The mismeasurement of science

Albert Einstein’s greatest scientific “blunder” (his word) came as a sequel to his greatest scientific achievement. That achievement was his theory of gravity, the general theory of relativity, which he introduced in 1915. Two years later, in 1917, Einstein ran into a problem while trying to apply general relativity to the Universe as a whole. At the time, Einstein believed that on large scales the Universe is static and unchanging. But he realized that general relativity predicts that such a Universe can’t exist: it would spontaneously collapse in on itself. To solve this problem, Einstein modified the equations of general relativity, adding an extra term involving what is called the “cosmological constant”, which, roughly speaking, is a type of pressure which keeps a static Universe from collapsing.

Twelve years later, in 1929, Edwin Hubble discovered that the Universe isn’t static and unchanging, but is actually expanding. Upon hearing the news, Einstein quickly realized that if he’d taken his original 1915 theory seriously, he could have used it to predict the expansion that Hubble had observed. That would have been one of the great theoretical predictions of all time! It was this realization that led Einstein to describe the cosmological constant as the “biggest blunder” of his life.

The story doesn’t end there. Nearly seven decades later, in 1998, two teams of astronomers independently made some very precise measurements of the expansion of the Universe, and discovered that there really is a need for the cosmological constant (ref,ref). Einstein’s “biggest blunder” was, in fact, one of his most prescient achievements.

The point of the story of the cosmological constant is not that Einstein was a fool. Rather, the point is that it’s very, very difficult for even the best scientists to accurately assess the value of scientific discoveries. Science is filled with examples of major discoveries that were initially underappreciated. Alexander Fleming abandoned his work on penicillin. Max Born won the Nobel Prize in physics for a footnote he added in proof to a paper – a footnote that explains how the quantum mechanical wavefunction is connected to probabilities. That’s perhaps the most important idea anyone had in twentieth century physics. Assessing science is hard.

The problem of measuring science

Assessing science may be hard, but it’s also something we do constantly. Countries such as the United Kingdom and Australia have introduced costly and time-consuming research assessment exercises to judge the quality of scientific work done in those countries. In just the past few years, many new metrics purporting to measure the value of scientific research have been proposed, such as the h-index, the g-index, and many more. In June of 2010, the journal Nature ran a special issue on such metrics. Indeed, an entire field of scientometrics is being developed to measure science, and there are roughly 1,500 professional scientometricians.

There’s a slightly surreal quality to all this activity. If even Einstein demonstrably made enormous mistakes in judging his own research, why are the rest of us trying to measure the value of science systematically, and even organizing the scientific systems of entire countries around these attempts? Isn’t the lesson of the Einstein story that we shouldn’t believe anyone who claims to be able to reliably assess the value of science? Of course, the problem is that while it may be near-impossible to accurately evaluate scientific work, as a practical matter we are forced to make such evaluations. Every time a committee decides to award or decline a grant, or to hire or not hire a scientist, they are making a judgement about the relative worth of different scientific work. And so our society has evolved a mix of customs and institutions and technologies to answer the fundamental question: how should we allocate resources to science? The answer we give to that question is changing rapidly today, as metrics such as citation count and the h-index take on a more prominent role. In 2006, for example, the UK Government proposed changing their research assessment exercise so that it could be done in a largely automated fashion, using citation-based metrics. The proposal was eventually dropped, but nonetheless the UK proposal is a good example of the rise of metrics.

In this essay I argue that heavy reliance on a small number of metrics is bad for science. Of course, many people have previously criticised metrics such as citation count or the h-index. Such criticisms tend to fall into one of two categories. In the first category are criticisms of the properties of particular metrics, for example, that they undervalue pioneer work, or that they unfairly disadvantage particular fields. In the second category are criticisms of the entire notion of quantitatively measuring science. My argument differs from both these types of arguments. I accept that metrics in some form are inevitable – after all, as I said above, every granting or hiring committee is effectively using a metric every time they make a decision. My argument instead is essentially an argument against homogeneity in the evaluation of science: it’s not the use of metrics I’m objecting to, per se, rather it’s the idea that a relatively small number of metrics may become broadly influential. I shall argue that it’s much better if the system is very diverse, with all sorts of different ways being used to evaluate science. Crucially, my argument is independent of the details of what metrics are being broadly adopted: no matter how well-designed a particular metric may be, we shall see that it would be better to use a more heterogeneous system.

