# The Bitcoin economy

The electronic currency Bitcoin has been in the news quite a bit lately since its value has risen from about $10 a year ago to over$650 today, hitting a peak of over \$1000 less than a month ago. I remember hearing Gavin Andresen, who is a Principal of the Bitcoin  Virtual Currency Project (no single person nor entity issues or governs Bitcoin) talk about Bitcoin on Econtalk two years ago and was astonished at how little he knew about basic economics much less monetary policy. Paul Krugman criticized Bitcoin today in his New York Times column and Timothy Lee responded in the Washington Post.

The principle behind Bitcoin is actually quite simple. There is a master list, called the block chain, which is an encrypted shared ledger in which all transactions are kept. The system uses public-key cryptography, where a public key can be used to encrypt a piece of information but a private key is required to decrypt it. Bitcoin owners each have a private key, and use it to update the ledger whenever a transaction takes place. The community at large then validates the transaction in a computationally intensive process called mining. The rewards for this work are Bitcoins, which are issued to the first computer to complete the computation. The intensive computations are integral to the system because it makes it difficult for attackers to falsify a transaction. As long as there are more honest participants than attackers then the attackers can never perform computations fast enough to falsify a transaction. The computations are also scaled so that Bitcoins are only issued every 10 minutes. Thus it does not matter how fast your computer is in absolute terms to mine Bitcoins, only that it is faster than everyone else’s computer. This article describes how people are creating businesses to mine Bitcoins.

Krugman’s post was about the ironic connection between Keynesian fiscal stimulus and gold. Although gold has some industrial use it is highly valued mostly because it is rare and difficult to dig up. Keyne’s theory of recessions and depressions is that there is a sudden collapse in aggregate demand, so the economy operates at below capacity, leading to excess unemployment. This situation was thought not to occur in classical economics because prices and wages should fall until equilibrium is restored and the economy operates at full capacity again. However, Keynes proposed that prices and wages are “sticky” and do not adjust very quickly. His solution was for the government to increase spending to take up the shortfall in demand and return the economy to full employment. He then jokingly proposed that the government could get the private sector to do the spending by burying money, which people could privately finance to dig out. He also noted that this was not that different from gold mining. Keyne’s point was that instead of wasting all that effort the government could simply print money and give it away or spend it. Krugman also points out that Adam Smith, often held up as a paragon of conservative principles, felt that paper money was much better for an economy to run smoothly than tying up resources in useless gold and silver. The connection between gold and Bitcoins is unmissable. Both have no intrinsic value and are a terrible waste of resources. Lee feels that Krugman misunderstands Bitcoin in that the intensive computations are integral to the functioning of the system and more importantly the main utility of Bitcoin is that it is a new form of payment network, which he feels is independent of monetary considerations.

Krugman and Lee have valid points but both are still slightly off the mark. I think we will definitely head towards some electronic monetary system in the future but it certainly won’t be Bitcoin in its current form. However, Bitcoin or at least something similar will also remain. The main problem with Bitcoin, as well as gold, is that its supply is constrained. The supply of Bitcoins is designed to cap out at 21 million with about half in circulation now. What this means is that the Bitcoin economy is subject to deflation. As the economy grows and costs fall, the price of goods denominated in Bitcoins must also fall. Andresen shockingly didn’t understand this important fact in the Econtalk podcast. The value of Bitcoins will always increase. Deflation is bad for economic growth because it encourages people to delay purchases and hoard Bitcoins. Of course if you don’t believe in economic growth then Bitcoins might be a good thing. Ideally, you want money to be neutral so the supply should grow along with the economy. This is why central banks target inflation around 2%. Hence, Bitcoin as it is currently designed will certainly fail as a currency and payment system but it would not take too much effort to fix its flaws. It may simply serve the role of the search engine Altavista to the eventual winner Google.

However, I think Bitcoin in its full inefficient glory and things like it will only proliferate. In our current era of high unemployment and slow growth, Bitcoin is serving as a small economic stimulus. As we get more economically efficient, fewer of us will be required for any particular sector of the economy. The only possible way to maintain full employment is to constantly invent new economic sectors. Bitcoin is economically useful because it is so completely useless.

