Information Geometry (Part 2)

Last time I provided some background to this paper:

• Gavin E. Crooks, Measuring thermodynamic length.

Now I’ll tell you a bit about what it actually says!

Remember the story so far: we’ve got a physical system that’s in a state of maximum entropy. I didn’t emphasize this yet, but that happens whenever our system is in thermodynamic equilibrium. An example would be a box of gas inside a piston. Suppose you choose any number for the energy of the gas and any number for its volume. Then there’s a unique state of the gas that maximizes its entropy, given the constraint that on average, its energy and volume have the values you’ve chosen. And this describes what the gas will be like in equilibrium!

Remember, by ‘state’ I mean mixed state: it’s a probabilistic description. And I say the energy and volume have chosen values on average because there will be random fluctuations. Indeed, if you look carefully at the head of the piston, you’ll see it quivering: the volume of the gas only equals the volume you’ve specified on average. Same for the energy.

More generally: imagine picking any list of numbers, and finding the maximum entropy state where some chosen observables have these numbers as their average values. Then there will be fluctuations in the values of these observables — thermal fluctuations, but also possibly quantum fluctuations. So, you’ll get a probability distribution on the space of possible values of your chosen observables. You should visualize this probability distribution as a little fuzzy cloud centered at the average value!

To a first approximation, this cloud will be shaped like a little ellipsoid. And if you can pick the average value of your observables to be whatever you’ll like, you’ll get lots of little ellipsoids this way, one centered at each point. And the cool idea is to imagine the space of possible values of your observables as having a weirdly warped geometry, such that relative to this geometry, these ellipsoids are actually spheres.

This weirdly warped geometry is an example of an ‘information geometry’: a geometry that’s defined using the concept of information. This shouldn’t be surprising: after all, we’re talking about maximum entropy, and entropy is related to information. But I want to gradually make this idea more precise.

Bring on the math!

We’ve got a bunch of observables X_1, \dots , X_n, and we’re assuming that for any list of numbers x_1, \dots , x_n, the system has a unique maximal-entropy state \rho for which the expected value of the observable X_i is x_i:

\langle X_i \rangle = x_i

This state \rho is called the Gibbs state and I told you how to find it when it exists. In fact it may not exist for every list of numbers x_1, \dots , x_n, but we’ll be perfectly happy if it does for all choices of

x = (x_1, \dots, x_n)

lying in some open subset of \mathbb{R}^n

By the way, I should really call this Gibbs state \rho(x) or something to indicate how it depends on x, but I won’t usually do that. I expect some intelligence on your part!

Now at each point x we can define a covariance matrix

\langle (X_i - \langle X_i \rangle)  (X_j - \langle X_j \rangle)  \rangle

If we take its real part, we get a symmetric matrix:

g_{ij} = \mathrm{Re} \langle (X_i - \langle X_i \rangle)  (X_j - \langle X_j \rangle)  \rangle

It’s also nonnegative — that’s easy to see, since the variance of a probability distribution can’t be negative. When we’re lucky this matrix will be positive definite. When we’re even luckier, it will depend smoothly on x. In this case, g_{ij} will define a Riemannian metric on our open set.

So far this is all review of last time. Sorry: I seem to have reached the age where I can’t say anything interesting without warming up for about 15 minutes first. It’s like when my mom tells me about an exciting event that happened to her: she starts by saying “Well, I woke up, and it was cloudy out…”

But now I want to give you an explicit formula for the metric g_{ij}, and then rewrite it in a way that’ll work even when the state \rho is not a maximal-entropy state. And this formula will then be the general definition of the ‘Fisher information metric’ (if we’re doing classical mechanics), or a quantum version thereof (if we’re doing quantum mechanics).

Crooks does the classical case — so let’s do the quantum case, okay? Last time I claimed that in the quantum case, our maximum-entropy state is the Gibbs state

\rho = \frac{1}{Z} e^{-\lambda^i X_i}

where \lambda^i are the ‘conjugate variables’ of the observables X_i, we’re using the Einstein summation convention to sum over repeated indices that show up once upstairs and once downstairs, and Z is the partition function

Z = \mathrm{tr} (e^{-\lambda^i X_i})

(To be honest: last time I wrote the indices on the conjugate variables \lambda^i as subscripts rather than superscripts, because I didn’t want some poor schlep out there to think that \lambda^1, \dots , \lambda^n were the powers of some number \lambda. But now I’m assuming you’re all grown up and ready to juggle indices! We’re doing Riemannian geometry, after all.)

