Azimuth

Is Life Improbable?

31 May, 2011

Mine? Yes. And maybe you’ve wondered just how improbable your life is. But that’s not really the question today…

Here at the Centre for Quantum Technologies, Dagomir Kaszlikowski asked me to give a talk on this paper:

• John Baez, Is life improbable?, Foundations of Physics 19 (1989), 91-95.

This was the second paper I wrote, right after my undergraduate thesis. Nobody ever seemed to care about it, so it’s strange—but nice—to finally be giving a talk on it.

My paper does not try to settle the question its title asks. Rather, it tries to refute the argument here:

• Eugene P. Wigner, The probability of the existence of a self-reproducing unit, Symmetries and Reflections, Indiana University Press, Bloomington, 1967, pp. 200-208.

According Wigner, his argument

purports to show that, according to standard quantum mechanical theory, the probability is zero for the existence of self-reproducing states, i.e., organisms.

Given how famous Eugene Wigner is (he won a Nobel prize, after all) and how earth-shattering his result would be if true, it’s surprising how little criticism his paper has received. David Bohm mentioned it approvingly in 1969. In 1974 Hubert Yockey cited it saying

for all physics has to offer, life should never have appeared and if it ever did it would soon die out.

As you’d expect, there are some websites mentioning Wigner’s argument as evidence that some supernatural phenomenon is required to keep life going. Wigner himself believed it was impossible to formulate quantum theory in a fully consistent way without referring to consciousness. Since I don’t believe either of these claims, I think it’s good to understand the flaw in Wigner’s argument.

So, let me start by explaining his argument. Very roughly, it purports to show that if there are many more ways a chunk of matter can be ‘dead’ than ‘living’, the chance is zero that we can choose some definition of ‘living’ and a suitable ‘nutrient’ state such that every ‘living’ chunk of matter can interact with this ‘nutrient’ state to produce two ‘living’ chunks.

In making this precise, Wigner considers more than just two chunks of matter: he also allows there to be an ‘environment’. So, he considers a quantum system made of three parts, and described by a Hilbert space

$H = H_1 \otimes H_1 \otimes H_2$

Here the first $H_1$ corresponds to a chunk of matter. The second $H_1$ corresponds to another chunk of matter. The space $H_3$ corresponds to the ‘environment’. Suppose we wait for a certain amount of time and see what the system does; this will be described by some unitary operator

$S: H \to H$

Wigner asks: if we pick this operator $S$ in a random way, what’s the probability that there’s some $n$ -dimensional subspace of ‘living organism’ states in $H_1$ , and some ‘nutrient plus environment’ state in $H_1 \otimes H_2$ , such that the time evolution sends any living organism together with the nutrient plus environment to two living organisms and some state of the environment?

A bit more precisely: suppose we pick $S$ in a random way. Then what’s the probability that there exists an $n$ -dimensional subspace

$V \subseteq H_1$

and a state

$w \in H_1 \otimes H_2$

such that $S$ maps every vector in $V \otimes \langle w \rangle$ to a vector in $V \otimes V \otimes H_2$ ? Here $\langle w \rangle$ means the 1-dimensional subspace spanned by the vector $w$ .

And his answer is: if

$\mathrm{dim}(H_1) \gg n$

then this probability is zero.

You may need to reread the last few paragraphs a couple times to understand Wigner’s question, and his answer. In case you’re still confused, I should say that $V \subseteq H_1$ is what I’m calling the space of ‘living organism’ states of our chunk of matter, while $w \in H_1 \otimes H_2$ is the ‘nutrient plus environment’ state.

Now, Wigner did not give a rigorous proof of his claim, nor did he say exactly what he meant by ‘probability’: he didn’t specify a probability measure on the space of unitary operators on $H$ . But if we use the obvious choice (called ‘normalized Haar measure’) his argument can most likely be turned into a proof.

So, I don’t want to argue with his math. I want to argue with his interpretation of the math. He concludes that

the chances are nil for the existence of a set of ‘living’ states for which one can find a nutrient of such nature that interaction always leads to multiplication.

The problem is that he fixed the decomposition of the Hilbert space $H$ as a tensor product

$H = H_1 \otimes H_1 \otimes H_2$

before choosing the time evolution operator $S$ . There is no good reason to do that. It only makes sense split up a physical into parts this way after we have some idea of what the dynamics is. An abstract Hilbert space doesn’t come with a favored decomposition as a tensor product into three parts!

If we let ourselves pick this decomposition after picking the operator $S$ , the story changes completely. My paper shows:

Theorem 1. Let $H$ , $H_1$ and $H_2$ be finite-dimensional Hilbert spaces with $H \cong H_1 \otimes H_1 \otimes H_2$ . Suppose $S : H \to H$ is any unitary operator, suppose $V$ is any subspace of $H_1$ , and suppose $w$ is any unit vector in $H_1 \otimes H_2$ Then there is a unitary isomorphism

$U: H \to H_1 \otimes H_1 \otimes H_2$

such that if we identify $H$ with $H_1 \otimes H_1 \otimes H_2$ using $U$ , the operator $S$ maps $V \otimes \langle w \rangle$ into $V \otimes V \otimes H_2$ .

In other words, if we allow ourselves to pick the decomposition after picking $S$ , we can always find a ‘living organism’ subspace of any dimension we like, together with a ‘nutrient plus environment’ state that allows our living organism to reproduce.

However, if you look at the proof in my paper, you’ll see it’s based on a kind of cheap trick (as I forthrightly admit). Namely, I pick the ‘nutrient plus environment’ state to lie in $V \otimes H_2$ , so the nutrient actually consists of another organism!

