I was just compiling the GNU plotutils library (version 2.6) using MSYS for MinGW, and had the following problematic error:

undefined reference to `signgam'

It turns out there are problems with the 'gauss' methods provided by various platforms: in the file specfun.c in the ode directory of the plotutils source, the following comment occurs:

/* The following gamma-related nonsense is necessary because (1) some
   vendors have lgamma(), some have gamma(), and some have neither [see
   include/sys-defines.h for further comments], (2) some vendors do not
   declare whichever function they have [e.g. Irix 5.3 requires an
   auxiliary preprocessing symbol to be defined for the declaration in
   math.h to be visible], and (3) some vendors supply broken versions which
   we can't use [e.g. AIX's libm.a gamma support is conspicuously broken],
   so we need to link in a replacement, but we can't use the same name for
   the external symbol `signgam'.  What a mess! -- rsm */

There are some threads discussing ways of solving this in old versions, and apparently only with Cygwin; however, I found that the simplest way to solve the problem is to make use of the C macro NO_SYSTEM_GAMMA referenced in the plotutils source - the package will then declare its own gamma functions for us. (So in fact, this isn't an error with plotutils, it's plotutils that provides the solution!)

The fix: make clean the directory and then run configure with an additional flag for the compiler defining the variable NO_SYSTEM_GAMMA. For example:

./configure --prefix=/where/to/install CFLAGS="-DNO_SYSTEM_GAMMA" --enable-libplotter

Then running make, make check and make install as usual, you should be good to go!

Remark: I actually had problems with the make check phase, which reported errors (8 of 11 failures, in fact!) when the relevant files (the .out and .xout files in the test directory) seemed to match. I recommend not worrying/bothering with these unless you have problems, or spot a potential future problem.

Other remark: The --enable-libplotter switch for the configure command is only needed if you want to use the C++ wrapper!

The following is taken from a 'Note' I wrote on Facebook in February 2009. I thought I'd repost it, since I'm less likely to forget about it here. It is, as I note at the end, very much an early-morning ramble or rant, but hey, it's helpful for thinking about things.

The question at hand (let's make this clear) is

What makes good art?

That's what makes GOOD art, not just what makes art. Nota bene.

---

Theories

Okay, the usual three-way division is still relevant here. What is at question is what we should consider when judging the quality of art. I'll outline the three main categories of viewpoint (there are others, some cynical and some highly romanticized) using Wikipedia articles to draw on structure and examples, and also to try and cover many bases.

Relativism

The thesis of the aesthetic relativist is that aesthetic quality (a.k.a. 'beauty', but more on that later) is sufficiently heavily dependent on individuals' feelings towards and experiences of other individuals that there can be no consistent objective assessment of art.

Many influential philosophers - people like Kant - subscribe(d) to this theory, based essentially on the strong evidence that more or less any two people will disagree about the quality of some piece of art. The most obvious influence on this variation is probably that of the culture. This is essentially the argument espoused by many analysts of the semiotics of ideal beauty. Semiotics is essentially the study of symbology, in which anthropologists (and other professionals, especially linguists) examine how various signs acquire their meaning - John Locke was one of the first people to clearly outline the study of the topic.

Objectivism

The general position of objectivists is that it is possible to salvage something of an objective value judgement to analyze art, and that although the objective position is still deeply tied to humanity, this tie does not restrict us to the purely relativistic beliefs. See A Primer on the Objectivist Esthetics for some introductory points. Ayn Rand, the foremother of objectivism herself, said that concepts and values are

determined by the nature of reality, but to be discovered by man's mind.

If differences of opinion are the 'obvious evidence' for relativism, then the 'obvious evidence' for objectivism is that "it is not a contradiction to say: 'This is a great work of art, but I don't like it'," in the words of Ayn Rand. Therefore, there exist some criteria which can be used to discriminate between good, bad and non- art.

A major premise of many objectivists is, in fact, frequently an appeal to semiotics and symbolism - in one very intriguing sense. This is that art is an efficient way of packaging various complex notions into - well, manageable chunks, almost (I really recommend "Why Do We Have Art?" at the above link.) As I suggested in my recent note on poetry, our ability to condense metaphysics, emotions and so on into small packages in fact elucidates them. After all, our cognitive ability is almost entirely associative in nature, and as such is greatly stimulated by complex, interwoven representations of topics. People do tend to love the way that a topic is composed of many different (often seemingly incompatible) strands which can be brought together into cohesive, pleasing (there's that aesthetic appeal) structures.