# Symplectic Integrators

Dynamical systems can be divided into two basic types: conservative and dissipative.  In biology, we almost always model dissipative systems and thus if we want to computationally simulate the system almost any numerical solver will do the job (unless the problem is stiff, which I’ll leave to another post). However, when simulating a conservative system, we must take care to conserve the conserved quantities. Here, I will give a very elementary review of symplectic integrators for numerically solving conservative systems.

# What counts as science?

Ever since the financial crisis of 2008 there has been some discussion about whether or not economics is a science. Some, like Russ Roberts of Econtalk, channelling Friedrich Hayek, do not believe that economics is a science. They think it’s more like history where we come up with descriptive narratives that cannot be proven. I think that one thing that could clarify this debate is to separate the goal of a field from its practice. A field could be a science although its practice is not scientific.

To me what defines a science is whether or not it strives to ask questions that have unambiguous answers. In that sense, most of economics is a science. We may never know what caused the financial crisis of 2008 but that is still a scientific question. Now, it is quite plausible that the crisis of 2008 had no particular cause just like there is no particular cause for a winter storm. It could have been just the result of a collection of random events but knowing that would be extremely useful. In this sense, parts of history can also be considered to be a science. I do agree that the practice of economics and history are not always scientific and can never be as scientific as a field like physics because controlled experiments usually cannot be performed. We will likely never find the answer for what caused World War I but there certainly was a set of conditions and events that led to it.

There are parts of economics that are clearly not science such as what constitutes a fair system. Likewise in history, questions regarding who was the best president or military mind are certainly  not science. Like art and ethics these questions depend on value systems. I would also stress that a big part of science is figuring out what questions can be asked. If it is true that recessions are random like winter storms then the question of when the next crisis will hit does not have an answer. There may be a short time window for some predictability but no chance of a long range forecast. However, we could possibly find some necessary conditions for recessions just like cold weather is necessary for a snow storm.

# Fred Sanger 1918 – 2013

Perhaps the greatest biologist of the twentieth century and two-time Nobel prize winner, Fred Sanger, has died at the age of 95. He won his first Nobel in 1958 for determining the amino acid sequence of insulin and his second in 1980 for developing a method to sequence DNA.  An obituary can be found here.

# Michaelis-Menten kinetics

This year is the one hundred anniversary of the Michaelis-Menten equation, which was published in 1913 by German born biochemist Leonor Michaelis and Canadian physician Maud Menten. Menten was one of the first women to obtain a medical degree in Canada and travelled to Berlin to work with Michaelis because women were forbidden from doing research in Canada. After spending a few years in Europe she returned to the US to obtain a PhD from the University of Chicago and spent most of her career at the University of Pittsburgh. Michaelis also eventually moved to the US and had positions at Johns Hopkins University and the Rockefeller University.

The Michaelis-Menten equation is one of the first applications of mathematics to biochemistry and perhaps the most important. These days people, including myself, throw the term Michaelis-Menten around to generally mean any function of the form

$f(x)= \frac {Vx}{K+x}$

although its original derivation was to specify the rate of an enzymatic reaction.  In 1903, it had been discovered that enzymes, which catalyze reactions, work by binding to a substrate. Michaelis took up this line of research and Menten joined him. They focused on the enzyme invertase, which catalyzes the breaking down (i.e. hydrolysis) of the substrate sucrose (i.e. table sugar) into the simple sugars fructose and glucose. They modelled this reaction as

$E + S \overset{k_f}{\underset{k_r}{\rightleftharpoons}} ES \overset{k_c}{\rightarrow }E +P$

where the enzyme E binds to a substrate S to form a complex ES which releases the enzyme and forms a product P. The goal is to calculate the rate of the appearance of P.

# Talk in Taiwan

I’m currently at the National Center for Theoretical Sciences, Math Division, on the campus of the National Tsing Hua University, Hsinchu for the 2013 Conference on Mathematical Physiology.  The NCTS is perhaps the best run institution I’ve ever visited. They have made my stay extremely comfortable and convenient.

