A Second Law for Open Markov Processes

15 November, 2014

guest post by Blake Pollard

What comes to mind when you hear the term ‘random process’? Do you think of Brownian motion? Do you think of particles hopping around? Do you think of a drunkard staggering home?

Today I’m going to tell you about a version of the drunkard’s walk with a few modifications. Firstly, we don’t have just one drunkard: we can have any positive real number of drunkards. Secondly, our drunkards have no memory; where they go next doesn’t depend on where they’ve been. Thirdly, there are special places, such as entrances to bars, where drunkards magically appear and disappear.

The second condition says that our drunkards satisfy the Markov property, making their random walk into a Markov process. The third condition is really what I want to tell you about, because it makes our Markov process into a more general ‘open Markov process’.

There are a collection of places the drunkards can be, for example:

V= \{ \text{bar},\text{sidewalk}, \text{street}, \text{taco truck}, \text{home} \}

We call this set V the set of states. There are certain probabilities associated with traveling between these places. We call these transition rates. For example it is more likely for a drunkard to go from the bar to the taco truck than to go from the bar to home so the transition rate between the bar and the taco truck should be greater than the transition rate from the bar to home. Sometimes you can’t get from one place to another without passing through intermediate places. In reality the drunkard can’t go directly from the bar to the taco truck: he or she has to go from the bar to sidewalk to the taco truck.

This information can all be summarized by drawing a directed graph where the positive numbers labelling the edges are the transition rates:

For simplicity we draw only three states: home, bar, taco truck. Drunkards go from home to the bar and back, but they never go straight from home to the taco truck.

We can keep track of where all of our drunkards are using a vector with 3 entries:

\displaystyle{ p(t) = \left( \begin{array}{c} p_h(t) \\ p_b(t) \\ p_{tt}(t) \end{array} \right) \in \mathbb{R}^3 }

We call this our population distribution. The first entry p_h is the number of drunkards that are at home, the second p_b is how many are at the bar, and the third p_{tt} is how many are at the taco truck.

There is a set of coupled, linear, first-order differential equations we can write down using the information in our graph that tells us how the number of drunkards in each place change with time. This is called the master equation:

\displaystyle{ \frac{d p}{d t} = H p }

where H is a 3×3 matrix which we call the Hamiltonian. The off-diagonal entries are nonnegative:

H_{ij} \geq 0, i \neq j

and the columns sum to zero:

\sum_i H_{ij}=0

We call a matrix satisfying these conditions infinitesimal stochastic. Stochastic matrices have columns that sum to one. If we take the exponential of an infinitesimal stochastic matrix we get one whose columns sum to one, hence the label ‘infinitesimal’.

The Hamiltonian for the graph above is

H = \left( \begin{array}{ccc} -2 & 5 & 10 \\ 2 & -12 & 0 \\ 0 & 7 & -10 \end{array} \right)

John has written a lot about Markov processes and infinitesimal stochastic Hamiltonians in previous posts.

Given two vectors p,q \in \mathbb{R}^3 describing the populations of drunkards which obey the same master equation, we can calculate the relative entropy of p relative to q:

\displaystyle{ S(p,q) = \sum_{ i \in V} p_i \ln \left( \frac{p_i}{q_i} \right) }

This is an example of a ‘divergence’. In statistics, a divergence a way of measuring the distance between probability distributions, which may not be symmetrical and may even not obey the triangle inequality.

The relative entropy is important because it decreases monotonically with time, making it a Lyapunov function for Markov processes. Indeed, it is a well known fact that

\displaystyle{ \frac{dS(p(t),q(t) ) } {dt} \leq 0 }

This is true for any two population distributions which evolve according to the same master equation, though you have to allow infinity as a possible value for the relative entropy and negative infinity for its time derivative.

Why is entropy decreasing? Doesn’t the Second Law of Thermodynamics say entropy increases?

Don’t worry: the reason is that I have not put a minus sign in my definition of relative entropy. Put one in if you like, and then it will increase. Sometimes without the minus sign it’s called the Kullback–Leibler divergence. This decreases with the passage of time, saying that any two population distributions p(t) and q(t) get ‘closer together’ as they get randomized with the passage of time.

