It was enlightening the first time one notices any kind of expression can be converted to a single tree of inlined anonymous functions and expressions.bar a b = a + b foo a b = bar a b + bar a b `foo` can be inlined as not to depend on `bar` foo = λa → λb → ((λc → λd → c + d) a b) + ((λc → λd → c + d) a b)
Might be obvious, but I don't know how to apply the same principle to types.data Bar a = B a a data Foo = Foo Bar Bar data Foo = forall a. Foo (B a a) (B a a) -- ?? Nonsense submitted by SrPeixinho
[link] [5 comments]
Create a tool to find type signatures that are less polymorphic than would be inferred by type inference.
This is a solution in search of a problem.
Earlier this week I gave a talk about the Curry-Howard isomorphism. Talks never go quite the way you expect. The biggest sticking point was my assertion that there is no function with the type a → b. I mentioned this as a throwaway remark on slide 7, assuming that everyone would agree instantly, and then we got totally hung up on it for about twenty minutes.
Part of this was my surprise at discovering that most of the audience (members of the Philly Lambda functional programming group) was not familiar with the Haskell type system. I had assumed that most of the members of a functional programming interest group would be familiar with one of Haskell, ML, or Scala, all of which have the same basic type system. But this was not the case. (Many people are primarily interested in Scheme, for example.)
I think the main problem was that I did not make clear to the audience what Haskell means when it says that a function has type a → b. At the talk, and then later on Reddit people asked
what about a function that takes an integer and returns a string: doesn't it have type a → b?
If you know one of the HM languages, you know that of course it doesn't; it has type Int → String, which is not the same at all. But I wasn't prepared for this confusion and it took me a while to formulate the answer. I think I underestimated the degree to which I have internalized the behavior of Hindley-Milner type systems after twenty years. Next time, I will be better prepared, and will say something like the following:
A function which takes an integer and returns a string does not have the type a → b; it has the type Int → String. You must pass it an integer, and you may only use its return value in a place that makes sense for a string. If f has this type, then 3 + f 4 is a compile-time type error because Haskell knows that f returns a string, and strings do not work with +.
But if f had the type a → b, then 3 + f 4 would be legal, because context requires that f return a number, and the type a → b says that it can return a number, because a number is an instance of the completely general type b. The type a → b, in contrast to Int → String, means that b and a are completely unconstrained.
Say function f had type a → b. Then you would be able to use the expression f x in any context that was expecting any sort of return value; you could write any or all of:3 + f x head(f x) "foo" ++ f x True && f x
and they would all type check correctly, regardless of the type of x. In the first line, f x would return a number; in the second line f would return a list; in the third line it would return a string, and in the fourth line it would return a boolean. And in each case f could be able to do what was required regardless of the type of x, so without even looking at x. But how could you possibly write such a function f? You can't; it's impossible.
Contrast this with the identity function id, which has type a → a. This says that id always returns a value whose type is the same as that if its argument. So you can write3 + id x
as long as x has the right type for +, and you can writehead(id x)
as long as x has the right type for head, and so on. But for f to have the type a → b, all those would have to work regardless of the type of the argument to f. And there is no way to write such an f.
Actually I wonder now if part of the problem is that we like to write a → b when what we really mean is the type ∀a.∀b.a → b. Perhaps making the quantifiers explicit would clear things up? I suppose it probably wouldn't have, at least in this case.
The issue is a bit complicated by the fact that the functionloop :: a -> b loop x = loop x
does have the type a → b, and, in a language with exceptions, throw has that type also; or consider Haskellfoo :: a -> b foo x = undefined
Unfortunately, just as I thought I was getting across the explanation of why there can be no function with type a → b, someone brought up exceptions and I had to mutter and look at my shoes. (You can also take the view that these functions have type a → ⊥, but the logical principle ⊥ → b is unexceptionable.)
In fact, experienced practitioners will realize, the instant the type a → b appears, that they have written a function that never returns. Such an example was directly responsible for my own initial interest in functional programming and type systems; I read a 1992 paper (“An anecdote about ML type inference”) by Andrew R. Koenig in which he described writing a merge sort function, whose type was reported (by the SML type inferencer) as [a] -> [b], and the reason was that it had a bug that would cause it to loop forever on any nonempty list. I came back from that conference convinced that I must learn ML, and Higher-Order Perl was a direct (although distant) outcome of that conviction.
Any discussion of the Curry-Howard isomorphism, using Haskell as an example, is somewhat fraught with trouble, because Haskell's type logic is utterly inconsistent. In addition to the examples above, in Haskell one can writefix :: (a -> a) -> a fix f = let x = fix f in f x
and as a statement of logic, is patently false. This might be an argument in favor of the Total Functional Programming suggested by D.A. Turner and others.