This goes to show that you have to be very careful about theorems like this. To prove that life is improbable, you need to find some necessary conditions for what counts as life, and show that these are improbable (in some sense, and of course it matters a lot what that sense is). Refuting such an argument does not prove that life is probable: for that you need some sufficient conditions for what counts as life. And either way, if you prove a theorem using a ‘cheap trick’, it probably hasn’t gotten to grips with the real issues.

I also show that as the dimension of $H$ approaches infinity, the probability approaches 1 that we can get reproduction with a 1-dimensional ‘living organism’ subspace and a ‘nutrient plus environment’ state that lies in orthogonal complement of $V \otimes H_2$ . In other words, the ‘nutrient’ is not just another organism sitting there all ready to go!

More precisely:

Theorem 2. Let $H$ , $H_1$ and $H_2$ be finite-dimensional Hilbert spaces with $\mathrm{dim}(H) = \mathrm{dim}(H_1)^2 \cdot \mathrm{dim}(H_2)$ . Let $\mathbf{S'}$ be the set of unitary operators $S: H \to H$ with the following property: there’s a unit vector $v \in H_1$ , a unit vector $w \in V^\perp \otimes H_2$ , and a unitary isomorphism

$U: H \to H_1 \otimes H_1 \otimes H_2$

such that if we identify $H$ with $H_1 \otimes H_1 \otimes H_2$ using $U$ , the operator $S$ maps $v \otimes w$ into $\langle v\rangle \otimes \langle v \rangle \otimes H_2$ . Then the normalized Haar measure of $\mathbf{S'}$ approaches 1 as $\mathrm{dim}(H) \to \infty$ .

Here $V^\perp$ is the orthogonal complement of $V \subseteq H_1$ ; that is, the space of all vectors perpendicular to $V$ .

I won’t include the proofs of these theorems, since you can see them in my paper.

Just to be clear: I certainly don’t think these theorems prove that life is probable! You can’t have theorems without definitions, and I think that coming up with a good general definition of ‘life’, or even supposedly simpler concepts like ‘entity’ and ‘reproduction’, is extremely tough. The formalism discussed here is oversimplified for dozens of reasons, a few of which are listed at the end of my paper. So far we’re only in the first fumbling stages of addressing some very hard questions.

All my theorems do is point out that Wigner’s argument has a major flaw: he’s choosing a way to divide the world into chunks of matter and the environment before choosing his laws of physics. This doesn’t make much sense, and reversing the order dramatically changes the conclusions.

By the way: I just started looking for post-1989 discussions of Wigner’s paper. So far I haven’t found any interesting ones. Here’s a more recent paper that’s somewhat related, which doesn’t mention Wigner’s work:

• Indranil Chakrabarty and Prashant, Non existence of quantum mechanical self replicating machine, 2005.

The considerations here seem more closely related to the Wooters–Zurek no-cloning theorem.

8 Comments | biology, physics, quantum technologies | Permalink
Posted by John Baez

The One Best Thing Everyone Could Do to Slow Climate Change

27 May, 2011

There’s a website called Quora where people can ask and answer questions of all sorts. Lots of people use it, so Curtis Faith suggested that we—that is, everyone here reading this blog—try answering some of the questions there. That sounded like a nice idea, so now there’s a ‘topic’ on Quora called Azimuth Project. The questions we tackle will be listed there, so people can easily find them.

To get the ball rolling, Curtis posted this question:

What is the one best thing everyone could do to slow down climate change?

If you’re like me, the first thing you’ll want to do is question the question. Are we really looking for the one best thing everyone could do? Everyone in the world, including the billion poorest people?

In that case, many answers that leap to mind are no good. We can’t say “take fewer airplane trips” because most of those people don’t take airplane trips to begin with. We can’t say “drive less” because most of those people don’t have cars. And so on. It’s no fair! We need an easier question!

Well… let’s not try to second-guess the question. It’s actually fun to take it seriously and try to answer it. It’ll force us to think about the world as a whole, instead of the sins of our rich neighbors.

Here are 50 tips for how to fight global warming from Global Warming Facts. Could any of these be the right answer? How many of these are things that everyone on this Earth can do?