That itself is a nice result, but I want to tell you what happens when you allow drunkards to appear and disappear at certain states. Drunkards appear at the bar once they’ve had enough to drink and once they are home for long enough they can disappear. The set of places where drunkards can appear or disappear B is called the set of boundary states.  So for the above process

B = \{ \text{home},\text{bar} \}

is the set of boundary states. This changes the way in which the population of drunkards changes with time!

The drunkards at the taco truck obey the master equation. For them,

\displaystyle{ \frac{dp_{tt}}{dt} = 7p_b -10 p_{tt} }

still holds. But because the populations can appear or disappear at the boundary states the master equation no longer holds at those states! Instead it is useful to define the flow of drunkards into the i^{th} state by

\displaystyle{ \frac{Dp_i}{Dt} = \frac{dp_i}{dt}-\sum_j H_{ij} p_j}

This quantity describes by how much the rate of change of the populations at the boundary states differ from that given by the master equation.

The reason why we are interested in open Markov processes is because you can take two open Markov processes and glue them together along some subset of their boundary states to get a new open Markov process! This allows us to build up or break down complicated Markov processes using open Markov processes as the building blocks.

For example we can draw the graph corresponding to the drunkards’ walk again, only now we will distinguish boundary states from internal states by coloring internal states blue and having boundary states be white:

Consider another open Markov process with states

V=\{ \text{home},\text{work},\text{bar} \}

where

B=\{ \text{home}, \text{bar}\}

are the boundary states, leaving

I=\{\text{work}\}

as an internal state:

Since the boundary states of this process overlap with the boundary states of the first process we can compose the two to form a new Markov process:

Notice the boundary states are now internal states. I hope any Markov process that could approximately model your behavior has more interesting nodes! There is a nice way to figure out the Hamiltonian of the composite from the Hamiltonians of the pieces, but we will leave that for another time.

We can ask ourselves, how does relative entropy change with time in open Markov processes? You can read my paper for the details, but here is the punchline:

\displaystyle{ \frac{dS(p(t),q(t) ) }{dt} \leq \sum_{i \in B} \frac{Dp_i}{Dt}\frac{\partial S}{\partial p_i} + \frac{Dq_i}{Dt}\frac{\partial S}{\partial q_i} }

This is a version of the Second Law of Thermodynamics for open Markov processes.

It is important to notice that the sum is only over the boundary states! This inequality tells us that relative entropy still decreases inside our process, but depending on the flow of populations through the boundary states the relative entropy of the whole process could either increase or decrease! This inequality will be important when we study how the relative entropy changes in different parts of a bigger more complicated process.

That is all for now, but I leave it as an exercise for you to imagine a Markov process that describes your life. How many states does it have? What are the relative transition rates? Are there states you would like to spend more or less time in? Are there states somewhere you would like to visit?

Here is my paper, which proves the above inequality:

• Blake Pollard, A Second Law for open Markov processes.

If you have comments or corrections, let me know!


Sensing and Acting Under Information Constraints

30 October, 2014

I’m having a great time at a workshop on Biological and Bio-Inspired Information Theory in Banff, Canada. You can see videos of the talks online. There have been lots of good talks so far, but this one really blew my mind:

• Naftali Tishby, Sensing and acting under information constraints—a principled approach to biology and intelligence, 28 October 2014.

Tishby’s talk wasn’t easy for me to follow—he assumed you already knew rate-distortion theory and the Bellman equation, and I didn’t—but it was great!

It was about the ‘action-perception loop':


This is the feedback loop in which living organisms—like us—take actions depending on our goals and what we perceive, and perceive things depending on the actions we take and the state of the world.

How do we do this so well? Among other things, we need to balance the cost of storing information about the past against the payoff of achieving our desired goals in the future.

Tishby presented a detailed yet highly general mathematical model of this! And he ended by testing the model on experiments with cats listening to music and rats swimming to land.

It’s beautiful stuff. I want to learn it. I hope to blog about it as I understand more. But for now, let me just dive in and say some basic stuff. I’ll start with the two buzzwords I dropped on you. I hate it when people use terminology without ever explaining it.

Rate-distortion theory

Rate-distortion theory is a branch of information theory which seeks to find the minimum rate at which bits must be communicated over a noisy channel so that the signal can be approximately reconstructed at the other end without exceeding a given distortion. Shannon’s first big result in this theory, the ‘rate-distortion theorem’, gives a formula for this minimum rate. Needless to say, it still requires a lot of extra work to determine and achieve this minimum rate in practice.