Also last time I claimed that it’s tremendously fun and enlightening to take the derivative of the logarithm of Z. The reason is that it gives you the mean values of your observables:

\langle X_i \rangle = - \frac{\partial}{\partial \lambda^i} \ln Z

But now let’s take the derivative of the logarithm of \rho. Remember, \rho is an operator — in fact a density matrix. But we can take its logarithm as explained last time, and the usual rules apply, so starting from

\rho = \frac{1}{Z} e^{-\lambda^i X_i}

we get

\mathrm{ln}\, \rho = - \lambda^i X_i - \mathrm{ln} \,Z

Next, let’s differentiate both sides with respect to \lambda^i. Why? Well, from what I just said, you should be itching to differentiate \mathrm{ln}\, Z. So let’s give in to that temptation:

\frac{\partial  }{\partial \lambda^i} \mathrm{ln}  \rho = -X_i + \langle X_i \rangle

Hey! Now we’ve got a formula for the ‘fluctuation’ of the observable X_i — that is, how much it differs from its mean value:

X_i - \langle X_i \rangle = - \frac{\partial \mathrm{ln} \rho }{\partial \lambda^i}

This is incredibly cool! I should have learned this formula decades ago, but somehow I just bumped into it now. I knew of course that \mathrm{ln} \, \rho shows up in the formula for the entropy:

S(\rho) = \mathrm{tr} ( \rho \; \mathrm{ln} \, \rho)

But I never had the brains to think about \mathrm{ln}\, \rho all by itself. So I’m really excited to discover that it’s an interesting entity in its own right — and fun to differentiate, just like \mathrm{ln}\, Z.

Now we get our cool formula for g_{ij}. Remember, it’s defined by

g_{ij} = \mathrm{Re} \langle (X_i - \langle X_i \rangle)  (X_j - \langle X_j \rangle)  \rangle

But now that we know

X_i - \langle X_i \rangle = -\frac{\partial \mathrm{ln} \rho }{\partial \lambda^i}

we get the formula we were looking for:

g_{ij} = \mathrm{Re} \langle \frac{\partial \mathrm{ln} \rho }{\partial \lambda^i} \; \frac{\partial \mathrm{ln} \rho }{\partial \lambda^j}  \rangle

Beautiful, eh? And of course the expected value of any observable A in the state \rho is

\langle A \rangle = \mathrm{tr}(\rho A)

so we can also write the covariance matrix like this:

g_{ij} = \mathrm{Re}\, \mathrm{tr}(\rho \; \frac{\partial \mathrm{ln} \rho }{\partial \lambda^i} \; \frac{\partial \mathrm{ln} \rho }{\partial \lambda^j} )

Lo and behold! This formula makes sense whenever \rho is any density matrix depending smoothly on some parameters \lambda^i. We don’t need it to be a Gibbs state! So, we can work more generally.

Indeed, whenever we have any smooth function from a manifold to the space of density matrices for some Hilbert space, we can define g_{ij} by the above formula! And when it’s positive definite, we get a Riemannian metric on our manifold: the Bures information metric.

The classical analogue is the somewhat more well-known ‘Fisher information metric’. When we go from quantum to classical, operators become functions and traces become integrals. There’s nothing complex anymore, so taking the real part becomes unnecessary. So the Fisher information metric looks like this:

g_{ij} = \int_\Omega p(\omega) \; \frac{\partial \mathrm{ln} p(\omega) }{\partial \lambda^i} \; \frac{\partial \mathrm{ln} p(\omega) }{\partial \lambda^j} \; d \omega

Here I’m assuming we’ve got a smooth function p from some manifold M to the space of probability distributions on some measure space (\Omega, d\omega). Working in local coordinates \lambda^i on our manifold M, the above formula defines a Riemannian metric on M, at least when g_{ij} is positive definite. And that’s the Fisher information metric!