Replace a regular incandescent light bulb with a compact fluorescent light bulb (cfl)
CFLs use 60% less energy than a regular bulb. This simple switch will save about 300 pounds of carbon dioxide a year.
We recommend you purchase your CFL bulbs at 1000bulbs.com, they have great deals on both screw-in and plug-in light bulbs.
Install a programmable thermostat
Programmable thermostats will automatically lower the heat or air conditioning at night and raise them again in the morning. They can save you $100 a year on your energy bill.
Move your thermostat down 2° in winter and up 2° in summer
Almost half of the energy we use in our homes goes to heating and cooling. You could save about 2,000 pounds of carbon dioxide a year with this simple adjustment.
Clean or replace filters on your furnace and air conditioner
Cleaning a dirty air filter can save 350 pounds of carbon dioxide a year.
Choose energy efficient appliances when making new purchases
Look for the Energy Star label on new appliances to choose the most energy efficient products
available.
Do not leave appliances on standby
Use the “on/off” function on the machine itself. A TV set that’s switched on for 3 hours a day (the average time Europeans spend watching TV) and in standby mode during the remaining 21 hours uses about 40% of its energy in standby mode.
Wrap your water heater in an insulation blanket
You’ll save 1,000 pounds of carbon dioxide a year with this simple action. You can save another 550 pounds per year by setting the thermostat no higher than 50°C.
Move your fridge and freezer
Placing them next to the cooker or boiler consumes much more energy than if they were standing on their own. For example, if you put them in a hot cellar room where the room temperature is 30-35ºC, energy use is almost double and causes an extra 160 kg of CO₂ emissions for fridges per year and 320 kg for freezers.
Defrost old fridges and freezers regularly
Even better is to replace them with newer models, which all have automatic defrost cycles and are generally up to two times more energy-efficient than their predecessors.
Don’t let heat escape from your house over a long period
When airing your house, open the windows for only a few minutes. If you leave a small opening all day long, the energy needed to keep it warm inside during six cold months (10ºC or less outside temperature) would result in almost 1 ton of CO₂ emissions.
Replace your old single-glazed windows with double-glazing
This requires a bit of upfront investment, but will halve the energy lost through windows and pay off in the long term. If you go for the best the market has to offer (wooden-framed double-glazed units with low-emission glass and filled with argon gas), you can even save more than 70% of the energy lost.
Get a home energy audit
Many utilities offer free home energy audits to find where your home is poorly insulated or energy inefficient. You can save up to 30% off your energy bill and 1,000 pounds of carbon dioxide a year. Energy Star can help you find an energy specialist.
Cover your pots while cooking
Doing so can save a lot of the energy needed for preparing the dish. Even better are pressure cookers and steamers: they can save around 70%!
Use the washing machine or dishwasher only when they are full
If you need to use it when it is half full, then use the half-load or economy setting. There is also no need to set the temperatures high. Nowadays detergents are so efficient that they get your clothes and dishes clean at low temperatures.
Take a shower instead of a bath
A shower takes up to four times less energy than a bath. To maximize the energy saving, avoid power showers and use low-flow showerheads, which are cheap and provide the same comfort.
Use less hot water
It takes a lot of energy to heat water. You can use less hot water by installing a low flow showerhead (350 pounds of carbon dioxide saved per year) and washing your clothes in cold or warm water (500 pounds saved per year) instead of hot.
Use a clothesline instead of a dryer whenever possible
You can save 700 pounds of carbon dioxide when you air dry your clothes for 6 months out of the year.
Insulate and weatherize your home
Properly insulating your walls and ceilings can save 25% of your home heating bill and 2,000 pounds of carbon dioxide a year. Caulking and weather-stripping can save another 1,700 pounds per year. Energy Efficient has more information on how to better insulate your home.
Be sure you’re recycling at home
You can save 2,400 pounds of carbon dioxide a year by recycling half of the waste your household generates.
Recycle your organic waste
Around 3% of the greenhouse gas emissions through the methane is released by decomposing bio-degradable waste. By recycling organic waste or composting it if you have a garden, you can help eliminate this problem! Just make sure that you compost it properly, so it decomposes with sufficient oxygen, otherwise your compost will cause methane emissions and smell foul.
Buy intelligently
One bottle of 1.5l requires less energy and produces less waste than three bottles of 0.5l. As well, buy recycled paper products: it takes less 70 to 90% less energy to make recycled paper and it prevents the loss of forests worldwide.
Choose products that come with little packaging and buy refills when you can
You will also cut down on waste production and energy use… another help against global warming.
Reuse your shopping bag
When shopping, it saves energy and waste to use a reusable bag instead of accepting a disposable one in each shop. Waste not only discharges CO2 and methane into the atmosphere, it can also pollute the air, groundwater and soil.
Reduce waste
Most products we buy cause greenhouse gas emissions in one or another way, e.g. during production and distribution. By taking your lunch in a reusable lunch box instead of a disposable one, you save the energy needed to produce new lunch boxes.
Plant a tree
A single tree will absorb one ton of carbon dioxide over its lifetime. Shade provided by trees can also reduce your air conditioning bill by 10 to 15%. The Arbor Day Foundation has information on planting and provides trees you can plant with membership.
Switch to green power
In many areas, you can switch to energy generated by clean, renewable sources such as wind and solar. In some of these, you can even get refunds by government if you choose to switch to a clean energy producer, and you can also earn money by selling the energy you produce and don’t use for yourself.
Buy locally grown and produced foods
The average meal in the United States travels 1,200 miles from the farm to your plate. Buying locally will save fuel and keep money in your community.
Buy fresh foods instead of frozen
Frozen food uses 10 times more energy to produce.
Seek out and support local farmers markets
They reduce the amount of energy required to grow and transport the food to you by one fifth. Seek farmer’s markets in your area, and go for them.
Buy organic foods as much as possible
Organic soils capture and store carbon dioxide at much higher levels than soils from conventional farms. If we grew all of our corn and soybeans organically, we’d remove 580 billion pounds of carbon dioxide from the atmosphere!
Eat less meat
Methane is the second most significant greenhouse gas and cows are one of the greatest methane emitters. Their grassy diet and multiple stomachs cause them to produce methane, which they exhale with every breath.
Reduce the number of miles you drive by walking, biking, carpooling or taking mass transit wherever possible
Avoiding just 10 miles of driving every week would eliminate about 500 pounds of carbon dioxide emissions a year! Look for transit options in your area.
Start a carpool with your coworkers or classmates
Sharing a ride with someone just 2 days a week will reduce your carbon dioxide emissions by 1,590 pounds a year. eRideShare.com runs a free service connecting North American commuters and travelers.
Don’t leave an empty roof rack on your car
This can increase fuel consumption and CO₂ emissions by up to 10% due to wind resistance and the extra weight – removing it is a better idea.
Keep your car tuned up
Regular maintenance helps improve fuel efficiency and reduces emissions. When just 1% of car owners properly maintain their cars, nearly a billion pounds of carbon dioxide are kept out of the atmosphere.
Drive carefully and do not waste fuel
You can reduce CO₂ emissions by readjusting your driving style. Choose proper gears, do not abuse the gas pedal, use the engine brake instead of the pedal brake when possible and turn off your engine when your vehicle is motionless for more than one minute. By readjusting your driving style you can save money on both fuel and car maintenance.
Check your tires weekly to make sure they’re properly inflated
Proper tire inflation can improve gas mileage by more than 3%. Since every gallon of gasoline saved keeps 20 pounds of carbon dioxide out of the atmosphere, every increase in fuel efficiency makes a difference!
When it is time for a new car, choose a more fuel efficient vehicle
You can save 3,000 pounds of carbon dioxide every year if your new car gets only 3 miles per gallon more than your current one. You can get up to 60 miles per gallon with a hybrid! You can find information on fuel efficiency on FuelEconomy and on GreenCars websites.
Try car sharing
Need a car but don’t want to buy one? Community car sharing organizations provide access to a car and your membership fee covers gas, maintenance and insurance. Many companies – such as Flexcar – offer low emission or hybrid cars too! Also, see ZipCar.
Try telecommuting from home
Telecommuting can help you drastically reduce the number of miles you drive every week. For more information, check out the Telework Coalition.
Fly less
Air travel produces large amounts of emissions so reducing how much you fly by even one or two trips a year can reduce your emissions significantly. You can also offset your air travel carbon emissions by investing in renewable energy projects.
Encourage your school or business to reduce emissions
You can extend your positive influence on global warming well beyond your home by actively encouraging other to take action.
Join the virtual march
The Stop Global Warming Virtual March is a non-political effort to bring people concerned about global warming together in one place. Add your voice to the hundreds of
thousands of other people urging action on this issue.
Encourage the switch to renewable energy
Successfully combating global warming requires a national transition to renewable energy sources such as solar, wind and biomass. These technologies are ready to be deployed more widely but there are regulatory barriers impeding them. U.S. citizens, take action to break down those barriers with Vote Solar.
Protect and conserve forest worldwide
Forests play a critical role in global warming: they store carbon. When forests are burned or cut down, their stored carbon is release into the atmosphere – deforestation now accounts for about 20% of carbon dioxide emissions each year. Conservation International has more information on saving forests from global warming.
Consider the impact of your investments
If you invest your money, you should consider the impact that your investments and savings will have on global warming. Check out SocialInvest and Ceres to can learn more about how to ensure your money is being invested in companies, products and projects that address issues related to climate change.
Make your city cool
Cities and states around the country have taken action to stop global warming by passing innovative transportation and energy saving legislation. If you’re in the U.S., join the cool cities list.
Tell Congress to act
The McCain Lieberman Climate Stewardship and Innovation Act would set a firm limit on carbon dioxide emissions and then use free market incentives to lower costs, promote efficiency and spur innovation. Tell your representative to support it.
Make sure your voice is heard!
Americans must have a stronger commitment from their government in order to stop global warming and implement solutions and such a commitment won’t come without a dramatic increase in citizen lobbying for new laws with teeth. Get the facts about U.S. politicians and candidates at Project Vote Smart and The League of Conservation Voters. Make sure your voice is heard by voting!
Share this list!
Spread this list worldwide and help people doing their part: the more people you will manage to enlighten, the greater YOUR help to save the planet will be (but please take action on first person too)! If you like, you are free to republish, adapt or translate the list and post it in your blog, website or forum as long as you give us credit with a link to the original source.