For the basic definitions and a statement of the theorem, try this:

• Natasha Devroye, Rate-distortion theory, course notes, University of Chicago, Illinois, Fall 2009.

One of the people organizing this conference is a big expert on rate-distortion theory, and he wrote a book about it.

• Toby Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression, Prentice–Hall, 1971.

Unfortunately it’s out of print and selling for $259 used on Amazon! An easier option might be this:

• Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, Chapter 10: Rate Distortion Theory, Wiley, New York, 2006.

The Bellman equation

The Bellman equation reduces the task of finding an optimal course of action to choosing what to do at each step. So, you’re trying to maximize the ‘total reward’—the sum of rewards at each time step—and Bellman’s equation says what to do at each time step.

If you’ve studied physics, this should remind you of how starting from the principle of least action, we can get a differential equation describing the motion of a particle: the Euler–Lagrange equation.

And in fact they’re deeply related. The relation is obscured by two little things. First, Bellman’s equation is usually formulated in a context where time passes in discrete steps, while the Euler–Lagrange equation is usually formulated in continuous time. Second, Bellman’s equation is really the discrete-time version not of the Euler–Lagrange equation but a more or less equivalent thing: the ‘Hamilton–Jacobi equation’.

Ah, another buzzword to demystify! I was scared of the Hamilton–Jacobi equation for years, until I taught a course on classical mechanics that covered it. Now I think it’s the greatest thing in the world!

Briefly: the Hamilton–Jacobi equation concerns the least possible action to get from a fixed starting point to a point q in space at time t. If we call this least possible action W(t,q), the Hamilton–Jacobi equation says

\displaystyle{ \frac{\partial W(t,q)}{\partial q_i} = p_i  }

\displaystyle{ \frac{\partial W(t,q)}{\partial t} = -E  }

where p is the particle’s momentum at the endpoint of its path, and E is its energy there.

If we replace derivatives by differences, and talk about maximizing total reward instead of minimizing action, we get Bellman’s equation:

Bellman equation, Wikipedia.

Markov decision processes

Bellman’s equation can be useful whenever you’re trying to figure out an optimal course of action. An important example is a ‘Markov decision process’. To prepare you for Tishby’s talk, I should say what this is.

In a Markov process, something randomly hops around from state to state with fixed probabilities. In the simplest case there’s a finite set S of states, and time proceeds in discrete steps. At each time step, the probability to hop from state s to state s' is some fixed number P(s,s').

This sort of thing is called a Markov chain, or if you feel the need to be more insistent, a discrete-time Markov chain.

A Markov decision process is a generalization where an outside agent gets to change these probabilities! The agent gets to choose actions from some set A. If at a given time he chooses the action \alpha \in A, the probability of the system hopping from state s to state s' is P_\alpha(s,s'). Needless to say, these probabilities have to sum to one for any fixed s.

That would already be interesting, but the real fun is that there’s also a reward R_\alpha(s,s'). This is a real number saying how much joy or misery the agent experiences if he does action \alpha and the system hops from s to s'.

The problem is to choose a policy—a function from states to actions—that maximizes the total expected reward over some period of time. This is precisely the kind of thing Bellman’s equation is good for!

If you’re an economist you might also want to ‘discount’ future rewards, saying that a reward n time steps in the future gets multiplied by \gamma^n, where 0 < \gamma \le 1 is some discount factor. This extra tweak is easily handled, and you can see it all here:

Markov decision process, Wikipedia.

Partially observable Markov decision processes

There’s a further generalization where the agent can’t see all the details of the system! Instead, when he takes an action \alpha \in A and the system hops from state s to state s', he sees something: a point in some set O of observations. He makes the observation o \in O with probability \Omega_\alpha(o,s').

(I don’t know why this probability depends on s' but not s. Maybe it ultimately doesn’t matter much.)

Again, the goal is to choose a policy that maximizes the expected total reward. But a policy is a bit different now. The action at any time can only depend on all the observations made thus far.

Partially observable Markov decision processes are also called POMPDs. If you want to learn about them, try these:

Partially observable Markov decision process, Wikipedia.