Crooks says more: he describes an experiment that would let you measure the length of a path with respect to the Fisher information metric — at least in the case where the state \rho(x) is the Gibbs state with \langle X_i \rangle = x_i. And that explains why he calls it ‘thermodynamic length’.

There’s a lot more to say about this, and also about another question: What use is the Fisher information metric in the general case where the states \rho(x) aren’t Gibbs states?

But it’s dinnertime, so I’ll stop here.

[Note: in the original version of this post, I omitted the real part in my definition g_{ij} = \mathrm{Re} \langle (X_i - \langle X_i \rangle)  (X_j - \langle X_j \rangle)  \rangle , giving a ‘Riemannian metric’ that was neither real nor symmetric in the quantum case. Most of the comments below are based on that original version, not the new fixed one.]

22 Responses to Information Geometry (Part 2)

  1. Gavin Crooks says:

    There’s an interesting generalization of these ideas in the followup paper:

    Far-From-Equilibrium Measurements Of Thermodynamic Length, E. H. Feng, G.E. Crooks Phys. Rev. E 79, 012104 (2009).

    The idea is that we think of the λs as the set of controllable parameters of the system. These could include conjugate variables, such as pressure, or mechanical variables such as volume. If you fix the pressure (or equivalently the average volume) then the volume fluctuates. If you fix the volume the pressure fluctuates. However, with fixed volume the second derivatives of the log partition function are no longer equal to the covariance matrix or Fisher information, but the Fisher information matrix is still equal to the covariance matrix.

    • John Baez says:

      I’ll check out that new article — thanks! It’s always pleasing but also a bit scary when the author of a paper I’m discussing shows up to comment. Luckily I mostly discuss papers I enjoyed, and that’s very much the case with the paper of yours I’m discussing here. It may not have been your main goal, but you gave a really quick and lucid explanation of the otherwise mysterious Fisher information metric

      g_{ij} = \int_\Omega p(\omega) \; \frac{\partial \mathrm{ln} p(\omega) }{\partial \lambda_i} \; \frac{\partial \mathrm{ln} p(\omega) }{\partial \lambda_j} \; d \omega

    • Tim van Beek says:

      Great! I’ll use this opportunity to ask a question whose answer is probably too obvious for the experts:

      I learned about the Fisher metric from this blog and Wikipedia only, and I would like to know some examples which kind of problems can be solved, and how, with this concept.

      In the paper you linked to, you wrote

      Moreover, the ability to measure thermodynamic length and free-energy change from out-of-equilibrium measurements indicates that these equilibrium properties influence the behavior of driven systems even far from equilibrium.

      So I gather that the thermodynamic length can be used to describe driven systems far from equilibrium – is there an example of a “concrete system” that can be studied in this way?

      • Gavin Crooks says:

        The people who developed thermodynamic length were interested in optimizing macroscopic thermodynamic processes, such as distillation columns, and what not. I’m interested in seeing if we can take those ideas and use them to understand the non-equilibrium behavior of microscopic machines; biological molecular motors, nanoscale solar cells, molecular pores and so on.

        For a concrete system I like to think about a single RNA hairpin being pulled apart using optical tweezers, since this is an experiment that can be done today. David Sivak and I are currently writing up a paper where we look at the Fisher information and metric for a (highly simplified) model of this experiment, and how the FI relates to the dissipation.

  2. Blake Stacey says:

    In this case, $g_{ij}$ will defines a Riemannian metric on our open set.

    I wonder how much the frustrations of doing equations on the Web have cumulatively held back the progress of science. . . .

    • Eric says:

      Well, Andrew at some point figured out how to get itex2mml to work on WordPress. This would allow the blog to have the same exact itex capability as the nCafe, nLab, and the nForum.

      If you ask him super nicely, he may be able to get it working here.