As a final word before we get to the details of the argument, I should perhaps mention my own prejudice about the evaluation of science, which is the probably not-very-controversial view that the best way to evaluate science is to ask a few knowledgeable, independent- and broad-minded people to take a really deep look at the primary research, and to report their opinion, preferably while keeping in mind the story of Einstein and the cosmological constant. Unfortunately, such a process is often not practically feasible.

Three problems with centralized metrics

I’ll use the term centralized metric as a shorthand for any metric which is applied broadly within the scientific community. Examples today include the h-index, the total number of papers published, and total citation count. I use this terminology in part because such metrics are often imposed by powerful central agencies – recall the UK government’s proposal to use a citation-based scheme to assess UK research. Of course, it’s also possible for a metric to be used broadly across science, without being imposed by any central agency. This is happening increasingly with the h-index, and has happened in the past with metrics such as the number of papers published, and the number of citations. In such cases, even though the metric may not be imposed by any central agency, it is still a central point of failure, and so the term “centralized metric” is appropriate. In this section, I describe three ways centralized metrics can inhibit science.

Centralized metrics suppress cognitive diversity: Over the past decade the complexity theorist Scott Page and his collaborators have proved some remarkable results about the use of metrics to identify the “best” people to solve a problem (ref,ref). Here’s the scenario Page and company consider. Suppose you have a difficult creative problem you want solved – let’s say, finding a quantum theory of gravity. Let’s also suppose that there are 1,000 people worldwide who want to work on the problem, but you have funding to support only 50 people. How should you pick those 50? One way to do it is to design a metric to identify which people are best suited to solve the problem, and then to pick the 50 highest-scoring people according to that metric. What Page and company showed is that it’s sometimes actually better to choose 50 people at random. That sounds impossible, but it’s true for a simple reason: selecting only the highest scorers will suppress cognitive diversity that might be essential to solving the problem. Suppose, for example, that the pool of 1,000 people contains a few mathematicians who are experts in the mathematical field of stochastic processes, but who know little about the topics usually believed to be connected to quantum gravity. Perhaps, however, unbeknownst to us, expertise in stochastic processes is actually critical to solving the problem of quantum gravity. If you pick the 50 “best” people according to your metric it’s likely that you’ll miss that crucial expertise. But if you pick 50 people at random you’ve got a chance of picking up that crucial expertise [1]. Richard Feynman made a similar point in a talk he gave shortly after receiving the Nobel Prize in physics (ref):

If you give more money to theoretical physics it doesn’t do any good if it just increases the number of guys following the comet head. So it’s necessary to increase the amount of variety… and the only way to do it is to implore you few guys to take a risk with your lives that you will never be heard of again, and go off in the wild blue yonder and see if you can figure it out.

What makes Page and company’s result so striking is that they gave a convincing general argument showing that this phenomenon occurs for any metric at all. They dubbed the result the diversity-trumps-ability theorem. Of course, exactly when the conclusion of the theorem applies depends on many factors, including the nature of the cognitive diversity in the larger group, the details of the problem, and the details of the metric. In particular, it depends strongly on something we can’t know in advance: how much or what type of cognitive diversity is needed to solve the problem at hand. The key point, though, is that it’s dangerously naive to believe that doing good science is just a matter of picking the right metric, and then selecting the top people according to that metric. No matter what the metric, it’ll suppress cognitive diversity. And that may mean suppressing knowledge crucial to solving the problem at hand.

Centralized metrics create perverse incentives: Imagine, for the sake of argument, that the US National Science Foundation (NSF) wanted to encourage scientists to use YouTube videos as a way of sharing scientific results. The videos could, for example, be used as a way of explaining crucial-but-hard-to-verbally-describe details of experiments. To encourage the use of videos, the NSF announces that from now on they’d like grant applications to include viewing statistics for YouTube videos as a metric for the impact of prior research. Now, this proposal obviously has many problems, but for the sake of argument please just imagine it was being done. Suppose also that after this policy was implemented a new video service came online that was far better than YouTube. If the new service was good enough then people in the general consumer market would quickly switch to the new service. But even if the new service was far better than YouTube, most scientists – at least those with any interest in NSF funding – wouldn’t switch until the NSF changed its policy. Meanwhile, the NSF would have little reason to change their policy, until lots of scientists were using the new service. In short, this centralized metric would incentivize scientists to use inferior systems, and so inhibit them from using the best tools.