Here are the slides for my talk on Correlations, Fluctuations, and Finite Size Effects in Neural Networks.  Here is a list of references that go with the talk

E. Hildebrand, M.A. Buice, and C.C. Chow, Kinetic theory of coupled oscillators,’ Physical Review Letters 98 , 054101 (2007) [PRL Online] [PDF]

M.A. Buice and C.C. Chow, Correlations, fluctuations and stability of a finite-size network of coupled oscillators’. Phys. Rev. E 76 031118 (2007) [PDF]

M.A. Buice, J.D. Cowan, and C.C. Chow, ‘Systematic Fluctuation Expansion for Neural Network Activity Equations’, Neural Comp., 22:377-426 (2010) [PDF]

C.C. Chow and M.A. Buice, ‘Path integral methods for stochastic differential equations’, arXiv:1009.5966 (2010).

M.A. Buice and C.C. Chow, `Effective stochastic behavior in dynamical systems with incomplete incomplete information.’ Phys. Rev. E 84:051120 (2011).

MA Buice and CC Chow. Dynamic finite size effects in spiking neural networks. PLoS Comp Bio 9:e1002872 (2013).

MA Buice and CC Chow. Generalized activity equations for spiking neural networks. Front. Comput. Neurosci. 7:162. doi: 10.3389/fncom.2013.00162, arXiv:1310.6934.

Here is the link to relevant posts on the topic.

# New paper on neural networks

Michael Buice and I have a new paper in Frontiers in Computational Neuroscience as well as on the arXiv (the arXiv version has fewer typos at this point). This paper partially completes the series of papers Michael and I have written about developing generalized activity equations that include the effects of correlations for spiking neural networks. It combines two separate formalisms we have pursued over the past several years. The first was a way to compute finite size effects in a network of coupled deterministic oscillators (e.g. see here, herehere and here).  The second was to derive a set of generalized Wilson-Cowan equations that includes correlation dynamics (e.g. see here, here, and here ). Although both formalisms utilize path integrals, they are actually conceptually quite different. The first formalism adapted kinetic theory of plasmas to coupled dynamical systems. The second used ideas from field theory (i.e. a two-particle irreducible effective action) to compute self-consistent moment hierarchies for a stochastic system. This paper merges the two ideas to generate generalized activity equations for a set of deterministic spiking neurons.

# Paper on compressed sensing and genomics

New paper on the arXiv. The next step after the completion of the Human Genome Project, was the search for genes associated with diseases such as autism or diabetes. However, after spending hundreds of millions of dollars, we find that there are very few common variants of genes with large effects. This doesn’t mean that there aren’t genes with large effect. The growth hormone gene definitely has a large effect on height. It just means that variations of genes that are common among people have small effects on the phenotype. Given the results of Fisher, Wright, Haldane and colleagues, this was probably expected as the most likely scenario and recent results measuring narrow-sense heritability directly from genetic markers (e.g. see this) confirms this view.

Current GWAS microarrays consider about a million or two markers and this is increasing rapidly. Narrow-sense heritability refers to the additive or linear genetic variance, which means the phenotype is given by the linear model $y= Z\beta + \eta$, where $y$ is the phenotype vector, $Z$ is the genotype matrix, $\beta$ are all the genetic effects we want to recover, and $\eta$ are all the nonadditive components including environmental effects. This is a classic linear regression problem. The problem comes when the number of coefficients $\beta$ far exceeds the number of people in your sample, which is the case for genomics. Compressed sensing is a field of high dimensional statistics that addresses this specific problem. People such as David Donoho, Emmanuel Candes and Terence Tao have proven under fairly general conditions that if the number of nonzero coefficients are sparse compared to the number samples, then the effects can be completely recovered using L1 penalized optimization algorithms such as the lasso or approximate message passing. In this paper, we show that these ideas can be applied to genomics.

Here is Steve Hsu’s summary of the paper

Application of compressed sensing to genome wide association studies and genomic selection

(Submitted on 8 Oct 2013)

We show that the signal-processing paradigm known as compressed sensing (CS) is applicable to genome-wide association studies (GWAS) and genomic selection (GS). The aim of GWAS is to isolate trait-associated loci, whereas GS attempts to predict the phenotypic values of new individuals on the basis of training data. CS addresses a problem common to both endeavors, namely that the number of genotyped markers often greatly exceeds the sample size. We show using CS methods and theory that all loci of nonzero effect can be identified (selected) using an efficient algorithm, provided that they are sufficiently few in number (sparse) relative to sample size. For heritability h2 = 1, there is a sharp phase transition to complete selection as the sample size is increased. For heritability values less than one, complete selection can still occur although the transition is smoothed. The transition boundary is only weakly dependent on the total number of genotyped markers. The crossing of a transition boundary provides an objective means to determine when true effects are being recovered. For h2 = 0.5, we find that a sample size that is thirty times the number of nonzero loci is sufficient for good recovery.