There are a lot of great ideas here, but if we look for those that everyone can do, there aren’t many.

What’s the most important item that was left off this list?

I’ll give my answer to Curtis’ question after I’ve heard some of yours. It’s a tough question but I have an idea. And no, it’s not “join the Azimuth Project”.

143 Comments | azimuth, climate, questions | Permalink
Posted by John Baez

Information Geometry (Part 8)

26 May, 2011

Now this series on information geometry will take an unexpected turn toward ‘green mathematics’. Lately I’ve been talking about relative entropy. Now I’ll say how this concept shows up in the study of evolution!

That’s an unexpected turn to me, at least. I learned of this connection just two days ago in a conversation with Marc Harper, a mathematician who is a postdoc in bioinformatics at UCLA, working with my friend Chris Lee. I was visiting Chris for a couple of days after attending the thesis defenses of some grad students of mine who just finished up at U.C. Riverside. Marc came by and told me about this paper:

• Marc Harper, Information geometry and evolutionary game theory.

and now I can’t resist telling you.

First of all: what does information theory have to do with biology? Let me start with a very general answer: biology is different from physics because biological systems are packed with information you can’t afford to ignore.

Physicists love to think about systems that take only a little information to describe. So when they get a system that takes a lot of information to describe, they use a trick called ‘statistical mechanics’, where you try to ignore most of this information and focus on a few especially important variables. For example, if you hand a physicist a box of gas, they’ll try to avoid thinking about the state of each atom, and instead focus on a few macroscopic quantities like the volume and total energy. Ironically, the mathematical concept of information arose first here—although they didn’t call it information back then; they called it ‘entropy’. The entropy of a box of gas is precisely the amount of information you’ve decided to forget when you play this trick of focusing on the macroscopic variables. Amazingly, remembering just this—the sheer amount of information you’ve forgotten—can be extremely useful… at least for the systems physicists like best.

But biological systems are different. They store lots of information (for example in DNA), transmit lots of information (for example in the form of biochemical signals), and collect a lot of information from their environment. And this information isn’t uninteresting ‘noise’, like the positions of atoms in a gas. The details really matter. Thus, we need to keep track of lots of information to have a chance of understanding any particular biological system.

So, part of doing biology is developing new ways to think about physical systems that contain lots of relevant information. This is why physicists consider biology ‘messy’. It’s also why biology and computers go hand in hand in the subject called ‘bioinformatics’. There’s no avoiding this: in fact, it will probably force us to automate the scientific method! That’s what Chris Lee and Marc Harper are really working on:

• Chris Lee, General information metrics for automated experiment planning, presentation in the UCLA Chemistry & Biochemistry Department faculty luncheon series, 2 May 2011.