The YouTube example is perhaps fanciful, at least today, but similar problems do already occur. At many institutions scientists are rewarded for publishing in “top-tier” journals, according to some central list, and penalized for publishing in “lower-tier” journals. For example, faculty at Qatar University are given a reward of 3,000 Qatari Rials (US $820) for each impact factor point of a journal they publish in. If broadly applied, this sort of incentive would creates all sorts of problems. For instance, new journals in exciting emerging fields are likely to be establishing themselves, and so have a lower impact factor. So the effect of this scheme will be to disincentivize scientists from participating in new fields; the newer the field, the greater the disincentive! Any time we create a centralized metric, we yoke the way science is done to that metric.

Centralized metrics misallocate resources: One of the causes of the financial crash of 2008 was a serious mistake made by rating agencies such as Moody’s, S&P, and Fitch. The mistake was to systematically underestimate the risk of investing in financial instruments derived from housing mortgages. Because so many investors relied on the rating agencies to make investment decisions, the erroneous ratings caused an enormous misallocation of capital, which propped up a bubble in the housing market. It was only after homeowners began to default on their mortgages in unusually large numbers that the market realized that the ratings agencies were mistaken, and the bubble collapsed. It’s easy to blame the rating agencies for this collapse, but this kind of misallocation of resources is inevitable in any system which relies on centralized decision-making. The reason is that any mistakes made at the central point, no matter how small, then spread and affect the entire system.

In science, centralization also leads to a misallocation of resources. We’ve already seen two examples of how this can occur: the suppression of cognitive diversity, and the creation of perverse incentives. The problem is exacerbated by the fact that science has few mechanisms to correct the misallocation of resources. Consider, for example, the long-term fate of many fashionable fields. Such fields typically become fashionable as the result of some breakthrough result that opens up many new research possiblities. Encouraged by that breakthrough, grant agencies begin to invest heavily in the field, creating a new class of scientists (and grant agents) whose professional success is tied not just to the past success of the field, but also to the future success of the field. Money gets poured in, more and more people pursue the area, students are trained, and go on to positions of their own. In short, the field expands rapidly. Initially this expansion may be justified, but even after the field stagnates, there are few structural mechanisms to slow continued expansion. Effectively, there is a bubble in such fields, while less fashionable ideas remain underfunded as a result. Furthermore, we should expect such scientific bubbles to be more common than bubbles in the financial market, because decision making is more centralized in science. We should also expect scientific bubbles to last longer, since, unlike financial bubbles, there are few forces able to pop a bubble in science; there’s no analogue to the homeowner defaults to correct the misallocation of resources. Indeed, funding agencies can prop up stagnant fields of research for decades, in large part because the people paying the cost of the bubble – usually, the taxpayers – are too isolated from the consequences to realize that their money is being wasted.

One metric to rule them all

No-one sensible would staff a company by simply applying an IQ test and employing whoever scored highest (c.f., though, ref). And yet there are some in the scientific community who seem to want to move toward staffing scientific institutions by whoever scores highest according to the metrical flavour-of-the-month. If there is one point to take away from this essay it is this: beware of anyone advocating or working toward the one “correct” metric for science. It’s certainly a good thing to work toward a better understanding of how to evaluate science, but it’s easy for enthusiasts of scientometrics to believe that they’ve found (or will soon find) the answer, the one metric to rule them all, and that that metric should henceforth be broadly used to assess scientific work. I believe we should strongly resist this approach, and aim instead to both improve our understanding of how to assess science, and also to ensure considerable heterogeneity in how decisions are made.

One tentative idea I have which might help address this problem is to democratize the creation of new metrics. This can happen if open science becomes the norm, so scientific results are openly accessible, online, making it possible, at least in principle, for anyone to develop new metrics. That sort of development will lead to a healthy proliferation of different ideas about what constitutes “good science”. Of course, if this happens then I expect it will lead to a certain amount of “metric fatigue” as people develop many different ways of measuring science, and there will be calls to just settle down on one standard metric. I hope those calls aren’t heeded. If science is to be anything more than lots of people following the comet head, we need to encourage people to move in different directions, and that means valuing many different ways of doing science.

Update: After posting this I Googled my title, out of curiosity to see if it had been used before. I found an interesting article by Peter Lawrence, which is likely of interest to anyone who enjoyed this essay.


Thanks to Jen Dodd and Hassan Masum for many useful comments. This is a draft of an essay to appear in a forthcoming volume on reputation systems, edited by Hassan Masum and Mark Tovey.


[1] Sometimes an even better strategy will be a mixed strategy, e.g., picking the top 40 people according to the metric, and also picking 10 at random. So far as I know this kind of mixed strategy hasn’t been studied. It’s difficult to imagine that the proposal to pick, say, one in five faculty members completely at random is going to receive much support at Universities, no matter how well founded the proposal may be. We have too much intuitive sympathy for the notion that the best way to generate global optima is to locally optimize. Incidentally, the success of such mixed strategies is closely related to the phenomenon of stochastic resonance, wherein adding a noise to a system can sometimes improve its performance.