• Tony Cassandra, Partially observable Markov decision processes.

The latter includes an introduction without any formulas to POMDPs and how to choose optimal policies. I’m not sure who would study this subject and not want to see formulas, but it’s certainly a good exercise to explain things using just words—and it makes certain things easier to understand (though not others, in a way that depends on who is trying to learn the stuff).

The action-perception loop

I already explained the action-perception loop, with the help of this picture from the University of Bielefeld’s Department of Cognitive Neuroscience:


Nafthali Tishby has a nice picture of it that’s more abstract:

We’re assuming time comes in discrete steps, just to keep things simple.

At each time t

• the world has some state W_t, and
• the agent has some state M_t.

Why the letter M? This stands for memory: it can be the state of the agent’s memory, but I prefer to think of it as the state of the agent.

At each time

• the agent takes an action A_t, which affects the world’s next state, and

• the world provides a sensation S_t to the agent, which affect’s the agent’s next state.

This is simplified but very nice. Note that there’s a symmetry interchanging the world and the agent!

We could make it fancier by having lots of agents who all interact, but there are a lot of questions already. The big question Tishby focuses on is optimizing how much the agent should remember about the past if they

• get a reward depending on the action taken and the resulting state of the world

but

• pay a price for the information stored from sensations.

Tishby formulates this optimization question as something like a partially observed Markov decision process, uses rate-distortion theory to analyze how much information needs to be stored to achieve a given reward, and uses Bellman’s equation to solve the optimization problem!

So, everything I sketched fits together somehow!

I hope what I’m saying now is roughly right: it will take me more time to get the details straight. If you’re having trouble absorbing all the information I just threw at you, don’t feel bad: so am I. But the math feels really natural and good to me. It involves a lot of my favorite ideas (like generalizations of the principle of least action, and relative entropy), and it seems ripe to be combined with network theory ideas.

For details, I highly recommend this paper:

• Naftali Tishby and Daniel Polani, Information theory of decisions and actions, in Perception-Reason-Action Cycle: Models, Algorithms and System. Vassilis, Hussain and Taylor, Springer, Berlin, 2010.

I’m going to print this out, put it by my bed, and read it every night until I’ve absorbed it.


Biodiversity, Entropy and Thermodynamics

27 October, 2014

 

I’m giving a short 30-minute talk at a workshop on Biological and Bio-Inspired Information Theory at the Banff International Research Institute.

I’ll say more about the workshop later, but here’s my talk, in PDF and video form:

Biodiversity, entropy and thermodynamics.

Most of the people at this workshop study neurobiology and cell signalling, not evolutionary game theory or biodiversity. So, the talk is just a quick intro to some things we’ve seen before here. Starting from scratch, I derive the Lotka–Volterra equation describing how the distribution of organisms of different species changes with time. Then I use it to prove a version of the Second Law of Thermodynamics.

This law says that if there is a ‘dominant distribution’—a distribution of species whose mean fitness is at least as great as that of any population it finds itself amidst—then as time passes, the information any population has ‘left to learn’ always decreases!

Of course reality is more complicated, but this result is a good start.

This was proved by Siavash Shahshahani in 1979. For more, see:

• Lou Jost, Entropy and diversity.

• Marc Harper, The replicator equation as an inference dynamic.

• Marc Harper, Information geometry and evolutionary game theory.

and more recent papers by Harper.


Entropy and Information in Biological Systems (Part 2)

4 July, 2014

John Harte, Marc Harper and I are running a workshop! Now you can apply here to attend:

Information and entropy in biological systems, National Institute for Mathematical and Biological Synthesis, Knoxville Tennesee, Wednesday-Friday, 8-10 April 2015.

Click the link, read the stuff and scroll down to “CLICK HERE” to apply. The deadline is 12 November 2014.

Financial support for travel, meals, and lodging is available for workshop attendees who need it. We will choose among the applicants and invite 10-15 of them.