But more about that some other day. Let me instead give another answer to the question of what information theory has to do with biology.

There’s an analogy between evolution and the scientific method. Simply put, life is an experiment to see what works; natural selection weeds out the bad guesses, and over time the better guesses predominate. This process transfers information from the world to the ‘experimenter’: the species that’s doing the evolving, or the scientist. Indeed, the only way the experimenter can get information is by making guesses that can be wrong.

All this is simple enough, but the nice thing is that we can make it more precise.

On the one hand, there’s a simple model of the scientific method called ‘Bayesian inference’. Assume there’s a set of mutually exclusive alternatives: possible ways the world can be. And suppose we start with a ‘prior probability distribution’: a preconceived notion of how probable each alternative is. Say we do an experiment and get a result that depends on which alternative is true. We can work out how likely this result was given our prior, and—using a marvelously simple formula called Bayes’ rule—we can use this to update our prior and obtain a new improved probability distribution, called the ‘posterior probability distribution’.

On the other hand, suppose we have a species with several different possible genotypes. A population of this species will start with some number of organisms with each genotype. So, we get a probability distribution saying how likely it is that an organism has any given genotype. These genotypes are our ‘mutually exclusive alternatives’, and this probability distribution is our ‘prior’. Suppose each generation the organisms have some expected number of offspring that depends on their genotype. Mathematically, it turns out this is just like updating our prior using Bayes’ rule! The result is a new probability distribution of genotypes: the ‘posterior’.

I learned about this from Chris Lee on the 19th of December, 2006. In my diary that day, I wrote:

The analogy is mathematically precise, and fascinating. In rough terms, it says that the process of natural selection resembles the process of Bayesian inference. A population of organisms can be thought of as having various ‘hypotheses’ about how to survive—each hypothesis corresponding to a different allele. (Roughly, an allele is one of several alternative versions of a gene.) In each successive generation, the process of natural selection modifies the proportion of organisms having each hypothesis, according to Bayes’ rule!

Now let’s be more precise:

Bayes’ rule says if we start with a ‘prior probability’ for some hypothesis to be true, divide it by the probability that some observation is made, then multiply by the ‘conditional probability’ that this observation will be made given that the hypothesis is true, we’ll get the ‘posterior probability’ that the hypothesis is true given that the observation is made.

Formally, the exact same equation shows up in population genetics! In fact, Chris showed it to me—it’s equation 9.2 on page 30 of this
book:

• R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation, section I.9: Selection at a single locus, Wiley, 2000.

But, now all the terms in the equation have different meanings!

Now, instead of a ‘prior probability’ for a hypothesis to be true, we have the frequency of occurrence of some allele in some generation of a population. Instead of the probability that we make some observation, we have the expected number of offspring of an organism. Instead of the ‘conditional probability’ of making the observation, we have the expected number of offspring of an organism given that it has this allele. And, instead of the ‘posterior probability’ of our hypothesis, we have the frequency of occurrence of that allele in the next generation.

(Here we are assuming, for simplicity, an asexually reproducing ‘haploid’ population – that is, one with just a single set of chromosomes.)

This is a great idea—Chris felt sure someone must have already had it. A natural context would be research on genetic programming, a machine learning technique that uses an evolutionary algorithm to optimize a population of computer programs according to a fitness landscape determined by their ability to perform a given task. Since there has also been a lot of work on Bayesian approaches to machine learning, surely someone has noticed their mathematical relationship?

I see at least one person found these ideas as new and exciting as I did. But I still can’t believe Chris was the first to clearly formulate them, so I’d still like to know who did.

Marc Harper actually went to work with Chris after reading that diary entry of mine. By now he’s gone a lot further with this analogy by focusing on the role of information. As we keep updating our prior using Bayes’ rule, we should be gaining information about the real world. This idea has been made very precise in the theory of ‘machine learning’. Similarly, as a population evolves through natural selection, it should be gaining information about its environment.

I’ve been talking about Bayesian updating as a discrete-time process: something that happens once each generation for our population. That’s fine and dandy, definitely worth studying, but Marc’s paper focuses on a continuous-time version called the ‘replicator equation’. It goes like this. Let $X$ be the set of alternative genotypes. For each $i \in X$ , let $P_i$ be the number of organisms that have the $i$ th genotype at time $t$ . Say that

$\displaystyle{ \frac{d P_i}{d t} = f_i P_i }$

where $f_i$ is the fitness of the $i$ th genotype. Let $p_i$ be the probability that at time $t$ , a randomly chosen organism will have the $i$ th genotype:

$\displaystyle{ p_i = \frac{P_i}{\sum_{i \in X} P_i } }$

Then a little calculus gives the replicator equation:

$\displaystyle{\frac{d p_i}{d t} = \left( f_i - \langle f \rangle \right) \, p_i }$

where

$\langle f \rangle = \sum_{i \in X} f_i p_i$

is the mean fitness of the organisms. So, the fraction of organisms of the $i$ th type grows at a rate proportional to the fitness of that type minus the mean fitness. It ain’t enough to be good: you gotta be better than average.

Note that all this works not just when each fitness $f_i$ is a mere number, but also when it’s a function of the whole list of probabilities $p_i$ . That’s good, because in the real world, the fitness of one kind of bug may depend on the fraction of bugs of various kinds.

But what does all this have to do with information?