 Comments: Main paper (27 pages, 4 figures) and Supplement (5 figures) combined Subjects: Genomics (q-bio.GN); Applications (stat.AP) Cite as: arXiv:1310.2264 [q-bio.GN] (or arXiv:1310.2264v1 [q-bio.GN] for this version)

# Happiness and divisive inhibition

The Wait But Why blog has an amusing post on why Generation Y yuppies (GYPSYS) are unhappy, which I found through the blog of Michigan economist  Miles Kimball. In short, it is because their expectations exceed reality and they are entitled. What caught my eye was that they defined happiness as “Reality-Expectations”. The key point being that this is a subtractive expression. My college friend Peter Lee, now Professor and Director of the University Manchester X-Ray imaging facility, used to define happiness as “desires fulfilled beyond expectations”. I always interpreted this as a divisive quantity, meaning “Reality/Expectations”.

Now, the definition does have implications if we actually try to use it as a model for how happiness would change with some quantity like money. For example, consider the model where reality and expectations are both proportional to money. Then happiness = a*money – b*money. As long as b is less than a, then money always buys happiness, but if a is less than b then more money brings more unhappiness. However, if we consider the divisive model of happiness then happiness = a*money/ b*money = a/b and happiness doesn’t depend on money at all.

However, the main reason I bring this up is because it is analogous to the two possible ways to model inhibition (or adaptation) in neuroscience. The neurons in the brain generally interact with each other through two types of synapses – excitatory and inhibitory. Excitatory synapses generally depolarize a neuron and make its potential get closer to threshold whereas inhibitory neurons hyperpolarize the neuron and make it farther from threshold (although there are ways this can be violated). For neurons receiving stationary asynchronous inputs, we can consider the firing rate to be some function of the excitatory E and inhibitory I inputs. In subtractive inhibition, the firing rate would have the abstract form f(E-I) whereas for divisive inhibition it would have the form f(E)/(I+C), where f is some thresholded gain function (i.e. zero below threshold, positive above threshold) and C is a constant to prevent the firing rate from reaching infinity. There are some critical differences between subtractive and divisive inhibition. Divisive inhibition works by reducing the gain of the neuron, i.e. it makes the slope of the gain function shallower while subtractive inhibition makes the threshold effectively higher. These properties have great computational significance, which I will get into in a future post.

# The cost of the shutdown and sequester

People may be wondering how the US government shutdown is affecting the NIH. I can’t speak for the rest of the institutes but I was instructed to not come to work and to not use my NIH email account or NIH resources. Two new fellows, who were supposed to begin on Oct 1, now have to wait and they will not be compensated for the missed time even if Congress does decides to give back pay to the furloughed employees. I really was hoping for them to start in August or September but that was pushed back because of the Sequester (have people forgotten about that?), which cut our budgets severely. In fact, because of the Sequester, I wasn’t able to hire one fellow because the salary requirements for their seniority exceeded my budget. We were just starting to get some really interesting psychophysics results on ambiguous stimuli but that had to be put on hold because we couldn’t immediately replace fellow Phyllis Thangaraj, who was running the experiments and left this summer to start her MD/PhD degree at Columbia. Now it will be delayed even further. I have several papers in the revision process that have also been delayed by the shutdown. All travel has been cancelled and I heard that people at conferences were ordered to return immediately, including those who were on planes on Oct 1. My quadrennial external review this week has now been postponed. All the flights for the committee and ad hoc members have to be cancelled and we now have to find another date where 20 or more people can agree on. All NIH seminars and the yearly NIH research festival has been cancelled. I was supposed to review an external NIH research proposal this week and that has been postponed indefinitely along with all other submitted proposals awaiting review. Academic labs, students and postdocs depending on their NIH grants this fiscal year will be without funding until the government is reopened. Personally, I will probably come out of this reasonably intact. However, I do worry how this will affect young people, who are the future.

