans

What this is really telling us is that different parts of the brain are integrating information differently. So how do we quantify the total integration of information of the brain? Which cut do we choose when evaluating the decrease in functioning?

Simple: We look at every possible way of partitioning the brain into two parts. For each one, we see how much the brain’s functioning is affected. Then we locate the minimum information partition, that is, the partition that results in the smallest change in brain functioning. The change in functioning that results from this particular partition is the integrated information!

Okay. Now, what exactly do we mean by “changes to the system’s functioning”? How do we measure this?

Answer: The functionality of a system is defined by the way in which the current state of the system constrains the past and future states of the system.

To make full technical sense of this, we have to dive a little deeper.

Level 3

How many possible states are there of a Connect Four board?

(I promise this is relevant)

The board is 6 by 7, and each spot can be either a red piece, a black piece, or empty.

Screen Shot 2018-04-20 at 1.03.04 AM

So a simple upper bound on the number of total possible board states is 342 (of course, the actual number of possible states will be much lower than this, since some positions are impossible to get into).

Now, consider what you know about the possible past and future states of the board if the board state is currently…

Screen Shot 2018-04-20 at 1.03.33 AM

Clearly there’s only one possible past state:

Screen Shot 2018-04-20 at 1.03.04 AM

And there are seven possible future states:

What this tells us is that the information about the current state of the board constrains the possible past and future states, selecting exactly one possible board out of the 342 possibilities for the past, and seven out of 342 possibilities for the future.

More generally, for any given system S we have a probability distribution over past and future states, given that the current state is X.

System

Pfuture(X, S) = Pr( Future state of S | Present state of S is X )
Ppast(X, S) = Pr( Past state of S | Present state of S is X )

For any partition of the system into two components, S1 and S2, we can consider the future and past distributions given that the states of the components are, respectively, X1 and X2, where X = (X1, X2).

System

Pfuture(X, S1, S2) = Pr( Future state of S1 | Present state of S1 is X1 )・Pr( Future state of S2 | Present state of S2 is X2 )
Ppast(X, S1, S2) = Pr( Past state of S1 | Present state of S1 is X1 )・Pr( Past state of S2 | Present state of S2 is X2 )

Now, we just need to compare our distributions before the partition to our distributions after the partition. For this we need some type of distance function D that assesses how far apart two probability distributions are. Then we define the cause information and the effect information for the partition (S1, S2).

Cause information = D( Ppast(X, S), Ppast(X, S1, S2) )
Effect information = D( Pfuture(X, S), Pfuture(X, S1, S2) )

In short, the cause information is how much the distribution over past states changes when you partition off your system into two separate systems And the future information is the change in the distribution over future states when you partition the system.

The cause-effect information CEI is then defined as the minimum of the cause information CI and effect information EI.

CEI = min{ CI, EI }

We’ve almost made it all the way to our full definition of ɸ! Our last step is to calculate the CEI for every possible partition of S into two pieces, and then select the partition that minimizes CEI (the minimum information partition MIP).

The integrated information is just the cause effect information of the minimum information partition!

ɸ = CEI(MIP)

Level 4

We’ve now semi-rigorously defined ɸ. But to really get a sense of how to calculate ɸ, we need to delve into causal diagrams. At this point, I’m going to assume familiarity with causal modeling. The basics are covered in a series of posts I wrote starting here.

Here’s a simple example system:

XOR AND.png

This diagram tells us that the system is composed of two variables, A and B. Each of these variables can take on the values 0 and 1. The system follows the following simple update rule:

A(t + 1) = A(t) XOR B(t)
B(t + 1) = A(t) AND B(t)

We can redraw this as a causal diagram from A and B at time 0 to A and B at time 1:

Causal Diagram

What this amounts to is the following system evolution rule:

    ABt → ABt+1
00        00
01       10
10       10
11       01

Now, suppose that we know that the system is currently in the state AB = 00. What does this tell us about the future and past states of the system?

Well, since the system evolution is deterministic, we can say with certainty that the next state of the system will be 00. And since there’s only one way to end up in the state 00, we know that the past state of the system 00.

We can plot the probability distributions over the past and future distributions as follows:

Probabilities Full System

This is not too interesting a distribution… no information is lost or gained going into the past or future. Now we partition the system:

XOR AND Cut

The causal diagram, when cut, looks like:

Causal Diagram Cut

Why do we have the two “noise” variables? Well, both A and B take two variables as inputs. Since one of these causal inputs has been cut off, we replace it with a random variable that’s equally likely to be a 0 or a 1. This procedure is called “noising” the causal connections across the partition.

According to this diagram, we now have two independent distributions over the two parts of the system, A and B. In addition, to know the total future state of a system, we do the following:

P(A1, B1 | A0, B0) = P(A1 | A0) P(B1 | B0)

We can compute the two distributions P(A1 | A0) and P(B1 | B0) straightforwardly, by looking at how each variable evolves in our new causal diagram.