Marc’s paper has a lot to say about this! But just to give you a taste, here’s a simple fact involving relative entropy, which was first discovered by Ethan Atkin. Suppose evolution as described by the replicator equation brings the whole list of probabilities $p_i$ —let’s call this list $p$ —closer and closer to some stable equilibrium, say $q$ . Then if a couple of technical conditions hold, the entropy of $q$ relative to $p$ keeps decreasing, and approaches zero.

Remember what I told you about relative entropy. In Bayesian inference, the entropy $q$ relative to $p$ is how much information we gain if we start with $p$ as our prior and then do an experiment that pushes us to the posterior $q$ .

So, in simple rough terms: as it approaches a stable equilibrium, the amount of information a species has left to learn keeps dropping, and goes to zero!

I won’t fill in the precise details, because I bet you’re tired already. You can find them in Section 3.5, which is called “Kullback-Leibler Divergence is a Lyapunov function for the Replicator Dynamic”. If you know all the buzzwords here, you’ll be in buzzword heaven now. ‘Kullback-Leibler divergence’ is just another term for relative entropy. ‘Lyapunov function’ means that it keeps dropping and goes to zero. And the ‘replicator dynamic’ is the replicator equation I described above.

Perhaps next time I’ll say more about this stuff. For now, I just hope you see why it makes me so happy.

First, it uses information geometry to make precise the sense in which evolution is a process of acquiring information. That’s very cool. We’re looking at a simplified model—the replicator equation—but doubtless this is just the beginning of a very long story that keeps getting deeper as we move to less simplified models.

Second, if you read my summary of Chris Canning’s talks on evolutionary game theory, you’ll see everything I just said meshes nicely with that. He was taking the fitness $f_i$ to be

$f_i = \sum_{j \in X} A_{i j} p_j$

where the payoff matrix $A_{i j}$ describes the ‘winnings’ of an organism with the $i$ th genotype when it meets an organism with the $j$ th genotype. This gives a particularly nice special case of the replicator equation.

Third, this particularly nice special case happens to be the rate equation for a certain stochastic Petri net. So, we’ve succeeded in connecting the ‘diagram theory’ discussion to the ‘information geometry’ discussion! This has all sort of implications, which will take quite a while to explore.

As the saying goes, in mathematics:

Everything sufficiently beautiful is connected to all other beautiful things.

105 Comments | biology, information and entropy | Permalink
Posted by John Baez

Outsourcing Carbon Emissions

25 May, 2011

George Monbiot points out that Britain is accomplishing some of its reductions in carbon emissions by the simple expedient of outsourcing them to other countries:

• George Monbiot, Pass the Parcel, 23 May, 2011.

This gets around the spirit but not the letter of the Kyoto Protocol, since some these other countries, notably China, aren’t required to limit their carbon emissions! He writes:

It could have been worse. After the Treasury and the business department tried to scupper the UK’s long-term carbon targets, David Cameron stepped in to rescue them. The government has now promised to cut greenhouse gases by 50% by 2027, which means that, with a following wind, the UK could meet its legally-binding target of 80% by 2050. For this we should be grateful. But the coalition has resolved the tension between green and growth in a less than convincing fashion: by dumping responsibility for the environmental impacts on someone else.

The carbon cut we have made so far, and the carbon cut we are likely to make by 2027, have been achieved by means of a simple device: allowing other countries, principally China, to run polluting industries on our behalf.

Officially, the UK’s greenhouse gas emissions have fallen from 788 million tonnes in 1990 to 566mt in 2009. Unofficially, another 253 megatonnes should be added to our account. That’s the difference between the greenhouse gases released when manufacturing the goods we export and those released when manufacturing the goods we import. The reason why our official figures look better than those of most other nations is that so much of our manufacturing industry has moved overseas. It is this which allows the government to meet its targets. If the stuff we buy is made in China, China gets the blame.

This would be less of an issue if China were obliged to restrict its emissions. But under the only global treaty in force at the moment—the Kyoto Protocol—developing countries have no need to reduce their impacts. That suits the governments of both rich and poorer nations. Governments like ours can pretend that there is no conflict between green and growth. They avoid unpopular decisions, allowing people to consume whatever they fancy, and they keep business sweet by promising endless expansion. Governments like China’s can keep supplying us with the goods we couldn’t produce at home without breaking our obligations.

The “unofficial” calculation of 253 extra megatonnes of CO₂ comes from here:

• Steven J. Davis and Ken Caldeira, Consumption-based accounting of CO₂ emissions, Proceedings of the National Academy of Sciences, 8 March 2011.

This paper claims that in wealthy countries such as Switzerland, Sweden, Austria, the United Kingdom, and France, more than 30% of consumption-related CO₂ emissions were “imported”. In other words, a lot of their CO₂ emissions weren’t actually done in those country: they happened during the production and shipping of goods that got imported to those countries!

You can see a bit of what’s going on from this picture (click to enlarge):

But be careful! For example: see the big fat arrow pointing from China to the US, with the number ‘395’ next to it? As far as I can tell, they got that number by working out how many megatonnes of CO₂ were created by manufacturing goods in China and shipping them to the US during the year 2004… but then subtracting the megatonnes of CO₂ created by manufacturing goods in the US and shipping them to China during that year.

So if I understand this correctly, there’s a lot of ‘cancellation’ going on in this picture. And that could fool the casual reader. After all, it’s not like CO₂ produced in the US while making goods for export to China really helps cancel out the CO₂ produced in China while making goods for export to the US! So, I’d prefer to see a picture that had labelled arrows pointing both ways between China and the US, and similarly for other countries or groups of countries.

(By the way, the EU is counted as one lump for the purposes of this picture.)

