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!


Network Theory Seminar (Part 4)

5 November, 2014

 

Since I was in Banff, my student Franciscus Rebro took over this week and explained more about cospan categories. These are a tool for constructing categories where the morphisms are networks such as electrical circuit diagrams, signal flow diagrams, Markov processes and the like. For some more details see:

• John Baez and Brendan Fong, A compositional framework for passive linear networks.

Cospan categories are really best thought of as bicategories, and Franciscus gets into this aspect too.


Network Theory (Part 33)

4 November, 2014

Last time I came close to describing the ‘black box functor’, which takes an electrical circuit made of resistors

and sends it to its behavior as viewed from outside. From outside, all you can see is the relation between currents and potentials at the ‘terminals’—the little bits of wire that poke out of the black box:

I came close to defining the black box functor, but I didn’t quite make it! This time let’s finish the job.

The categories in question

The black box functor

\blacksquare : \mathrm{ResCirc} \to \mathrm{LinRel}

goes from the category \mathrm{ResCirc}, where morphisms are circuits made of resistors, to the category \mathrm{LinRel}, where morphisms are linear relations. Let me remind you how these categories work, and introduce a bit of new notation.

Here is the category \mathrm{ResCirc}:

• an object is a finite set;

• a morphism from X to Y is an isomorphism class of cospans

in the category of graphs with edges labelled by resistances: numbers in (0,\infty). Here we think of the finite sets X and Y as graphs with no edges. We call X the set of inputs and Y the set of outputs.

• we compose morphisms in \mathrm{ResCirc} by composing isomorphism classes of cospans.

And here is the category \mathrm{LinRel}:

• an object is a finite-dimensional real vector space;

• a morphism from U to V is a linear relation R : U \leadsto V, meaning a linear subspace R \subseteq U \times V;

• we compose a linear relation R \subseteq U \times V and a linear relation S \subseteq V \times W in the usual way we compose relations, getting:

SR = \{(u,w) \in U \times W : \; \exists v \in V \; (u,v) \in R \mathrm{\; and \;} (v,w) \in S \}

In case you’re wondering: I’ve just introduced the wiggly arrow notation

R : U \leadsto V

for a linear relation from U to V, because it suggests that a relation is a bit like a function but more general. Indeed, a function is a special case of a relation, and composing functions is a special case of composing relations.

The black box functor

Now, how do we define the black box functor?

Defining it on objects is easy. An object of \mathrm{ResCirc} is a finite set S, and we define

\blacksquare{S} = \mathbb{R}^S \times \mathbb{R}^S

The idea is that S could be a set of inputs or outputs, and then

(\phi, I) \in \mathbb{R}^S \times \mathbb{R}^S

is a list of numbers: the potentials and currents at those inputs or outputs.

So, the interesting part is defining the black box functor on morphisms!

For this we start with a morphism in \mathrm{ResCirc}:

The labelled graph \Gamma consists of:

• a set N of nodes,

• a set E of edges,

• maps s, t : E \to N sending each edge to its source and target,

• a map r : E \to (0,\infty) sending each edge to its resistance.

The cospan gives maps

i: X \to N, \qquad o: Y \to N

These say how the inputs and outputs are interpreted as nodes in the circuit. We’ll call the nodes that come from inputs or outputs ‘terminals’. So, mathematically,

T = \mathrm{im}(i) \cup \mathrm{im}(o) \subseteq N

is the set of terminals: the union of the images of i and o.

In the simplest case, the maps i and o are one-to-one, with disjoint ranges. Then each terminal either comes from a single input, or a single output, but not both! This is a good picture to keep in mind. But some subtleties arise when we leave this simplest case and consider other cases.

Now, the black box functor is supposed to send our circuit to a linear relation. I’ll call the circuit \Gamma for short, though it’s really the whole cospan

So, our black box functor is supposed to send this circuit to a linear relation

\blacksquare(\Gamma) : \mathbb{R}^X \times \mathbb{R}^X \leadsto \mathbb{R}^Y \times \mathbb{R}^Y

This is a relation between the potentials and currents at the input terminals and the potentials and currents at the output terminals! How is it defined?

I’ll start by outlining how this works.