This is actually quite an inclusive, if no doubt dissatisfyingly vague, definition - it includes art reflecting on (read: containing - most artists spend a LONG TIME thinking about how their art relates to their subject matter, and so do consequently most critics - but this is not necessary for the full enjoyment of the piece) natural beauty, and manufactured beauty (as I say, more on beauty later), probably speeches and anything containing metaphor, and so on. And we can also conceive of ways of comparing quality - according to the criteria in the above link, for example. However, personally I find these criteria rather vague and limiting. Mainly "Clarity", which I think is of debatable value.

Of course, this label has little to say about, for example, true abstract art, or many types of post-modernist art. It also has an interesting problem with photography, which I suspect is of interest to the readers of this essay. Namely, that people often regard photographs as art when they were, in reality, little more than snaps - a phenomenon particularly prevalent in the field of wildlife photography. This also touches upon the problems of art without the artist, and leads us naturally on to natural beauty and such topics... But there's no space for that here. See below!

As far as abstract art goes, I find it unintuitive that it can be included in this category - and music has a similar problem. This is where we discuss whether art must be representational - i.e., whether it must be a symbol of something. First, we turn once more to Rand:

As a re-creation of reality, a work of art has to be representational; its freedom of stylization is limited by the requirement of intelligibility; if it does not present an intelligible subject, it ceases to be art.

This certainly seems to say no. I suspect Ben might sympathize with the viewpoint expressed in the above statement, since he has a fairly pragmatic approach to such high-brow concepts (I don't think it is necessarily high-brow) as art, and since he considers things which need study and background to be less worthwhile.

But -- let me look at music first. If you read books like The Rest is Noise: Listening to the Twentieth Century (by Alex Ross), or biographies of most 'classical' music composers, then you start to find out about lots and lots of symbolism buried in the history and personal stories of the composers. There are obvious and subtle motifs, and the question as to whether such cultural and subtle representations are deemed acceptable is moot. I don't see how you can deny it. Decent music with lyrics tends to be as metaphorical as poetry, if not as considered.

Abstract art is a more awkward prospect. I personally don't rate a lot of abstract art very highly, so I won't discuss it.

Realism

This viewpoint is held by those who insist that aesthetic value - often termed beauty (see below) - is a concept completely independent of humans.

Beauty

To quote the Simpsons, "Beauty is truth, truth beauty" (Keats) --

But the truth can be harsh and disturbing! How can that be considered beautiful?

Often, we abbreviate much of aesthetic quality into the term 'beauty', in a rather uppity sense. Take the first line of Wikipedia:

Beauty is a characteristic of a person, place, object, or idea that provides a perceptual experience of pleasure, meaning, or satisfaction

Or the OED, which gives this as an additional definition to visual attractiveness:

That quality or combination of qualities which affords keen pleasure to other senses (e.g. that of hearing), or which charms the intellectual or moral faculties, through inherent grace, or fitness to a desired end

These get at the empathetic nature of beauty, to my mind. The point is that the object in question evokes some degree of some variety of resonant satisfaction in our mind (I use 'resonant' advisedly; memory and thought seem to work by resonating neuron networks.)

Anyway, the interesting thing is what makes beauty; and I think that there are different categories of beauty. Here are some, mainly focusing on the visual arts:

Beauty of the human (and to some degree other animal) form

This kind of beauty - a respect for various qualities ranging from elegance, healthiness, calmness, and happiness to that found in sexually provocative material - is probably derived largely from the evolutionary advantage of being able to pick out fertile and healthy mates - there is conclusive evidence, for instance, that in women, a waist-to-hip ratio of 0.7 is a good predictor of both attractiveness and fertility. In animals, it makes sense that we look for similar attributes - like the elegance and pure colour of the wings of the dove I snapped above - in judging beauty.

Beauty of animals and the inanimate object (including devices or tools, and statues)

This type of beauty often has appeals between several of the other categories here - as well as the appeal discussed under the human/animal form discussion, we see the appeal of patterns and so on.