# Heritability and additive genetic variance

Most people have an intuitive notion of heritability being the genetic component of why close relatives tend to resemble each other more than strangers. More technically, heritability is the fraction of the variance of a trait within a population that is due to genetic factors. This is the pedagogical post on heritability that I promised in a previous post on estimating heritability from genome wide association studies (GWAS).

One of the most important facts about uncertainty and something that everyone should know but often doesn’t is that when you add two imprecise quantities together, while the average of the sum is the sum of the averages of the individual quantities, the total error (i.e. standard deviation) is not the sum of the standard deviations but the square root of the sum of the square of the standard deviations or variances. In other words, when you add two uncorrelated noisy variables, the variance of the sum is the sum of the variances. Hence, the error grows as the square root of the number of quantities you add and not linearly as it had been assumed for centuries. There is a great article in the American Scientist from 2007 called The Most Dangerous Equation giving a history of some calamities that resulted from not knowing about how variances sum. The variance of a trait can thus be expressed as the sum of the genetic variance and environmental variance, where environment just means everything that is not correlated to genetics. The heritability is the ratio of the genetic variance to the trait variance.

# Government shutdown

As of today, I am officially furloughed without pay since the NIH is officially closed and nonessential employees like myself are barred from working without pay by the Antideficiency Act of 1884. However, given that blogging is not considered an official duty, I can continue to post to Scientific Clearing House. Those who are not up on American politics may be wondering why the US government has shutdown. The reason is that the US fiscal year begins on Oct 1 and according to the the US Constitution, only Congress can appropriate funds for the functioning of government and they did not pass a budget for the new fiscal year by midnight of September 30. Actually, Congress has not passed a budget on time in recent years but has passed Continuing Resolutions that to keep the government going. So why have they not passed a budget or a CR this year? Well, currently the US government is divided with the Democratic party controlling the Senate and Presidency and the Republican party controlling the House of Representatives. All three entities must agree for a law to pass. Three years ago, the Democrats controlled the Congress, which includes both the House and Senate, and passed the Affordable Care Act, also known as Obamacare, which the President signed into law.   The Republicans took control of the House in 2011 and have been trying to repeal the ACA ever since but have been stopped by the Senate. This year they decided to try a new tactic, which was to pass a budget that withholds funding for the ACA. The Senate did not agree, passed a budget with the ACA and sent it back to the House, which then took out funding for the ACA again with some modifications and sent it back. This went on back and forth without converging to an agreement and thus we are closed today.

# David Wright, 1963-2013

I guess this is just a week for sadness.  My high school friend, David Wright, died this morning at the age of 50.  He collapsed at his desk, probably of a heart attack. Dave was a natural athlete and story teller. He was always there for you if you needed someone.  He will be missed.

# Epic comeback

Well, I spoke too soon in my earlier post on the America’s Cup.  Oracle Team USA has since won 7 races in a row and now it is 8-8 in the best of 17 match (although they have already had 18 races). The final race to determine the winner is today. Check out the action here.  In the past, America’s Cup races had usually been best of 3 or best of 5 matches. In this new format, the races are much shorter, taking less than an hour rather than several, and they try to get in two a day if the weather permits. In the beginning New Zealand had the faster boat. They had already been racing for over a month in the challenger series and were just better than Oracle. However, the long format and some weather delays has given Oracle a chance to get up to speed and now they are definitely the faster boat. Yesterday, they flew by New Zealand on the upwind leg. The only chance New Zealand has to win today is if Oracle makes a mistake.

# Richard Azuma, 1930 – 2013

I was saddened to learn that Richard “Dick” Azuma, who was a professor in the University of Toronto Physics department from 1961 to 1994 and emeritus after that, passed yesterday. He was a nuclear physicist par excellence and chair of the department when I was there as an undergraduate in the early 80’s. I was in the Engineering Science (physics option) program, which was an enriched engineering program at UofT. I took a class in nuclear physics with Professor Azuma during my third year. He brought great energy and intuition to the topic. He was one of the few professors I would talk to outside of class and one day I asked if he had any open summer jobs. He went out of his way to secure a position for me at the nuclear physics laboratory TRIUMF in Vancouver in 1984. That was the best summer of my life. The lab was full of students from all over Canada and I remain good friends with many of them today. I worked on a meson scattering experiment and although I wasn’t of much use to the experiment I did get to see first hand what happens in a lab. I wrote a 4th year thesis on some of the results from that experiment. I last saw Dick in 2010 when I went to Toronto to give a physics colloquium. He was still very energetic and as engaged in physics as ever. We will all miss him greatly.