    • Eric says:

      Here is a test WordPress site that uses itex2mml

      Serving MathML

    • John Baez says:

      Andrew offered to have the Azimuth blog on a website that did itex2mml, but I’ve always felt that the bother of installing fonts limited the readership of the n-Café. I know plenty of mathematicians who never got around to installing those fonts. Some of them even suffered through n-Café articles interspersed with gobbledygook where there should have been equations, while others (no doubt) just stayed away.

      In the case of the n-Café, which was trying to attract a small but committed audience, that was probably okay. But for this blog, the goal of attracting a big audience of people interested in ‘saving the planet’ overshadows the goal of getting really nifty-looking equations.

      Of course one might argue that these posts on information geometry, full of equations, do not belong on Azimuth! They may even scare away the eco-minded people I’m trying to attract. Or, they may diffuse the thrust of this blog.

      But so far I’ve been hoping these posts attract mathematicians and physicists, who then may become interested in ecological issues. It seems to be working, sort of.

  3. With the appearance of informational geometric notions in quantum statistics and machine learning, how should that lead us to answer the question raised in this post as to how much of physical theory is inference? Ariel Caticha, in The Information Geometry of Space and Time, goes so far as “The laws of physics could be mere rules for processing information about nature.”

    • John Baez says:

      When someone says something like “The laws of physics could be mere rules for processing information about nature”, the only controversial bit is the word “mere”, which suggests all sorts of deep mysteries, but doesn’t actually explain them.

      Caticha’s paper looks interesting, but it’s hard to tell how interesting. It seeks to describe points in space as “merely” labels for probability distributions, getting a Riemannian metric on space which is “merely” the Fisher information metric.

      The fun starts when Caticha cooks up a dynamics that describes how the geometry of space evolves in time. This is reminiscent of Wheeler’s “geometrodynamics”: a formulation of Einstein’s equation of general relativity that focuses on how the geometry of space changes with time. Caticha points this out… but alas, leaves open the question of whether his dynamics matches the usual dynamics in general relativity!

      If it worked, it would be nice. This would also be a way of sidestepping Eric Forgy’s desire for a Lorentzian metric. I don’t see any way to get a Lorentzian metric from a Fisher information metric: the latter is always Riemannian (i.e., positive definite).

      But I think this whole program of getting general relativity from information theory is a bit over-speculative… until someone gets it to work.

    • phorgyphynance says:

      If it worked, it would be nice. This would also be a way of sidestepping Eric Forgy’s desire for a Lorentzian metric. I don’t see any way to get a Lorentzian metric from a Fisher information metric: the latter is always Riemannian (i.e., positive definite).

      For what its worth, I never suggested you could (or that you should even try to) get a Lorentzian metric from the Fisher information metric. What I suggested is what I put forward here. Nothing more nothing less.

      The Fisher information metric defines the spatial component of your Lorentzian metric where “space” is the probabilistic space whose length is standard deviation and cosines of angles are correlations.

      The “time” component corresponds to the deterministic part of your observable where length corresponds to the mean. It is most easily seen as a differential

      dX = \sigma dW + \mu dt,

      where \sigma is standard deviation and \mu is mean.

      The Fisher information metric sees the “spatial” component g(\sigma dW,\sigma dW) = \sigma^2. The “temporal” component of the metric sees the deterministic part g(\mu dt,\mu dt) = -\frac{\mu^2}{c^2}.

      The full inner product of your observable is then

      g(dX,dX) = \sigma^2 - \frac{\mu^2}{c^2},

      where c is a parameter that relates to “risk aversion” in the financial interpretation.

      The mathematics is very neat. The algebra gives meaningful results that have clear statistical meaning.

  4. Michael Eilenberg says:

    Often, I would like to print posts from this site, but unfortunately the result is always disastrous; at least in Firefox. Typically the output is a union of the first page of {page header, article, comments, page footer}.

    I solve it by copy&paste into OpenOffice and then print. Is there a better way?


  5. […] suppose we’re working in the special case discussed in Part 2, where our manifold is an open subset of , and at the point is the Gibbs state with . Then all […]

  6. […] Part 1     • Part 2     • Part 3     • Part 4     • Part […]

  7. […] found the above in the nice blog of John Baez where Vasileios Anagnostopoulos says about that in the […]

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