As discussed last time there's a deep-seated problem with performativity as a theory of social categorization. Specifically, it puts the focus on the wrong thing. That our actions are performative in nature gives us important insight into the role agency plays both in forming our own identities and in defending those identities against silencing, marginalization, oppression, and colonialism. But, by centering discussions of identity on our own personal agency we miss out on other important facets of the issue. When we say that someone belongs to a category, we do so because we've decided they belong to the category, or because we think they belong to the category. The statement that they belong to the category is not merely true (or false), we are projecting it to be true (or false). That is, we do not passively observe people's gender, race, class, etc; instead we actively project our own notions of gender, race, class, etc upon them. This projecting of beliefs onto others is called projectivism.
Interestingly, by localizing "truth" as the beliefs we hold to be true, the projective act is itself performative: by projecting something to be true, one comes to believe that it is true. And yet there is no reason to suppose these beliefs are correct (local truths need not be global truths), nor that they will agree with others' beliefs (local truths need not be true in other locales). Crucially, in the case of categorizing or identifying ourselves, we have access to our own personal thoughts, feelings, memories, subconscious inclinations, etc. Whereas, when others are categorizing us, they do not; they can only observe our bodies, our actions, and the results of our actions. Thus arises the discrepancy in cases like transgenderism. When self-identifying, we may well prize our internal observations over our externally observable state. Nevertheless, others will continue to project their categorizations upon us, regardless of our self-identification.
Not only do people project categories onto others, we do it compulsively. Our persistent and ubiquitous gendering of others is an especially powerful example, but it is in no way unique. Projecting race is another example. And in professional cultures where there are sharply contested borders between "tribes" (e.g., academia and hacker culture), projecting these "tribes" is yet another. This compulsive projectivism —or, more particularly, our unconsciousness of it— is where issues arise.
When we are not typically confronted with evidence that our projections are mistaken, our projectivism becomes almost unconscious. Once there, we fail to notice the fact that we are actively projecting and we come to believe we're passively observing truths about the world. So when our projections turn out to be mistaken, we get a feeling of betrayal, we feel like the person whose identity we were mistaken about was "lying" to us. This subsequent projection that they were "lying" stems from the fact that we mistook our earlier projections for mere observations. Thus, because of an original error on our part, we end up imputing that others are being dishonest or deceptive.
When the identity one desires to be seen as (which may differ from the identity they claim for themselves) is often or easily at odds with the identities projected upon them, they understandably become concerned about trying to avoid these projections of "lying". If one can successfully avoid projections of "lying" they are said to "pass", terminology which turns around and suggests that they were in fact lying the whole time and only managed not to get caught. This terminology is, of course, deeply problematic.
Simply acknowledging compulsive projectivism is not enough. To undo the damage caused by misgendering, racial profiling, stereotyping, and other misprojections, we must lift this knowledge up and remain consciously aware that the beliefs we project onto others are not an observation of their identities. We must denaturalize the projectivist assumption that our beliefs are others' truths, by discontinuing the use words like "passing" which rely on that assumption. And when we feel betrayed we must locate that feeling within ourselves and stop projecting it in bad faith. The performative theory highlights the positive role of agency in our lives, but agency alone is not enough. The projectivistic theory extends this to highlight the negative role of agency when used to deny or overwhelm the agency of others.
 I do not mean this terminology to be the same as Hume's notion of projectivism, though of course both terms have the same etymology. Hume's projectivism is popular in the ethics literature, with which I am largely unfamiliar; thus, my use of the term here is not meant to entail whatever baggage it may have accrued in that literature.
 While it is not usually presented as such, Austin's original definition of performative speech acts should also only hold up to localized truth. In the classical example "I now pronounce you married", by saying the words one does the deed of pronouncing the couple to be married. However, the pronouncement of marriage does not cause the couple to be married in a universal sense; it only causes them to be married in the current jurisdiction, and a different jurisdiction may or may not recognize that marriage as valid. Because the marriage must be localized, therefore the pronouncement of marriage must be localized: one can't pronounce a couple to be married (everywhere), they can only pronounce them to be married (here, or there, or wherever). Thus, the deed performed by the utterance of the words is a localized deed: the pronouncement of a localized wedding.Twitter Facebook Google+ Tumblr
I've forgotten how to write. Somewhere along the way I've forgotten how to say, what I mean. Little sticks and thistles, they burrow under your skin like dry wind and the leaves you brush from your faces. And you find yourself there, looking over, looking out, and turn to tell another how you came to this place, this pretty place, and all you find are tangled weeds and hills and where was the path where you left that friend you thought had come with you
I have half a dozen half written posts, if half written means written and my mind keeps telling me to edit to edit to go over once more, unable to let go, unable to let slip a word lest it falls all out and i somehow say what i somehow mean and someone takes offense. Offence. That word of our times, that police baton with which we beat the helpless, refuse to listen to the stories, those stories once heard we proclaim have "set us free" but we leave the authors beaten, unwilling to look at their lives lest we feel too closely the grip of that truncheon in our fist.