Beauty of the landscape, skyscape, seascape... etc.

This is a much more subtle form of beauty. There is a definite need to look to the multifarious views of our philosophical betters. For example, look at Burke, who divides aesthetic appeal into masculine and feminine components (being a man...) He declares the sublime to be the masculine element, which overwhelms the viewer with power, scale, uniformity, abruptness, difficulty, intense colours, etc. and argues that the language of the sublime is muddled, not descriptive, emotionally abstract, and full of conflicting sensations. Contrastingly, he defines beauty as the feminine element, which overwhelms the viewer with the wish to protect the subject, and is characterized by symmetry, intricateness, smoothness, mildness, attractiveness (in a metrosexual way!) and so on. Burke wondered if the beautiful was appealing to evolutionary elements of sexual desire, and the sublime - but terrifying - to our instinct for self-preservation in some way.

A highly important concept in aesthetics (particularly the 18th century) is the picturesque, literally that which is fit to be made into a picture or which is in the style of a picture - it is a term considered to unite the sublime and the beautiful as Burke considered his extremes, and classically taking the form of the idealized rural landscape painting, with stern elements like crags, ruins or torrents, and softer elements like a quiet - vulnerable - cottage, a gentle winding river or a soft forest. This concept of the picturesque sheds some light, I feel, on how we relate to landscapes and many other works of art.

Beauty of colour, patterns and shape

The subtleties of these forms are essentially those of abstract art, and as such are in some ways difficult to pin down. However, using the tools developed using our landscape analysis, perhaps some elements are already separated.

The human instinct to spot patterns - and the satisfaction we experience when we use this ability - seems to help us gain pleasure from fractal-based images like that above. We seem to particularly like subtle or obscured patterns. This seems to be reasonable, given we have evolved highly specialized symbolic and pattern-spotting mechanisms, and since our culture is set up to teach young children that spotting things is good.

Beauty of abstract consistency and bringing together of disparate thoughts (particularly mathematical beauty)

(See the Wikipedia articles on Mathematical Beauty and Mathematics and Art.)

It is surely a related type of satisfaction which we can gain from understanding the mysterious patterns and intricacies of a consistent theory. We do love it like a good story when things come together, in quite a literal sense.

Summary

So why have I gone on about beauty, and where does it leave us?

Well, my argument is that the aesthetic value of a piece of art qua art is based entirely on its beauty, as defined above:

That quality or combination of qualities which affords keen pleasure to the senses, or which charms the intellectual or moral faculties, through inherent grace, or fitness to a desired end

And as such it is essentially a relativist concept, and every individual can have their own response to any of the terms in the above definition. However, I dispute that this makes the concept of art useless. Apart from giving us this definition of the value of art (I'll discuss value in just a minute) this discussion also indicates how culturally dependent and how dependent on human nature the notion of art is, since (as we observed above) various elements of beauty are a consequence of biological (neurological) or cultural influences.

As far as investigating 'good art', i.e. judging its quality, goes, there are two obvious possible types of criterion to judge how beautiful art is, given the above context.

• Breadth of appeal
• Intensity of appeal

That is, on how many 'levels' it works (sensually, intellectually, morally, functionally...), and how strong the sensation is. Mix and match to your own taste. I think a 50-50 split sounds good. Simple, really.

Note: You can attach value to the amount of work put into something, if you want, but be aware that split-second dives for wildlife photographs become worthless in the category, whilst a child intently (but poorly) copying trees into what I like to call a 'crapscape' gets a heck of a lot of credit.
Note: My definition of art clearly includes film, novels, songs, whatever. I can live with that. See that overly sentimental emotional flicks and childish books, whilst they may evoke strong emotions, do not really have the variety of effects the above evaluation looks for, since they are working purely on one level.
Note: I've been watching TV and stuff whilst doing this, and it's now 2:17 in the morning. I'm pretty tired and this is probably full of incomplete stuff and bollocks. Enjoy.

---

Wow, I wasn't kidding with that third note. Well, I can see omissions and problems already. Maybe I'll sort this out sometime.

 

The Conservatives are proposing a form of free market education, based largely according to them on the Swedish (and also American and Canadian) model of the last 20 years. The problem is, there's no convincing argument what they're doing is constructive.