First, our circuit picks out a subspace

dQ \subseteq \mathbb{R}^T \times \mathbb{R}^T

This is the subspace of allowed potentials and currents on the terminals. I’ll explain this and why it’s called dQ a bit later. Briefly, it comes from the principle of minimum power, described last time.

Then, the map

i: X \to T

gives a linear relation

S(i) : \mathbb{R}^X \times \mathbb{R}^X \leadsto \mathbb{R}^T \times \mathbb{R}^T

This says how the potentials and currents at the inputs are related to those at the terminals. Similarly, the map

o: Y \to T

gives a linear relation

S(o) : \mathbb{R}^Y \times \mathbb{R}^Y \leadsto \mathbb{R}^T \times \mathbb{R}^T

This says how the potentials and currents at the outputs are related to those at the terminals.

Next, we can ‘turn around’ any linear relation

R : \mathbb{R}^Y \times \mathbb{R}^Y \leadsto \mathbb{R}^T \times \mathbb{R}^T

to get a relation

R^\dagger : \mathbb{R}^T \times \mathbb{R}^T  \leadsto \mathbb{R}^Y \times \mathbb{R}^Y

defined by

R^\dagger = \{(\phi',-I',\phi,-I) : (\phi, I, \phi', I') \in R \}

Here we are just switching the input and output potentials, but when we switch the currents we also throw in a minus sign. The reason is that we care about the current flowing in to an input, but out of an output.

Finally, one more trick: given a linear subspace

L \subseteq V

of a vector space V we get a linear relation

1|_L : V \leadsto V

called the identity restricted to L, defined like this:

1|_L = \{ (v, v) :\; v \in L \} \subseteq V \times V

If L is all of V this relation is actually the identity function on V. Otherwise it’s a partially defined function that’s defined only on L, and is the identity there. (A partially defined function is an example of a relation.) My notation 1|_L is probably bad, but I don’t know a better one, so bear with me.

Let’s use all these ideas to define

\blacksquare(\Gamma) : \mathbb{R}^X \times \mathbb{R}^X \leadsto \mathbb{R}^Y \times \mathbb{R}^Y

To do this, we compose three linear relations:

1) We start with

S(i) : \mathbb{R}^X \times \mathbb{R}^X \leadsto \mathbb{R}^T \times \mathbb{R}^T

2) We compose this with

1|_{dQ} : \mathbb{R}^T \times \mathbb{R}^T \leadsto \mathbb{R}^T \times \mathbb{R}^T

3) Then we compose this with

S(o)^\dagger : \mathbb{R}^T \times \mathbb{R}^T \leadsto \mathbb{R}^Y \times \mathbb{R}^Y

Note that:

1) says how the potentials and currents at the inputs are related to those at the terminals,

2) picks out which potentials and currents at the terminals are actually allowed, and

3) says how the potentials and currents at the terminals are related to those at the outputs.

So, I hope all makes sense, at least in some rough way. In brief, here’s the formula:

\blacksquare(\Gamma) = S(o)^\dagger \; 1|_{dQ} \; S(i)

Now I just need to fill in some details. First, how do we define S(i) and S(o)? They work exactly the same way, by ‘copying potentials and adding currents’, so I’ll just talk about one. Second, how do we define the subspace dQ? This uses the principle of minimum power.

Duplicating potentials and adding currents

Any function between finite sets

i: X \to T

gives a linear map

i^* : \mathbb{R}^T \to \mathbb{R}^X

Mathematicians call this linear map the pullback along i, and for any \phi \in \mathbb{R}^T it’s defined by

i^*(\phi)(x) = \phi(i(x))

In our application, we think of \phi as a list of potentials at terminals. The function i could map a bunch of inputs to the same terminal, and the above formula says the potential at this terminal gives the potential at all those inputs. So, we are copying potentials.

We also get a linear map going the other way:

i_* : \mathbb{R}^X \to \mathbb{R}^T

Mathematicians call this the pushforward along i, and for any I \in \mathbb{R}^X it’s defined by

\displaystyle{ i_*(I)(t) = \sum_{x \; : \; i(x) = t } I(x) }

In our application, we think of I as a list of currents entering at some inputs. The function i could map a bunch of inputs to the same terminal, and the above formula says the current at this terminal is the sum of the currents at all those inputs. So, we are adding currents.