A0 = 0 ⇒ A1 = 0, 1 (½ probability each)
B0 = 0 ⇒ B1 = 0

A0 = 0 ⇒ A-1 = 0, 1 (½ probability each)
B0 = 0 ⇒ B-1 = 0, 1 (probabilities ⅔ and ⅓)

This implies the following probability distribution for the partitioned system:

Partitioned System

I recommend you go through and calculate this for yourself. Everything follows from the updating rules that define the system and the noise assumption.

Good! Now we have two distributions, one for the full system and one for the partitioned system. How do we measure the difference between these distributions?

There are a few possible measures we could use. My favorite of these is the Kullback-Leibler divergence DKL. Technically, this metric is only used in IIT 2.0, not IIT 3.0 (which uses the earth-mover’s distance). I prefer DKL, as it has a nice interpretation as the amount of information lost when the system is partitioned. I have a post describing DKL here.

Here’s the definition of DKL:

DKL(P, Q) = ∑ Pi log(Pi / Qi)

We can use this quantity to calculate the cause information and the effect information:

Cause information = log(3) ≈ 1.6
Effect information = log(2) = 1

These values tell us that our partition destroys about .6 more bits of information about the past than it does the future. For the purpose of integrated information, we only care about the smaller of these two (for reasons that I don’t find entirely convincing).

Cause-effect information = min{ 1, 1.6 } = 1

Now, we’ve calculated the cause-effect information for this particular partition. And since our system has only two variables, this is the only possible partition.

The integrated information is the cause-effect information of the minimum information partition. Since our system only has two components, the partition we’ve examined is the only possible partition, meaning that it must be the minimum information partition. And thus, we’ve calculated ɸ for our system!

ɸ = 1

Level 5

Let’s now define ɸ in full generality.

Our system S consists of a vector of N variables X = (X1, X2, X3, …, XN), each an element in some space 𝒳. Our system also has an updating rule, which is a function f: 𝒳N → 𝒳N. In our previous example, 𝒳 = {0, 1}, N = 2, and f(x, y) = (x XOR y, x AND y).

More generally, our updating rule f can map X to a probability distribution p:  𝒳N → . We’ll denote P(Xt+1 | Xt) as the distribution over the possible future states, given the current state. P is defined by our updating rule: P(Xt+1 | Xt) = f(Xt). The distribution over possible past states will be denoted P(Xt-1 | Xt). We’ll obtain this using Bayes’ rule: P(Xt-1 | Xt) = P(Xt | Xt-1) P(Xt-1) / P(Xt) = f(Xt-1) P(Xt-1) / P(Xt).

A partition of the system is a subset of {1, 2, 3, …, N}, which we’ll label A. We define B = {1, 2, 3, …, N} \ A. Now we can define XA = ( X)a∈A, and XB = ( X)b∈B. Loosely speaking, we can say that X = (XA, XB), i.e. that the total state is just the combination of the two partitions A and B.

We now define the distributions over future and past states in our partitioned system:

Q(Xt+1 | Xt) = P(XA, t+1 | XA, t) P(XB, t+1 | XB, t)
Q(Xt-1 | Xt) = P(XA, t-1 | XA, t) P(XB, t-1 | XB, t).

The effect information EI of the partition defined by A is the distance between P(Xt+1 | Xt) and Q(Xt+1 | Xt), and the cause information CI is defined similarly. The cause-effect information is defined as the minimum of these two.

CI(f, A, Xt) = D( P(Xt-1 | Xt), Q(Xt-1 | Xt) )
EI(f, A, Xt) = D( P(Xt+1 | Xt), Q(Xt+1 | Xt) )

CEI(f, A, Xt) = min{ CI(f, A, Xt), EI(f, A, Xt) }

And finally, we define the minimum information partition (MIP) and the integrated information:

MIP = argminA CEI(f, A, Xt)
ɸ(f, Xt) = minA CEI(f, A, Xt)
= CEI(f, MIP, Xt)

And we’re done!

Notice that our final result is a function of f (the updating function) as well as the current state of the system. What this means is that the integrated information of a system can change from moment to moment, even if the organization of the system remains the same.

By itself, this is not enough for the purposes of integrated information theory. Integrated information theory uses ɸ to define gradations of consciousness of systems, but the relationship between ɸ and consciousness isn’t exactly one-to-on (briefly, consciousness resides in non-overlapping local maxima of integrated information).

But this post is really meant to just be about integrated information, and the connections to the theory of consciousness are actually less interesting to me. So for now I’ll stop here! 🙂

The problem with philosophy

(Epistemic status: I have a high credence that I’m going to disagree with large parts of this in the future, but it all seems right to me at present. I know that’s non-Bayesian, but it’s still true.)