They want to allow parents to effectively choose which school they want their portion of the schools' budget to go to, in essence forcing schools to compete for students.

There are two main problems with the Conservatives' position, perhaps symbolic of the current trends in politics - firstly, their policy is not right-wing enough, because it fails to allows schools to turn a profit (as Swedish schools can) and hence will likely lead to charitable institutions and small collabratives setting up minor that have little impact on the system; and secondly, it is far, far too right-wing, because it totally fails to promote equality and fairness in the educational system.

If you know me, you'll not be massively surprised to hear that the second issue is where my personal bone of contention is to be found. Newsnight recently found the Swedish model to be unimpressive as a piece of pioneering social organization, noting that in recent years "not only are standards generally down, there are strong indications that the new schools have increased social segregation".

It seems to me blindingly obvious that this is precisely what you would expect from such a crude attempt to make the school system more dynamic.

Obviously, you won't hear me disagreeing with the Conservatives' statement that giving

many more children access to the kind of education that is currently only available to the well-off: safe classrooms, talented and specialist teachers, access to the best curriculum and exams, and smaller schools run by teachers who know the children’s names

is a good thing (although I might well dispute, based on personal experience and the teachers who I know, the claim that one only gets such attention in the public and private school system). And it's trivially true that opening more schools will improve class sizes and, assuming that their budget is not effectively reduced during this process, also improve teaching quality and time (we could look here to up-and-coming countries like China, where 7AM-6PM state-funded schooling is reality for many people, as Ken Livingstone pointed out earlier this week).

Of course,

The Swedish people generally approve of the new system. About 10% of all students of compulsory school age now attend the new schools, and in the upper secondary level it is about 20%.

In fact, the Conservatives' draft schools' manifesto has much in the way of increased spending recommendations and increased powers, intervention and supervision: they will

• allocate increased pay budget control for headteachers (allowing them to "pay good teachers more", rather than the converse, it seems implied);
• "pay the student loan repayments for top maths and science graduates for as long as they remain teachers";
• "legislate so that teachers can ban any items that cause disruption in the classroom";
• "reinforce powers of discipline";
• "set up technical Academies across England, starting in at least the twelve biggest cities";
• "fund 400,000 new apprenticeship, preapprenticeship, college and other training places over two years"

and so on. Hardly the anti-nanny state, fund-cutting image we're used to, is it? The point is the Conservatives are basically in favour of creating more schools and educational institutions and allocating more funds to the school sector. Sure, the usual pro-discipline measures are there, but what's that really but increasing how heavily then government intervenes in day-to-day life? (I'm not against it, but many Conservatives seem to claim to be.)

But really, the basic principle of allocating funds in an essentially market-controlled manner is going to do exactly what it does in every other market - create some really huge, bloated organisations (Tesco, Whitehall, I could go on), crows out many smaller specialist instutitions, and result in bizarre competitions that end up not being beneficial to student, or (more likely) teachers. Put in this light, the apparently reasonable approach of having

more unannounced inspections, and failing schools will be inspected more often – with the best schools visited less frequently [and] any school that is in special measures for more than a year will be taken over immediately by a successful Academy provider.

seems somewhat more likely to result in the rapid construction of very large, monocultured schools, with presumably the same old problems with discipline, management, class sizes, and inefficiency.

Surely this will result in 'good' schools (presumably judged by the local media as much as anyone more qualified) receiving far too many applications to cope with, and hence either taking in too many pupils to manage adequately, or turning away vast swathes of candidates (based on what, one wonders) who would be forced to turn to 'worse' schools, creating significantly more inequality in the educational system, in the short and medium term at least.

Impressively, the Conservatives appear intent on doing this without even offering a monetary incentives to the people owning the schools. Regardless, having schools compete for students might also make for a considerably more high-pressured environment, both for students and teachers.

And even the other celebrated Conservative goals (particularly of 'liberating' schools from the tyranny of the 'politicised' national curriculum) seem thin when you consider even their idols Sweden enforce their curriculum.

Also, an interesting comment is to be found in this draft manifesto - the Conservatives will

end the bias towards the inclusion of children with special needs in mainstream schools.

I don't know what everyone else makes of this, but it sounds fascinatingly dangerous to me.

I went to see Ken Livingstone today.