Putting these together, our map

i : X \to T

gives a linear relation

S(i) : \mathbb{R}^X \times \mathbb{R}^X \leadsto \mathbb{R}^T \times \mathbb{R}^T

where the pair (\phi, I) \in \mathbb{R}^X \times \mathbb{R}^X is related to the pair (\phi', I') \in \mathbb{R}^T \times \mathbb{R}^T iff

\phi = i^*(\phi')

and

I' = i_*(I)

So, here’s the rule of thumb when attaching the points of X to the input terminals of our circuit: copy potentials, but add up currents. More formally:

\begin{array}{ccl} S(i) &=& \{ (\phi, I, \phi', I') : \; \phi = i^*(\phi') , \; I' = i_*(I) \}  \\ \\  &\subseteq& \mathbb{R}^X \times \mathbb{R}^X \times \mathbb{R}^T \times \mathbb{R}^T \end{array}

The principle of minimum power

Finally, how does our circuit define a subspace

dQ \subseteq \mathbb{R}^T \times \mathbb{R}^T

of allowed potential-current pairs at the terminals? The trick is to use the ideas we discussed last time. If we know the potential at all nodes of our circuit, say \phi \in \mathbb{R}^N, we know the power used by the circuit:

P(\phi) = \displaystyle{ \sum_{e \in E} \frac{1}{r_e} \big(\phi(s(e)) - \phi(t(e))\big)^2 }

We saw last time that if we fix the potentials at the terminals, the circuit will choose potentials at the other nodes to minimize this power. We can describe the potential at the terminals by

\psi \in \mathbb{R}^T

So, the power for a given potential at the terminals is

Q(\psi) = \displaystyle{ \frac{1}{2} \min_{\phi \in \mathbb{R}^N \; : \; \phi|_T = \psi} \sum_{e \in E} \frac{1}{r_e} \big(\phi(s(e)) - \phi(t(e))\big)^2 }

Actually this is half the power: I stuck in a factor of 1/2 for some reason we’ll soon see. This Q is a quadratic function

Q : \mathbb{R}^T \to \mathbb{R}

so its derivative is linear. And, our work last time showed something interesting: to compute the current J_x flowing into a terminal x \in T, we just differentiate Q with respect to the potential at that terminal:

\displaystyle{ J_x = \frac{\partial Q(\psi)}{\partial \psi_x} }

This is the reason for the 1/2: when we take the derivative of Q, we bring down a 2 from differentiating all those squares, and to make that go away we need a 1/2.

The space of allowed potential-current pairs at the terminals is thus the linear subspace

dQ = \{ (\psi, J) : \; \displaystyle{ J_x = \frac{\partial Q(\psi)}{\partial \psi_x} \}  \subseteq \mathbb{R}^T \times \mathbb{R}^T }

And this completes our precise description of the black box functor!

The hard part is this:

Theorem. \blacksquare : \mathrm{ResCirc} \to \mathrm{LinRel} is a functor.

In other words, we have to prove that it preserves composition:

\blacksquare(fg) = \blacksquare(f) \blacksquare(g)

For that, read our paper:

• John Baez and Brendan Fong, A compositional framework for passive linear networks.


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.


Network Theory Seminar (Part 3)

21 October, 2014

 

This time we use the principle of minimum power to determine what a circuit made of resistors actually does. Its ‘behavior’ is described by a functor sending circuits to linear relations between the potentials and currents at the input and output terminals. We call this the ‘black box’ functor, since it takes a circuit:

and puts a metaphorical ‘black box’ around it:

hiding the circuit’s internal details and letting us see only how it acts as viewed ‘from outside’.

For more, see the lecture notes here:

Network theory (part 32).

http://johncarlosbaez.wor


Network Theory (Part 32)

20 October, 2014

Okay, today we will look at the ‘black box functor’ for circuits made of resistors. Very roughly, this takes a circuit made of resistors with some inputs and outputs:

and puts a ‘black box’ around it:

forgetting the internal details of the circuit and remembering only how the it behaves as viewed from outside. As viewed from outside, all the circuit does is define a relation between the potentials and currents at the inputs and outputs. We call this relation the circuit’s behavior. Lots of different choices of the resistances R_1, \dots, R_6 would give the same behavior. In fact, we could even replace the whole fancy circuit by a single edge with a single resistor on it, and get a circuit with the same behavior!