# America’s Cup 2013

Today may be the last race for the America’s Cup yacht series between the US and New Zealand.   Here are the highlights from the last race.

It is a best of 17 series and New Zealand has 8 wins so today may be the last chance to watch these hundred million dollar multihull yachts fly around San Francisco harbour at close to 50 miles per hour.  All the races are posted on You Tube.

# Coase and the Nature of the Firm

Economist and Noble Laureate Ronald Coase died earlier this month just three months short of his 103rd birthday. Coase is mostly famous for two papers: “The Nature of the Firm” (1937) and “The Problem of Social Cost” (1960). He came up with many of the ideas for the first paper when he was just 21. Coase asked the simple question of why companies exist. According to Adam Smith it should actually be more cost-effective for a person to contract out work rather than hire people. Coases’s answer was that there are always transaction costs or frictions that make a firm more cost-effective. In other words, the market (i.e. price mechanism) is not always the most efficient way to organize production. The size of a firm is determined by the point when the extra (marginal) cost of organizing an extra employee balances the transaction costs of obtaining her services on the free market. Hence, the great irony of modern capitalism is that its main pillar, the firm, is a paragon of central planning. Firms in essence are totalitarian regimes where the citizens are free to leave.

Conservative and libertarian leaning individuals generally prize  private companies and free markets over governments. They argue that many of the functions of government, such as schools and healthcare, would be more efficient if privatized. The question then is why are private firms more efficient than government? When we hand over functions formerly performed by a democratically elected government, we are in essence making society less democratic. One could argue that firms are more efficient because they are subject to competition. That is why we want to break up monopolies. However, that should be true of government too. If we don’t like the government we have then we can always elect another one. We can even change the constitution to our liking. In principle, no one is under more competition than our elected officials. It is the job of the citizenry to ensure that they are doing their job.

# TB, streptomycin, and who gets credit

The Science Show did a feature story recently about the discovery of streptomycin,  the first antibiotic to treat tuberculosis, which had killed 2 billion people in the 18th and 19th centuries. Streptomycin was discovered by graduate student Albert Schatz in 1943, who worked in the lab of Professor Selman Waksman at Rutgers. Waksman was the sole winner of the 1952 Nobel Prize for this work. The story is narrated by the author of the book Experiment Eleven, who paints Waksman as the villain and Schatz as the victim. Evidently, Waksman convinced Schatz to sign away his patent rights to Rutgers but secretly negotiated a deal to obtain 20% of the royalties. When Schatz discovered this, he sued Waksman and obtained a settlement. However, this turned the scientific community against him and he forced him out of microbiology into science education. To me, this is just more evidence that prizes and patents are incentives for malfeasance.

# New paper on population genetics

James Lee and I just published a paper entitled “The causal meaning of Fisher’s average effect” in the journal Genetics Research. The paper can be obtained here. This paper is the brainchild of James and I just helped him out with some of the proofs.  James’s take on the paper can be read here. The paper resolves a puzzle about the incommensurability of Ronald Fisher’s two definitions of the average effect noted by population geneticist D.S. Falconer three decades ago.

Fisher was well known for both brilliance and obscurity and people have long puzzled over the meaning of some of his work.  The concept of the average effect is extremely important for population genetics but it is not very well understood. The field of population genetics was invented in the early twentieth century by luminaries such as Fisher, Sewall Wright, and JBS Haldane to reconcile Darwin’s theory of evolution with Mendelian genetics. This is a very rich field that has been somewhat forgotten. People in mathematical,  systems, computational, and quantitative biology really should be fully acquainted with the field.