The idea

Information theory and entropy methods are becoming powerful tools in biology, from the level of individual cells, to whole ecosystems, to experimental design, model-building, and the measurement of biodiversity. The aim of this investigative workshop is to synthesize different ways of applying these concepts to help systematize and unify work in biological systems. Early attempts at “grand syntheses” often misfired, but applications of information theory and entropy to specific highly focused topics in biology have been increasingly successful. In ecology, entropy maximization methods have proven successful in predicting the distribution and abundance of species. Entropy is also widely used as a measure of biodiversity. Work on the role of information in game theory has shed new light on evolution. As a population evolves, it can be seen as gaining information about its environment. The principle of maximum entropy production has emerged as a fascinating yet controversial approach to predicting the behavior of biological systems, from individual organisms to whole ecosystems. This investigative workshop will bring together top researchers from these diverse fields to share insights and methods and address some long-standing conceptual problems.

So, here are the goals of our workshop:

• To study the validity of the principle of Maximum Entropy Production (MEP), which states that biological systems – and indeed all open, non-equilibrium systems – act to produce entropy at the maximum rate.

• To familiarize all the participants with applications to ecology of the MaxEnt method: choosing the probabilistic hypothesis with the highest entropy subject to the constraints of our data. We will compare MaxEnt with competing approaches and examine whether MaxEnt provides a sufficient justification for the principle of MEP.

• To clarify relations between known characterizations of entropy, the use of entropy as a measure of biodiversity, and the use of MaxEnt methods in ecology.

• To develop the concept of evolutionary games as “learning” processes in which information is gained over time.

• To study the interplay between information theory and the thermodynamics of individual cells and organelles.

For more details, go here.

If you’ve got colleagues who might be interested in this, please let them know. You can download a PDF suitable for printing and putting on a bulletin board by clicking on this:



Network Theory III

16 March, 2014

 

In the last of my Oxford talks I explain how entropy and relative entropy can be understood using certain categories related to probability theory… and how these categories also let us understand Bayesian networks!

The first two parts are explanations of these papers:

• John Baez, Tobias Fritz and Tom Leinster, A characterization of entropy in terms of information loss

• John Baez and Tobias Fritz, A Bayesian characterization of relative entropy.

Somewhere around here the talk was interrupted by a fire drill, waking up the entire audience!

By the way, in my talk I mistakenly said that relative entropy is a continuous functor; in fact it’s just lower semicontinuous. I’ve fixed this in my slides.

The third part of my talk was my own interpretation of Brendan Fong’s master’s thesis:

• Brendan Fong, Causal Theories: a Categorical Perspective on Bayesian Networks.

I took a slightly different approach, by saying that a causal theory \mathcal{C}_G is the free category with products on certain objects and morphisms coming from a directed acyclic graph G. In his thesis he said \mathcal{C}_G was the free symmetric monoidal category where each generating object is equipped with a cocommutative comonoid structure. This is close to a category with finite products, though perhaps not quite the same: a symmetric monoidal category where every object is equipped with a cocommutative comonoid structure in a natural way (i.e., making a bunch of squares commute) is a category with finite products. It would be interesting to see if this difference hurts or helps.

By making this slight change, I am claiming that causal theories can be seen as algebraic theories in the sense of Lawvere. This would be a very good thing, since we know a lot about those.

You can also see the slides of this talk. Click on any picture in the slides, or any text in blue, and get more information!


Relative Entropy (Part 4)

16 February, 2014

In recent posts by Manoj Gopalkrishnan and Marc Harper we’ve seen how not just entropy but relative entropy—the entropy of a probability distribution relative to the equilibrium distribution—is a driving force in chemistry and evolution. Now Tobias Fritz and I have finally finished our paper on this subject:

A Bayesian characterization of relative entropy.

Very roughly, we proved that any reasonable measure of the information you gain when you to update your assumptions about the world based on discovering what a system is really doing must be some constant c times the relative entropy.

I’ve blogged about this here before:

Relative Entropy (Part 1): how various structures important in probability theory arise naturally when you do linear algebra using only the nonnegative real numbers.

Relative Entropy (Part 2): a category related to statistical inference, \mathrm{FinStat}, and how relative entropy defines a functor on this category.

Relative Entropy (Part 3): statement of our main theorem, which characterizes relative entropy up to a constant multiple as the only functor F : \mathrm{FinStat} \to [0,\infty) with a few nice properties.

Now that the paper is done, we’re having a nice conversation about it on the n-Category Café. Since I don’t want to splinter the conversation, I won’t enable comments here—please go there and join the fun!