The idea is that when we use a circuit to do something, all we care about is its behavior: what it does as viewed from outside, not what it’s made of.

Furthermore, we’d like the behavior of a system made of parts to depend in a simple way on the external behaviors of its parts. We don’t want to have to ‘peek inside’ the parts to figure out what the whole will do! Of course, in some situations we do need to peek inside the parts to see what the whole will do. But in this particular case we don’t—at least in the idealization we are considering. And this fact is described mathematically by saying that black boxing is a functor.

So, how do circuits made of resistors behave? To answer this we first need to remember what they are!

Review

Remember that for us, a circuit made of resistors is a mathematical structure like this:

It’s a cospan where:

\Gamma is a graph labelled by resistances. So, it consists of a finite set N of nodes, a finite set E of edges, two functions

s, t : E \to N

sending each edge to its source and target nodes, and a function

r : E \to (0,\infty)

that labels each edge with its resistance.

i: I \to \Gamma is a map of graphs labelled by resistances, where I has no edges. A labelled graph with no edges has nothing but nodes! So, the map i is just a trick for specifying a finite set of nodes called inputs and mapping them to N. Thus i picks out some nodes of \Gamma and declares them to be inputs. (However, i may not be one-to-one! We’ll take advantage of that subtlety later.)

o: O \to \Gamma is another map of graphs labelled by resistances, where O again has no edges, and we call its nodes outputs.

The principle of minimum power

So what does a circuit made of resistors do? This is described by the principle of minimum power.

Recall from Part 27 that when we put it to work, our circuit has a current I_e flowing along each edge e \in E. This is described by a function

I: E \to \mathbb{R}

It also has a voltage across each edge. The word ‘across’ is standard here, but don’t worry about it too much; what matters is that we have another function

V: E \to \mathbb{R}

describing the voltage V_e across each edge e.

Resistors heat up when current flows through them, so they eat up electrical power and turn this power into heat. How much? The power is given by

\displaystyle{ P = \sum_{e \in E} I_e V_e }

So far, so good. But what does it mean to minimize power?

To understand this, we need to manipulate the formula for power using the laws of electrical circuits described in Part 27. First, Ohm’s law says that for linear resistors, the current is proportional to the voltage. More precisely, for each edge e \in E,

\displaystyle{ I_e = \frac{V_e}{r_e} }

where r_e is the resistance of that edge. So, the bigger the resistance, the less current flows: that makes sense. Using Ohm’s law we get

\displaystyle{ P = \sum_{e \in E} \frac{V_e^2}{r_e} }

Now we see that power is always nonnegative! Now it makes more sense to minimize it. Of course we could minimize it simply by setting all the voltages equal to zero. That would work, but that would be boring: it gives a circuit with no current flowing through it. The fun starts when we minimize power subject to some constraints.

For this we need to remember another law of electrical circuits: a spinoff of Kirchhoff’s voltage law. This says that we can find a function called the potential

\phi: N \to \mathbb{R}

such that

V_e = \phi_{s(e)} - \phi_{t(e)}

for each e \in E. In other words, the voltage across each edge is the difference of potentials at the two ends of this edge.

Using this, we can rewrite the power as

\displaystyle{ P = \sum_{e \in E} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)})^2 }

Now we’re really ready to minimize power! Our circuit made of resistors has certain nodes called terminals:

T \subseteq N

These are the nodes that are either inputs or outputs. More precisely, they’re the nodes in the image of

i: I \to \Gamma

or

o: O \to \Gamma

The principle of minimum power says that:

If we fix the potential \phi on all terminals, the potential at other nodes will minimize the power

\displaystyle{ P(\phi) = \sum_{e \in E} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)})^2 }

subject to this constraint.

This should remind you of all the other minimum or maximum principles you know, like the principle of least action, or the way a system in thermodynamic equilibrium maximizes its entropy. All these principles—or at least, most of them—are connected. I could talk about this endlessly. But not now!