But that’s a small nitpick: this article is full of interesting things. For example, the authors say that the surge of carbon emissions since 2000 has been driven

not only by growth of the global population and per-capita GDP, but also by unanticipated global increases in the energy intensity of GDP (energy per unit GDP) and the carbon intensity of energy (emissions per unit energy).

And, they say that in 2004, 23% of world-wide CO₂ emissions, or 6.2 gigatonnes of carbon dioxide, were associated to international trade, primarily exports from China and other developing countries to rich countries.

(As you can see, the numbers labelling those arrows in the picture above don’t add up to anything like 6,200. That’s what made me suspect that there’s a lot of ‘cancellation’ going on in that picture.)

15 Comments | carbon emissions | Permalink
Posted by John Baez

The Melting of Greenland and West Antarctica

19 May, 2011

Ice is melting at an accelerating pace in Greenland and the Antarctic. You may know all about this. But maybe like me you’re still just catching up on the basics. If so, here’s a quick intro.

If all the ice in Antarctica melted, it would raise the sea by 61 meters! Such mammoth sea level changes do happen:

But it seems they take millennia. In the shorter term, meaning the next century or two, people tend to focus their concern on the West Antarctic Ice Sheet, or WAIS:

If the entire WAIS melted, it would make sea levels rise 4.8 meters. Another region of concern is Greenland: if all the ice there melted, it would cause a global sea level rise of 7.2 meters. This would inundate many of the world’s coastal cities.

Luckily, it seems no reputable glaciologists think the WAIS or Greenland will completely melt in the coming century. But amount of melting of these ice sheets has been a big challenge to predict.

The last International Panel on Climate Change report, back in 2007, took a pretty conservative stance, and assumed these ice sheets would melt at a slow and more or less constant rate until 2100. Their conclusion was that about 75% of sea level rise would be caused by the oceans expanding as they warmed. The melting of small glaciers, ice caps and Greenland would account for most of the rest. The Antarctic, they believed, would actually provide a small net reduction in sea levels, with increases in snowfall more than enough to outweigh the effects of melting. They predicted an overall sea level rise of between 0.18 and 0.59 meters, with most of the uncertainty arising from different assumptions about what the world economy will do.

However, almost as soon as the 4th IPCC report was released, evidence started accumulating that the melting of Greenland and the West Antarctic Ice Sheet were speeding up. For example:

This graph, taken from Skeptical Science, shows Isabella Velicogna’s estimates of the mass of the Greenland ice sheet. Unfiltered data are blue crosses. Data filtered to eliminate seasonal variations are shown as red crosses. The best fit by a quadratic function is shown in green. The data came from the Gravity Recovery and Climate Experiment—or GRACE, for short. This remarkable project uses a pair of satellites to accurately measure small variations from place to place in the Earth’s gravitational field. When ice sheets melt, GRACE can detect it.

The big news, of course, was that the melting is speeding up. Here’s the same sort of graph for Antarctica, again created by Velicogna:

• I. Velicogna, Increasing rates of ice mass loss from the Greenland and Antarctic ice sheets revealed by GRACE, Geophysical Research Letters, 36 (2009), L19503.

More recently, Eric Rignot et al compared GRACE data to another way of keeping track of these ice sheets:

• Eric Rignot, Acceleration of the contribution of the Greenland and Antarctic ice sheets to sea level rise, Geophysical Research Letters 38, L05503.

Satellites and radio echo soundings measure ice leaving these sheets, while regional atmospheric climate model data can be used to estimate the amount of snow being added. The difference should be the overall loss of ice.

These graphs show Rignot’s results:

Graph a is Greenland, graph b is Antarctica and graph c is the total of both. These graphs show not the amount of ice, but the rate at which the amount of ice is changing, in gigatonnes per year. So, a line sloping down would mean that the ice loss is accelerating at a constant rate.

By fitting a line to satellite and atmospheric data, Rignot’s team found that over the last 18 years, Greenland has been losing an average of 22 gigatonnes more ice each year. Antarctica has been losing an average of 14.5 gigatonnes more each year.

But also note the black versus the red on the top two graphs! The GRACE data is in red. The other approach is in black. They match fairly well, though of course not perfectly.

The upshot? Rignot says:

That ice sheets will dominate future sea level rise is not surprising—they hold a lot more ice mass than mountain glaciers. What is surprising is this increased contribution by the ice sheets is already happening. If present trends continue, sea level is likely to be significantly higher than levels projected by the United Nations Intergovernmental Panel on Climate Change in 2007.

But how much sea level rise, exactly? Opinions still vary. A recent National Academy of Sciences report said at least 0.6 meters by 2100. But this still doesn’t include any melting of Greenland or the Antarctic!

This paper tries to take Greenland and the Antarctic into account:

• S. Jevrejeva, J. C. Moore and A. Grinsted, How will sea level respond to changes in natural and anthropogenic forcings by 2100?, Geophysical Research Letters 37 (2010), L07703.

The authors say their estimates are in line with past sea level responses to temperature change, and they suggest that estimates based on ice and ocean thermal responses alone may be misleading. With six different IPCC scenarios they estimate a sea level rise of 0.6–1.6 meters by 2100, and are confident the rise will be between 0.59 and 1.8 meters.

This paper suggests an upper bound on sea level rise 2 meters per century (if you max out everything) and a more realistic upper bound of 1 meter/century for this century (it could accelerate later):

• W. T. Pfeffer, J. T. Harper and S. O’Neel, Kinematic constraints on glacier contributions to 21st-century sea-level rise, Science 321 (2008), 1340-1343.

So, except for James Hansen, it sounds like most people would agree on an upper bound of about 2 meters of sea level rise by 2100. This is considerably more than the 4th IPCC report: remember, that gave an upper bound of about 0.6 meters.