Ken Livingstone is an interesting man. He is variously described as outspoken, "bold and imaginative", a disgrace, and Trotskyist. Whatever you think of him, I believe that more politicians like him could be a good thing.

I don't mean more left-leaning, or more environmentalist, or more liberal (well, actually, I do think these are good things, but that's really not what I'm getting at) - what I mean is forthright, open and frank. I like Vince Cable. I also have massively more respect for people on the right, like maybe Boris Johnson, and hell, even John Bolton, who state their (frequently wrong...) views openly and, yes, divisively, than those who studiously avoid the controversial in what they have to say, even when it is massively important to the electorate.

Yes, this is heavily directed at Cameron and Osborne and their ilk, people who leave you confused about their long-term vision, and their commitment to their policies.

But it is equally heavily directed to the Blairite school (although I certainly feel Cameron and Osborne - and Clegg, come to that) and the whole culture of 'professional' politicians (I use the word in the same way one describes 'professional fouls' in football) who exist, publically, as sort of ethereal ghosts in the machine, who do not represent with any confidence any values or schools of thought, but rather just give glimpses into an unknowable Whitehall existence.

Perhaps the world would be a better place if politicians (in fact, this is largely directed at higher-ranking ministers and their shadow counterparts) could step away from not just party line, but from the strange brand of modern political pseudo-realism which permeates all attempts at political debate. It is endlessly refreshing to hear people talking coherently about their own personal views, and putting forward reasoned, historical or just plain true arguments about important stuff.

Just my two cents.

Ken Livingstone was speaking at the Cambridge Union on Wednesday 17^th February 2010. He was mayor of London from 2000 to 2008, winning his first election as an independent candidate, and returning as a Labout party candidate in 2004. His achievements include introducing the first same-sex civil partnership register, before the government followed suit 3 years later, overhauling the public transport system, and introducing the congestion charge and other environmental measures. He was succeeded by Boris Johnson, but plans to run again for the role next term. Previously, he had served from 1981 as Leader of the greater London Council until the post was abolished by Margaret Thatcher in 1986. He is currently an adviser on urban planning in Venezuela, where we works with Hugo Chávez who he says is "just like me".

How sure are you?

Oddly, even though it's apparently one of the most fundamental questions to ask, we are often not very good at accurately judging and reporting our certainty of our own knowledge of the world.

I just got back from a stimulating and entertaining talk by the excellent Professor David Spiegelhalter entitled Quantifying Epistemic Uncertainty at the Trinity Mathematical Society.

He talked variously about the vagaries of different ways of interpreting probabilities (particularly the frequentist, Bayesian and 'metaphysical' interpretations, of which more below) and how they relate to our understanding of information in crucial areas like healthcare. He then went on to illustrate (with an excellent practical example in which he involved the whole group) exactly how our understanding of certainty can be quantified (or at least put in a useful frame) and used for the training and adjustment of accurate forecasts.

Metaphysical Understanding

The 'metaphysical' interpretation involves conceiving of possible futures, in an attempt to personalize risk. The article linked to above references this interactive demonstration which you might like to play around with, and maybe even use as the basis for a little experiment of your own. His argument is that the best way of getting people to understand the risks involved in individual lifestyle choices (like the purchase and use of statins) is to phrase it in terms of what could happen to them over 'multiple lives'.

If I am told there is a 10% chance I will suffer a heart attack or stroke in the next 10 years, lowered to 8% by taking statins, then my understanding of this changes according to whether I think of this in terms of, say, a 20% cut in the chance I suffer such an event; or for every 50 people taking statins, one 1 avoid a heart attack.

It is always clear that the mode of presentation of statistical information is of prime importance, and juxtaposition of statistics allows intuitive understanding of relative magnitudes to dominate over any attempt at a full understanding of, say, the worth of a treatment.

He contests that thinking in terms of 'future mes' is the most natural and valuable way to present what are by their very nature subjective statistics and risks. In 2 out of every 100 lives I lead, I would avoid a heart attack because I took statins.

Scoring Rules

The main body of the talk, however, was devoted to understanding the concept of a scoring rule. The idea is that, given some questions, and a subject who is answering them and providing an estimate of their own confidence in the result, you want to know how to reward (or penalise) the subject in order to get the most reliable information out of them about their own confidence.