My book “Reinventing Discovery” will be released in 2011. It’s about the way open online collaboration is revolutionizing science. A summary of many of the themes in the book is available in this essay. If you’d like to be notified when the book is available, please send a blank email to with the subject “subscribe book”. You can subscribe to my blog here, and to my Twitter account here.

More on funding

Chad Orzel has some thoughtful comments on my earlier questions about research funding. Here’s a few excerpts and some further thoughts:

… a good deal of the image problems that science in general has at the moment can be traced to a failure to grapple more directly with issues of funding and the justification of funding… In the latter half of the 20th century, we probably worked out the quantum details of 1000 times as many physical systems as in the first half, but that sort of thing feels a little like stamp collecting– adding one new element to a mixture and then re-measuring the band structure of the resulting solid doesn’t really seem to be on the same level as, say, the Schrödinger equation, but I’m at a loss for how to quantify the difference… The more important question, though, is should we really expect or demand that learning be proportional to funding?

This really gets to the nub of it. In research, as in so many other things, funding may hit a point of diminishing returns beyond which what we learn becomes more and more marginal. However, it is by no means obvious where the threshold is beyond which society as a whole would be better off allocating its resources to other more worthy causes.

And what, exactly, do we as a society expect to get out of fundamental research?

For years, the argument has been based on technology– that fundamental research is necessary to understand how to build the technologies of the future, and put a flying car in every garage. This has worked well for a long time, and it’s still true in a lot of fields, but I think it’s starting to break down in the really big-ticket areas. You can make a decent case that, say, a major neutron diffraction facility will provide materials science information that will allow better understanding of high-temperature superconductors, and make life better for everyone. It’s a little harder to make that case for the Higgs boson, and you’re sort of left with the Tang and Velcro argument– that working on making the next generation of whopping huge accelerators will lead to spin-off technologies that benefit large numbers of people. It’s not clear to me that this is a winning argument– we’ve gotten some nice things out of CERN, the Web among them, but I don’t know that the return on investment really justifies the expense.

The spinoff argument also has the problem that it’s hard to argue that these things wouldn’t have happened anyway. No disrespect to Tim Berners-Lee’s wonderful work, but it’s hard to believe that if he hadn’t started the web, some MIT student in a dorm room wouldn’t have done so shortly thereafter.

Of course, it’s not like I have a sure-fire argument. Like most scientists, I think that research is inherently worth funding– it’s practically axiomatic. Science is, at a fundamental level, what sets us apart from other animals. We don’t just accept the world around us as inscrutable and unchangeable, we poke at it until we figure out how it works, and we use that knowledge to our advantage. No matter what poets and musicians say, it’s science that makes us human, and that’s worth a few bucks to keep going. And if it takes millions or billions of dollars, well, we’re a wealthy society, and we can afford it.

We really ought to have a better argument than that, though.

As for the appropriate level of funding, I’m not sure I have a concrete number in mind. If we’ve got half a trillion to piss away on misguided military adventures, though, I think we can throw a few billion to the sciences without demanding anything particular in return.

One could attempt to frame this in purely economic terms: what’s the optimal rate at which to invest in research in order to maximize utility, under reasonable assumptions? This framing misses some of the other social benefits that Chad alludes to – all other things being equal, I’d rather live in a world where we understand general relativity, just because – but has the benefit of being at less passably well posed. I don’t know a lot about their conclusions, but I believe this kind of question has recently come under a lot of scrutiny from economists like Paul Romer, under the name endogeneous growth theory.

Categorized as Science

The Future of Science

How is the web going to impact science?

At present, the impact of the web on science has mostly been to make access to existing information easier, using tools such as online journals and databases such as the ISI Web of Knowledge and Google Scholar. There have also been some interesting attempts at developing other forms of tools, although so far as I am aware none of them have gained a lot of traction with the wider scientific community. (There are signs of exceptions to this rule on the horizon, especially some of the tools being developed by Timo Hannay’s team at Nature.)

The contrast with the internet at large is striking. Ebay, Google, Wikipedia, Facebook, Flickr and many others are new types of institution enabling entirely new forms of co-operation. Furthermore, the rate of innovation in creating such new institutions is enormous, and these examples only scratch the surface of what will soon be possible.