I guess one moral is: stay tuned for further developments.

Everything you just read here, and more, was put together by the Azimuth Project team:

• Sea level rise, Azimuth Library.

I thank everyone who contributed, especially Staffan Liljgeren and Frederik de Roo.

25 Comments | climate | Permalink
Posted by John Baez

Conferences on Math and Climate Change

15 May, 2011

Here are some conferences on climate change and related issues, specially designed to get mathematicians interacting with scientists who work on these things! If you know of any more coming up, please let me know. These ones are sponsored by the Mathematics and Climate Research Network, a US-based organization, but there are probably others.

• Society for Industrial and Applied Mathematics (SIAM) Conference on Applications of Dynamical Systems, Snowbird, Utah, US, 22-26 May, 2011. Organized by Jonathan Dawes and Vivien Kirk.

Climate modeling and data assimilation are among the themes of this conference, which is aimed at starting communication between mathematicians who develop dynamical systems techniques and the applied scientists who use them.

• Mathematical Biosciences Institute (MBI) Workshop on Ocean Ecologies and their Physical Habitats in a Changing Climate, Columbus, Ohio, US, June 2011. Organized by Ken Golden, Chris Jones, Hans Kaper, and Mary Lou Zeeman.

The goal of this workshop is to bring together biologists studying ocean and polar ecologies; oceanographers, biogeochemists, and climate scientists studying the changing physical habitats; and mathematicians with ecological and physical expertise. The interactions between ocean ecological systems and their physical environments may dramatically impact both marine biodiversity and the planetary response to the changing atmosphere. The types of mathematics used to model ecological and physical processes are typically quite different. The team organizing this workshop anticipates interesting new mathematical challenges arising from combining these different approaches. The workshop will focus on two main themes:

1) polar and sea ice ecologies;

2) phytoplankton and the carbon cycle.

• Minisymposium on the Dynamics of the Earth’s Climate, as part of the International Congress on Industrial and Applied Mathematics (ICIAM), Vancouver, British Columbia, July 2011.
Organized by Hans G. Kaper, Mary C. Silber and Mary Lou Zeeman.

The speakers in this mini-symposium will highlight some interesting mathematical problems that have come from climate science and can be addressed with techniques developed in the dynamical systems community.

• Institute of Mathematics and its Applications (IMA) Conference on Mathematics of the Climate System, University of Reading, United Kingdom, 12-15 September, 2011. Organized by Paul Williams, Colin Cotter, Mike Cullen, Mike Davey, Christopher Ferro, John Huthnance and David Stainforth.

This conference is about the construction and use of mathematical models of the climate system. The conference will focus on three related topics:

1) the extraction of mathematical models from climate data and climate-model output (homogenisation, stochastic model reduction, bistability and metastable states, low frequency variability, data-driven coarse-graining, set-oriented methods, trend identification, time-series analysis);

2) reduced models and their dynamics (linear response theory, bifurcations, extreme events, uncertainty);

3) testing hypotheses about the climate system using statistical frameworks (emulators, Bayesian methods, nonparametric methods, equitability).

Leave a Comment » | climate, conferences, mathematics | Permalink
Posted by John Baez

A Question About Graduate Schools

12 May, 2011

I got an email from a physics major asking for some advice about graduate programs. He said it would be okay if I posted it here. Maybe you can help out!

Prof. Baez,

My name is Blake Pollard, I am an undergraduate physics major at Columbia University graduating next week. I agree very much with the premise of and the need for the Azimuth Project and would like to help out. Though my passion is physics, most of my undergraduate research has been in climate and sustainability. I would really like to find a graduate program enabling me to do both physics and something useful for the environmental movement, hence I haven’t committed to a Ph.D. program in pure physics. I studied category theory a bit here at Columbia from Lauda, and took some representation theory with Khovanov, but I think (at least at this point in time) my calling in physics is geometrical algebras. I was planning on spending a year off reading on my own, trying to do some work, and making the decision between environmental, physics, or mathematics graduate studies. Your blog served me well as a guidepost in my early college years for reading good stuff, and would appreciate any advice you have on:

1) graduate programs where I could do work on both mathematical physics and the environment

2) good people/places/projects that I could participate in in the coming year.

The Azimuth Project web resources have already been helpful in finding people to reach out to, but I figured you might have something or someone popping out of your head in particular.

I have programming experience in data mining, numerical simulations, remote sensing, and just having fun programming; decent math/physics background; and really just want to find a good place where good people are working hard. Like Göttingen way back in the day. Sorry for the longish email.

Thank you in advance for your time,

Blake S. Pollard
Applied Physics 2011

In a later email he added a bit more detail:

I think most people, though, would associate my goals with doing some physical modeling/analysis of environmental systems/problems, maybe a statistical-physical hybrid. That is probably what I would do in a PhD program on the environmental side of things. But I’m more thinking of having an advisor who does research in mathematical physics, while applying myself on the side to some problem related to the environment, like Google’s 20% projects. Probably it’s a bit too inter-departmental and too flexible for there to be a formal program for this (plus I’d likely be too busy!).

It seems though the answer might be simply doing a PhD in environmental sciences and doing physics in my spare time. Just organizing my own thoughts.

Are there any good grad programs that involve a mix of mathematical or theoretical physics and environmental science? I’ll take any good answers I get and add them to the Azimuth Wiki.

21 Comments | azimuth, physics | Permalink
Posted by John Baez

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Azimuth

Is Life Improbable?

The One Best Thing Everyone Could Do to Slow Climate Change

Information Geometry (Part 8)

Outsourcing Carbon Emissions

The Melting of Greenland and West Antarctica

Conferences on Math and Climate Change

A Question About Graduate Schools

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