For those who are unacquainted with genetics, here is a quick primer to understand the paper. Certain traits, like eye colour or the ability to roll your tongue, are affected by your genes. Prior to the discovery of the structure of DNA, it was not clear what genes were, except that they were the primary discrete unit of genetic inheritance. These days it usually refers to some region on the genome. Mendel’s great insight was that genes come in pairs, which we now know to correspond to the two copies of each of the 23 chromosomes we have.  A variant of a particular gene is called an allele.  Traits can depend on genes (or more accurately genetic loci) linearly or nonlinearly. Consider a quantitative trait that depends on a single genetic locus that has two alleles, which we will call a and A. This means that a person will have one of three possible genotypes: 1) homozygous in A (i.e. have two A alleles), 2) heterozygous (have one of each), or 3) homozygous in a (i.e. have no A alleles). If the locus is linear then if you plot the measure of the trait (e.g. height) against the number of A alleles, you will get a straight line. For example, suppose allele A contributes a tenth of a centimetre to height. Then people with one A allele will be on average one tenth of a centimetre taller than those with no A alleles and those with two A alleles will be two tenths taller. The familiar notion of dominance is a nonlinear effect. So for example, the ability to roll your tongue is controlled by a single gene. There is a dominant rolling allele and a recessive nonrolling allele. If you have at least one rolling allele, you can roll your tongue.

The average effect of a gene substitution is the average change in a trait if one allele is substituted for another. A crucial part of population genetics is that you always need to consider averages. This is because genes are rarely completely deterministic. They can be influenced by the environment or other genes. Thus, in order to define the effect of the gene, you need to average over these other influences. This then leads to a somewhat ambiguous definition of average effect and Fisher actually came up with two. The first, and as James would argue the primary definition, is a causal one in that we want to measure the average effect of a gene if you experimentally substituted one allele for another prior to development and influence by the environment. A second correlation definition would simply be to plot the trait against the number of alleles as in the example above. The slope would then be the average effect. This second definition looks at the correlation between the gene and the trait but as the old saying goes “correlation does not imply causation”. For example, the genetic loci may not have any effect on the trait but happens to be strongly correlated with a true causal locus (in the population you happen to be examining). Distinguishing between genes that are merely associated with a trait from ones that are actually causal remains an open problem in genome wide association studies.

Our paper goes over some of the history and philosophy of the tension between these two definitions. We wrote the paper because these two definitions do not always agree and we show under what conditions they do agree. The main reason they don’t agree is that averages will depend on the background over which you average. For a biallelic gene, there are 2 alleles but 3 genotypes. The distribution of alleles in a population is governed by two parameters. It’s not enough to specify the frequency of one allele. You also need to know the correlation between alleles. The regression definition matches the causal definition if a particular function representing this correlation is held fixed while the experimental allele substitutions under the causal definition are carried out. We also considered the multi-allele and multi-loci case in as much generality as we could.

# The problem with democracy

Winston Churchill once said that “Democracy is the worst form of government, except for all those other forms that have been tried from time to time.” The current effectiveness of the US government does make one wonder if that is even true. The principle behind democracy is essentially utilitarian – a majority or at least a plurality decides on the course of the state. However, implicit in this assumption is that the utility function for individuals match their participation function.

For example, consider environmental regulation. The utility function for the amount of allowable emissions of some harmful pollutant like mercury for most people will be downward sloping – most people would increase their utility the less the pollutant is emitted. However, for a small minority of polluters it will be upward sloping with a much steeper slope. Let’s say that the sum of the utility gained for the bulk of the population for strong regulation is greater than that gained by the few polluters for weak regulation. If the democratic voice one has in affecting policy is proportional to the summed utility then the smaller gain for the many will outweigh the larger gain to the few. Unfortunately, this is not usually case. More often, the translation of utility to legislation and regulation is not proportional but passes through a very nonlinear participation function with a sharp threshold. The bulk of the population is below the threshold so they provide little or no voice on the issue. The minority utility is above the threshold and provides a very loud voice which dominates the result. Our laws are thus systematically biased to protecting the interests of special interest groups.

The way out of this trap is to either align everyone’s utility functions or to linearize the participation functions. We could try to use regulation to dampen the effectiveness of minority participation functions or use public information campaigns to change utility functions or increase the participation functions of the silent majority. Variations of these methods have been tried with varying degrees of success. Then there is always the old install a benevolent dictator who respects the views of the majority. That one really doesn’t have a good track record though.