Over the past few months I’ve drafted a short book on how I think science will change over the next few years as a result of the web. Although I’m still revising and extending the book, over the next few weeks I’ll be posting self-contained excerpts here that I think might be of some interest. Thoughtful feedback, argument, and suggestions are very welcome!

A few of the things I discuss in the book and will post about here include:

  • Micropublication: Allowing immediate publication in small incremental steps, both of conventional text, and in more diverse media formats (e.g. commentary, code, data, simulations, explanations, suggestions, criticism and correction). All are to be treated as first class fully citable publications, creating an incentive for people to contribute far more rapidly and in a wider range of ways than is presently the case.
  • Open source research: Using version control systems to open up scientific publications so they can be extended, modified, reused, refactored and recombined by other users, all the while preserving a coherent and citable record of who did what, and when.
  • The future of peer review: The present quality assurance system relies on refereeing as a filtering system, prior to publication. Can we move to a system where the filtering is done after publication?
  • Collaboration markets: How can we fully leverage individual expertise? Most researchers spend much of their time reinventing the wheel, or doing tasks at which they have relatively little comparative advantage. Can we provide mechanisms to easily outsource work like this?
  • Legacy systems and migration: Why is it that the scientific community has been so slow to innovate on the internet? Many of the ideas above no doubt look like pipedreams. Nonetheless, I believe that by carefully considering and integrating with today’s legacy incentive systems (citation, peer review, and journal publication), it will be possible to construct a migration path that incentivizes scientists to make the jump to new tools for doing research.

The Research Funding “Crisis”

If you talk with academics for long, sooner or later you’ll hear one of them talk about a funding crisis in fundamental research (e.g. Google and Cosmic Variance).

There are two related questions that bother me.

First, how much funding is enough for fundamental research? What criterion should be used to decide how much money is the right amount to spend on fundamental research?

Second, the human race spent a lot lot more on fundamental research in the second half of the twentieth century than it did in the first. It’s hard to get a good handle on exactly how much, in part because it depends on what you mean by fundamental research. At a guess, I’d say at least 1000 times as much was spent in the second half of the twentieth century. Did we learn 1000 times as much? In fact, did we learn as much, even without a multiplier?

Optimal photons

Optimal photons for quantum information processing, joint with Peter Rohde and Tim Ralph.

Producing single photons is hard work, and there’s no really good single photon sources available, but a plethora of theoretical proposals for sources.

Perhaps somewhat surprisingly, not all photons are created equal. In particular, getting the kind of interference effects necessary for quantum information processing depends a lot on the shape of the wavepacket produced by the source: if you have the wrong shape wavepacket, even a tiny amount of noise may destroy the interference effects. For this reason, it’s important to understand which sources produce photons for which interferenceis stable against the noise.

That’s the subject of this paper. In particular, the main result is to show that for a wide variety of possible applications, Gaussian wavepackets produce the most stable interference effects, with the implication that people designing sources should look for sources which produce Gaussian or near Gaussian wavepackets.

The Solovay-Kitaev algorithm

New paper: “The Solovay-Kitaev algorithm”, jointly with Chris Dawson, submitted as a tutorial / review article to the journal “Quantum Information and Computation”. I expect the paper to appear at the preprint arxiv this coming Monday, as I write.

The Solovay-Kitaev theorem is an important result in quantum computing. In a classical computer, it’s well-known that the same computation can be built up out of many different basic gate sets. For example, you can build up a circuit to add two numbers either using OR and NOT gates, or out of NAND gates alone. Furthermore, you can use OR and NOT gates to simulate NAND gates (or vice versa), so, roughly speaking, these two different circuits can have the same size, up to some constant of proportionality.

In the case of quantum computers, things are more complicated. The set of possible quantum gates forms a continuum, and it’s not necessarily possible to use one gate set to simulate another exactly. Instead, some approximation may be necessary.

The Solovay-Kitaev theorem guarantees that different universal gate sets can simulate one another exceedingly quickly, to a very good approximation. From a practical point of view, this is important in compiling quantum algorithms (like Shor’s) into a form that can be implemented fault-tolerantly. From a more mathematical point of view, the Solovay-Kitaev theorem is a remarkable general statement about how quickly the group SU(d) is “filled in” by a universal set of gates.

This paper aims to provide a simple exposition of the Solovay-Kitaev theorem, which we hope is accessible to people who’ve just started working on quantum computing. We explain the proof in the form of an algorithm (just nine simple lines of pseudocode!) which, given a goal unitary operation U, builds up a good approximation to U using some prespecified set of universal gates. Chris has also prepared an online implementation of the algorithm.