Philosophy is great. Some of the clearest thinkers and most rational people I know come out of philosophy, and many of my biggest worldview-changing moments have come directly from philosophers. So why is it that so many scientists seem to feel contempt towards philosophers and condescension towards their intellectual domain? I can actually occasionally relate to the irritation, and I think I understand where some of it comes from.

Every so often, a domain of thought within philosophy breaks off from the rest of philosophy and enters the sciences. Usually when this occurs, the subfield (which had previously been stagnant and unsuccessful in its attempts to make progress) is swiftly revolutionized and most of the previous problems in the field are promptly solved.

Unfortunately, what also often happens is that the philosophers that were previously working in the field are often unaware of or ignore the change in their field, and end up wasting a lot of time and looking pretty silly. Sometimes they even explicitly challenge the scientists at the forefront of this revolution, like Henri Bergson did with Einstein after he came out with his pesky new theory of time that swept away much of the past work of philosophers in one fell swoop.

Next you get a generation of philosophy students that are taught a bunch of obsolete theories, and they are later blindsided when they encounter scientists that inform them that the problems they’re working on have been solved decades ago. And by this point the scientists have left the philosophers so far in the dust that the typical philosophy student is incapable of understanding the answers to their questions without learning a whole new area of math or something. Thus usually the philosophers just keep on their merry way, asking each other increasingly abstruse questions and working harder and harder to justify their own intellectual efforts. Meanwhile scientists move further and further beyond them, occasionally dropping in to laugh at their colleagues that are stuck back in the Middle Ages.

Part of why this happens is structural. Philosophy is the womb inside which develops the seeds of great revolutions of knowledge. It is where ideas germinate and turn from vague intuitions and hotbeds of conceptual confusion into precisely answerable questions. And once these questions are answerable, the scientists and mathematicians sweep in and make short work of them, finishing the job that philosophy started.

I think that one area in which this has happened is causality.

Statisticians now know how to model causal relationships, how to distinguish them from mere regularities, how to deal with common causes and causal pre-emption, how to assess counterfactuals and assign precise probabilities to these statements, and how to compare different causal models and determine which is most likely to be true.

(By the way, guess where I came to be aware of all of this? It wasn’t in the metaphysics class in which we spent over a month discussing the philosophy of causation. No, it was a statistician friend of mine who showed me a book by Judea Pearl and encouraged me to get up to date with modern methods of causal modeling.)

Causality as a subject has firmly and fully left the domain of philosophy. We now have a fully fleshed out framework of causal reasoning that is capable of answering all of the ancient philosophical questions and more. This is not to say that there is no more work to be done on understanding causality… just that this work is not going to be done by philosophers. It is going to be done by statisticians, computer scientists, and physicists.

Another area besides causality where I think this has happened is epistemology. Modern advances in epistemology are not coming out of the philosophy departments. They’re coming out of machine learning institutes and artificial intelligence researchers, who are working on turning the question of “how do we optimally come to justified beliefs in a posteriori matters?” into precise code-able algorithms.

I’m thinking about doing a series of posts called “X for philosophers”, in which I take an area of inquiry that has historically been the domain of philosophy, and explain how modern scientific methods have solved or are solving the central questions in this area.

For instance, here’s a brief guide to how to translate all the standard types of causal statements philosophers have debated for centuries into simple algebra problems:

Causal model

An ordered triple of exogenous variables, endogenous variables, and structural equations for each endogenous variable

Causal diagram

A directed acyclic graph representing a causal model, whose nodes represent the endogenous variables and whose edges represent the structural equations

Causal relationship

A directed edge in a causal diagram

Causal intervention

A mutilated causal diagram in which the edges between the intervened node and all its parent nodes are removed

Probability of A if B

P(A | B)

Probability of A if we intervene on B

P(A | do B) = P(AB)

Probability that A would have happened, had B happened

P(AB | -B)

Probability that B is a necessary cause of A

P(-A-B | A, B)

Probability that B is a sufficient cause of A

P(AB | -A, -B)

Right there is the guide to understanding the nature of causal relationships, and assessing the precise probabilities of causal conditional statements, counterfactual statements, and statements of necessary and sufficient causation.

To most philosophy students and professors, what I’ve written is probably chicken-scratch. But it is crucially important for them in order to not become obsolete in their causal thinking.

There’s an unhealthy tendency amongst some philosophers to, when presented with such chicken-scratch, dismiss it as not being philosophical enough and then go back to reading David Lewis’s arguments for the existence of possible worlds. It is this that, I think, is a large part of the scientist’s tendency to dismiss philosophers as outdated and intellectually behind the times. And it’s hard not to agree with them when you’ve seen both the crystal-clear beauty of formal causal modeling, and also the debates over things like how to evaluate the actual “distance” between possible worlds.

Artificial intelligence researcher extraordinaire Stuart Russell has said that he knew immediately upon reading Pearl’s book on causal modeling that it was going to change the world. Philosophy professors should either teach graph theory and Bayesian networks, or they should not make a pretense of teaching causality at all.