Now let’s just use the principle of minimum power. Let’s see what it tells us about the behavior of an electrical circuit.

Let’s imagine changing the potential \phi by adding some multiple of a function

\psi: N \to \mathbb{R}

If this other function vanishes at the terminals:

\forall n \in T \; \; \psi(n) = 0

then \phi + x \psi doesn’t change at the terminals as we change the number x.

Now suppose \phi obeys the principle of minimum power. In other words, supposes it minimizes power subject to the constraint of taking the values it does at the terminals. Then we must have

\displaystyle{ \frac{d}{d x} P(\phi + x \psi)\Big|_{x = 0} }

whenever

\forall n \in T \; \; \psi(n) = 0

This is just the first derivative test for a minimum. But the converse is true, too! The reason is that our power function is a sum of nonnegative quadratic terms. Its graph will look like a paraboloid. So, the power has no points where its derivative vanishes except minima, even when we constrain \phi by making it lie on a linear subspace.

We can go ahead and start working out the derivative:

\displaystyle{ \frac{d}{d x} P(\phi + x \psi)! = ! \frac{d}{d x} \sum_{e \in E} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)} + x(\psi_{s(e)} -\psi_{t(e)}))^2  }

To work out the derivative of these quadratic terms at x = 0, we only need to keep the part that’s proportional to x. The rest gives zero. So:

\begin{array}{ccl} \displaystyle{ \frac{d}{d t} P(\phi + x \psi)\Big|_{x = 0} } &=& \displaystyle{ \frac{d}{d x} \sum_{e \in E} \frac{x}{r_e} (\phi_{s(e)} - \phi_{t(e)}) (\psi_{s(e)} - \psi_{t(e)}) \Big|_{x = 0} } \\ \\  &=&   \displaystyle{  \sum_{e \in E} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) (\psi_{s(e)} - \psi_{t(e)}) }  \end{array}

The principle of minimum power says this is zero whenever \psi : N \to \mathbb{R} is a function that vanishes at terminals. By linearity, it’s enough to consider functions \psi that are zero at every node except one node n that is not a terminal. By linearity we can also assume \psi(n) = 1.

Given this, the only nonzero terms in the sum

\displaystyle{ \sum_{e \in E} \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) (\psi_{s(e)} - \psi_{t(e)}) }

will be those involving edges whose source or target is n. We get

\begin{array}{ccc} \displaystyle{ \frac{d}{d x} P(\phi + x \psi)\Big|_{x = 0} } &=& \displaystyle{ \sum_{e: \; s(e) = n}  \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)})}  \\  \\        && -\displaystyle{ \sum_{e: \; t(e) = n}  \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) }   \end{array}

So, the principle of minimum power says precisely

\displaystyle{ \sum_{e: \; s(e) = n}  \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) = \sum_{e: \; t(e) = n}  \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) }

for all nodes n that aren’t terminals.

What does this mean? You could just say it’s a set of linear equations that must be obeyed by the potential \phi. So, the principle of minimum power says that fixing the potential at terminals, the potential at other nodes must be chosen in a way that obeys a set of linear equations.

But what do these equations mean? They have a nice meaning. Remember, Kirchhoff’s voltage law says

V_e = \phi_{s(e)} - \phi_{t(e)}

and Ohm’s law says

\displaystyle{ I_e = \frac{V_e}{r_e} }

Putting these together,

\displaystyle{ I_e = \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) }

so the principle of minimum power merely says that

\displaystyle{ \sum_{e: \; s(e) = n} I_e = \sum_{e: \; t(e) = n}  I_e }

for any node n that is not a terminal.

This is Kirchhoff’s current law: for any node except a terminal, the total current flowing into that node must equal the total current flowing out! That makes a lot of sense. We allow current to flow in or out of our circuit at terminals, but ‘inside’ the circuit charge is conserved, so if current flows into some other node, an equal amount has to flow out.

In short: the principle of minimum power implies Kirchoff’s current law! Conversely, we can run the whole argument backward and derive the principle of minimum power from Kirchhoff’s current law. (In both the forwards and backwards versions of this argument, we use Kirchhoff’s voltage law and Ohm’s law.)