For example, consider a meteoriologist forecasting the weather - if they are predicting weather with some stated percentage of confidence, how do you give them the incentive to be truthful and accurate and assess their accuracy?

The basic idea, stripped down to the simplest case of a binary event (e.g. will it rain tomorrow?), is to allocate some score S to every a probabilitistic forecast (or set of forecasts). In fact, we always assume we have several foecasts. Then the overall score should be higher when the forecaster is most reliably accurate at declaring uncertainty, and lowest when the forecaster routinely makes incorrect forecasts with high confidence.

Specifically, we call a scoring rule a proper scoring rule if the expected score is maximised (or more generally optimised) if and only if the actual probability of the event is exactly equal to the forecast probability.

For example, take a test with a quadratic (Brier, after the man who created the score around 1950) score, , where x is 0 or 1 according to whether or not the event occurs and q is the estimated probability it does. Then if the forecaster believes that there is a probability p that it does occur, the expected score (from his point of view) is which is equal to . So clearly, the forecaster maximizes his expected payoff by setting his projected likelihood q to exactly what he truly believes it to be, p.

Note that not all scoring rules are proper - not even all likely seeming candidates are. For example, you can check encourages forecasters to exaggerate (i.e. to pick q to be always either 0 or 1).

You can easily conduct an experiment using a scoring rule with any group of people - give them a multiple-choice test (two options each time for these simple rules) and get them to write down a 'confidence' probability for their answer - say 5/10 for a total guess and 10/10 if they are entirely certain. Then give them the corresponding score according to whether their answer was correct and their stated confidence, and you are left with a set of overall scores indicating who was the most honest and reliable answerer. This works very well (especially with punishing negative scores for high-confidence mistakes) as both a party game and statistical exercise.

A sample (quadratic) set of numbers are (listed as confidence, score given correct, score given incorrect):

• 5: +0, -0
• 6: +9, -11
• 7: +16, -24
• 8: +21, -39
• 9: 24, -56
• 10: 25, -75

A few different scoring rules are illustrated here.

In the long run, i.e. over several forecasts and experiments, we can 'decompose' (see this article also) the sum of the Brier score above into three components: Uncertainty, Reliability and Resolution. (Note the linked article uses a Brier score which is lower for better forecasts - this corresponds to dropping the leading 1 term in the above definition and making the remaining expression positive.)

1. Uncertainty provides a measure of how naturally diverse the events are (maximised when the event occurs 1/2 of the time for a binary event like we are considering). The Brier score is higher (worse) when the uncertainty is naturally larger, reflecting the fact that it is more difficult to give accurate predictions in this case.
2. Reliability shows how far from the truth the forecasts were, on average; so a figure of 0 indicates perfect accuracy, whilst larger values show more deviation.
3. Resolution shows how much the forecasts differ from the overall average score - note that one can achieve very high levels of reliability simply by constantly predicting the climatic average; resolution is higher when the forecasts are 'more definite' than this, and so is substracted from the score to improve it for higher resolution. It precisely equals (and hence cancels out) the uncertainty in the score when the predictions are always definite (0 or 1).

Spiegelhalter also noted that one can actually justify the construction of a semi-frequentist, semi-Bayesian theory of probability - and in fact derive the standard axioms of probability as theorems in this alternative, 'deeper' formulation of the subject.

It's possibly worth considering the slightly more fickle aspects of this method - encroaching on the field of game theory, in fact, as one considers less local tactics and maybe even psychological aspects to the situation. Most notably, it may be the case that maximizing expected score leads to non-optimal behaviours, and with human error and tendency to bias confidence included the situation becomes a lot less clear. Scoring rules are fairly heavily used (as suggested by the terminology and examples) in meteorology and similar areas, but the field seems to be studied rather less than one might hope. Still, something to do over the holidays...

David Spiegelhalter is the Winton Professor of the Public Understanding of Risk here at the University of Cambridge. He works largely with public-sector outreach and educational projects and the media, but also teaches an advanced course in Applied Bayesian Statistics, and retains a research interest in several practical areas of statistics, especially healthcare. He also runs the website Understanding Uncertainty, a resource for easy to read statistical treatments of various topics. He was talking at the Trinity Mathematical Society (TMS) on Monday 15^th of February 2010.