But having blogged about this thrice before, what’s new?

One thing is that our conversation is getting more deeply into the category-theoretic aspects. Read the long parenthetical remarks in my post on the n-Café to get up to speed on that aspect.

Another is that by looking at our paper, you can actually see the proof of our result. As I mention on the n-Café.

The proof is surprisingly hard. Or maybe we’re just surprisingly bad at proving things. But the interesting thing is this: the proof is swift and effective in the ‘generic’ case—the case where the support of the probability measures involved is the whole set they’re living on, and the constant c is finite.

It takes some more work to handle the case where the probability measures have smaller support.

But the really hard work starts when we handle the case that, in the end, has c = \infty. Then the proof becomes more like analysis than what you normally expect in category theory. We slowly corner the result, blocking off all avenues of escape. Then we close in, grab its neck, and strangle it, crushing its larynx ever tighter, as it loses the will to fight back and finally expires… still twitching.

We haven’t gotten into discussing this much yet, perhaps because the mathematicians on the n-Café are too dainty and civilized. But someone into analysis might be able to find a more efficient proof.

That would make me a bit sad—since why didn’t we find it?—but mainly happy—since this subject deserves to be clean and elegant. We really need a category-theoretic formulation of the second law of thermodynamics that’s suitable for studying complex networks: that’s the long-term goal here.


Bio-Inspired Information Theory

31 January, 2014

 

There will be a 5-day workshop on Biological and Bio-Inspired Information Theory at BIRS from Sunday the 26th to Friday the 31st of October, 2014. It’s being organized by

Toby Berger (University of Virginia)
Andrew Eckford (York University)
Peter Thomas (Case Western Reserve University)

BIRS is the Banff International Research Station, a conference venue in a rather wild part of Alberta, in the mountains west of Calgary.

Description

Here’s the workshop proposal on the BIRS website:

Currently, research in the community is organized into three streams:

• Information theory and biochemistry (including information theory and intercellular communication);

• Information theory and neuroscience; and

• Information-theoretic analysis of biologically-inspired communication systems (including nano-networking and design of biologically implemented information processing networks).

We propose a BIRS workshop to explore these streams, focusing on mathematical open problems that cut across the streams. The main objectives of this workshop would be: to bring together the most prominent researchers in this field; to discuss and review the current state of mathematical research in this field; to promote cross-pollination among the various streams of research to find common problems; and to collectively identify key future directions or open problems that would bring the greatest impact to this field over the next few years.

Expected impact

A BIRS workshop involving the field’s leading researchers would allow a review of the current state of the art, and would promote cross-pollination among these three streams of research. We expect to have these leading researchers in attendance. For example, Prof. Toby Berger (U. Virginia), a widely recognized pioneer in this field and a recipient of the Shannon Award (the top prize awarded by the IEEE Information Theory Society), is one of the co-organizers of the workshop. Moreover, we have approached many of the field’s most prominent mathematicians and scientists: a complete list is found elsewhere in this proposal, but among the most prominent confirmed participants are: Prof. Tadashi Nakano (Osaka U.), one of the earliest researchers in engineered molecular communication; Dr. Thomas D. Schneider (NIH – National Cancer Institute), inventor of the sequence logo and prominent researcher in genetic information theory; and Profs. William Bialek (Princeton U.) and Naftali Tishby (Hebrew U.), prominent experts on information theory in neural coding.

Although the focus of our workshop is on mathematical fundamentals, our list of expected participants includes a few experimental scientists, e.g. Raymond Cheong and Andre Levchenko (both from Johns Hopkins U.), in addition to mathematical scientists. This is because quantitative application of information theoretic analysis to biological systems typically requires empirical estimation of joint probability distributions for multiple input and output variables, often posing daunting data collection challenges which pioneered the use of high-throughput experimental methods to collect large data sets quantifying the input/output relationships for a specific biochemical signaling pathway). We believe a blended approach, emphasizing mathematics but including experimental perspectives, will enhance the impact of our workshop and increase the usefulness to our participants.

Given that publications in these research areas have achieved prominence in the past few years, the time is right for a general meeting among the key researchers to review the state of the field and develop future directions. Thus, our proposed workshop is timely and would be expected to have a tremendous impact on the field over the next several years.


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