When the node n is a terminal, the quantity

\displaystyle{  \sum_{e: \; s(e) = n} I_e \; - \; \sum_{e: \; t(e) = n}  I_e }

need not be zero. But it has an important meaning: it’s the amount of current flowing into that terminal!

We’ll call this I_n, the current at the terminal n \in T. This is something we can measure even when our circuit has a black box around it:

So is the potential \phi_n at the terminal n. It’s these currents and potentials at terminals that matter when we try to describe the behavior of a circuit while ignoring its inner workings.

Black boxing

Now let me quickly sketch how black boxing becomes a functor.

A circuit made of resistors gives a linear relation between the potentials and currents at terminals. A relation is something that can hold or fail to hold. A ‘linear’ relation is one defined using linear equations.

A bit more precisely, suppose we choose potentials and currents at the terminals:

\psi : T \to \mathbb{R}

J : T \to \mathbb{R}

Then we seek potentials and currents at all the nodes and edges of our circuit:

\phi: N \to \mathbb{R}

I : E \to \mathbb{R}

that are compatible with our choice of \psi and J. Here compatible means that

\psi_n = \phi_n

and

J_n = \displaystyle{  \sum_{e: \; s(e) = n} I_e \; - \; \sum_{e: \; t(e) = n}  I_e }

whenever n \in T, but also

\displaystyle{ I_e = \frac{1}{r_e} (\phi_{s(e)} - \phi_{t(e)}) }

for every e \in E, and

\displaystyle{  \sum_{e: \; s(e) = n} I_e \; = \; \sum_{e: \; t(e) = n}  I_e }

whenever n \in N - T. (The last two equations combine Kirchoff’s laws and Ohm’s law.)

There either exist I and \phi making all these equations true, in which case we say our potentials and currents at the terminals obey the relation… or they don’t exist, in which case we say the potentials and currents at the terminals don’t obey the relation.

The relation is clearly linear, since it’s defined by a bunch of linear equations. With a little work, we can make it into a linear relation between potentials and currents in

\mathbb{R}^I \oplus \mathbb{R}^I

and potentials and currents in

\mathbb{R}^O \oplus \mathbb{R}^O

Remember, I is our set of inputs and O is our set of outputs.

In fact, this process of getting a linear relation from a circuit made of resistors defines a functor:

\blacksquare : \mathrm{ResCirc} \to \mathrm{LinRel}

Here \mathrm{ResCirc} is the category where morphisms are circuits made of resistors, while \mathrm{LinRel} is the category where morphisms are linear relations.

More precisely, here is the category \mathrm{ResCirc}:

• an object of \mathrm{ResCirc} is a finite set;

• a morphism from I to O is an isomorphism class of circuits made of resistors:

having I as its set of inputs and O as its set of outputs;

• we compose morphisms in \mathrm{ResCirc} by composing isomorphism classes of cospans.

(Remember, circuits made of resistors are cospans. This lets us talk about isomorphisms between them. If you forget the how isomorphism between cospans work, you can review it in Part 31.)

And here is the category \mathrm{LinRel}:

• an object of \mathrm{LinRel} is a finite-dimensional real vector space;

• a morphism from U to V is a linear relation R \subseteq U \times V, meaning a linear subspace of the vector space U \times V;

• we compose a linear relation R \subseteq U \times V and a linear relation S \subseteq V \times W in the usual way we compose relations, getting:

SR = \{(u,w) \in U \times W : \; \exists v \in V \; (u,v) \in R \mathrm{\; and \;} (v,w) \in S \}

Next steps

So far I’ve set up most of the necessary background but not precisely defined the black boxing functor

\blacksquare : \mathrm{ResCirc} \to \mathrm{LinRel}

There are some nuances I’ve glossed over, like the difference between inputs and outputs as elements of I and O and their images in N. If you want to see the precise definition and the proof that it’s a functor, read our paper:

• John Baez and Brendan Fong, A compositional framework for passive linear networks.

The proof is fairly long: there may be a much quicker one, but at least this one has the virtue of introducing a lot of nice ideas that will be useful elsewhere.

Next time I’ll define the black box functor more carefully.


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