Half a dozen half written posts, weeks of thoughts writ out, on programs and mathematics and words and history. Thoughts I cannot set free. They haunt me, they call me beckoning to spill once again that mental blood to pore and pore over them and wring them dry of every drip of humanity so I can hang out the scraps and let others see how terribly clever i am. I never wanted to be clever, never wanted to be seen like that. I only wanted, once, to be free. From the heartache of a harrowing life, from the illusions and false idols, from my own ignorance. And now these thoughts tie me up in clever little knots, and have me writing bad poetryTwitter Facebook Google+ Tumblr
I'd like to take this moment to point out that all forms of binarism are bad. (Including the binarist notion that all things are either "good" or "bad".) I feel like this has to be pointed out because we, every one of us, has a nasty habit: in our overzealousness to tear down one binary, we do so by reinforcing other binaries. So let me say again. All forms of binarism are bad.
It's well-known that I've had a long, fraught history with certain "feminist" communities, due to which I have heretofore disavowed that label. Because of these persistent conflicts, around ten years ago I retreated from feminist circles and communities. However, over the past year I have rejoined a number of feminist circles— or rather, I have joined womanist, black feminist, transfeminist, and queer feminist circles. And thanks to this reinvolvement with feminist activism I have come, once again, to feel a certain attachment to that word: "feminist". The attachment feels strange to me now, having disavowed it for so long in favor of "womanism", "black feminism", "transfeminism", and "queer feminism". But because of this attachment I feel, once more, the need to reclaim feminism away from those "feminist" communities whose philosophy and political methods I continue to disavow.
So, to piss everyone off once more: ( a manifesto. )
Edit 2014.07.13: Added footnotes  and .Twitter Facebook Google+ Tumblr
Although the words have superficially similar pronunciations, performativity and performance are two extremely different notions. In her book Gender Trouble (1990) and its sequel Bodies That Matter (1993), Judith Butler put forth the thesis that gender identity is performative. Over the last decade performance-based theories of gender and identity have become popular, even mainstream, despite a number of deep-seated and readily-apparent flaws. Unfortunately, these latter performance-based theories are often portrayed as successors of Butlerean performativity. They're not.
To understand performativity one should go back to Austin's original definition of performative speech acts. Whenever we speak, we speak for a reason. Austin was interested in explaining these reasons— in particular, explaining the contrast between what we say and why we say it. When we ask "could you pass the salt?" we are not literally interested in whether the addressee is capable of moving the salt shaker, we're making a request. When we ask "how do you do?" or "what's up?" we do not actually want an answer, we are merely greeting someone. It is within this context of discussing the why behind what we say that Austin became interested in performative speech acts: speech acts which through their very utterance do what it is they say, or speech acts which are what it is they mean. When the right person in the right context utters "I now pronounce you married", that vocalization is in fact the pronouncement itself. To state that you pronounce something, is itself to make the proclamation. In just the same way, when under the right circumstances someone says they promise such-and-so, they just did.
There are a number of interesting details about what it means to be a performative speech act. For instance, just uttering the words is not enough: if a random stranger comes up to you and pronounces you married, that does not actually mean you're married. For the performative speech act to have any force it must be uttered in a felicitous context (e.g., the words must be spoken with the proper intent, the pronouncer of marriage must be ordained with the ability to marry people, the partners must be willing, the pairing must be of an appropriate sort according to the bigotry of the times, etc). Another detail is that performative speech acts do more than just enact what they say, they also create something: pronouncing a marriage constructs the marriage itself, declaring war brings the war into existence, giving a promise makes the promise, sentencing someone creates the sentence, etc. Because of details like these, claiming that a particular speech act is performative says a heck of a lot more than just saying the act was performed (i.e., spoken).
On the other hand, a performance is the enactment of a particular variety of artistic expression ranging from theatrical plays, to musical opuses, to religious ceremonies, to performance art, and so on. Whether a particular act is performative is independent of whether it is (a part of) a performance. Many performative speech acts are of a ceremonial nature (e.g., marriages, divorces, christenings, declarations of war, etc) and consequently we like to make a big affair of it. Thus, these particular acts tend to be both: they're performative performances. However, many other performative speech acts are executed with little fanfare: ordering food in a restaurant, apologizing, accepting apologies, resigning from a game, etc. These are all performative acts, and yet there's absolutely no need for any sort of performance behind them. Indeed we often find it humorous, or rude, or severe, when someone chooses to turn these performative acts into performances.
The distinction between performativity and performance is crucial to understanding the thesis Butler put forth. We can expand the idea of performativity to include not just speech acts, but other acts as well. Doing so, Butler's thesis is that one's identity as a particular gender is not something which exists a priori, but rather that it is constructed by the enactment —and especially the continuous ritualistic re-enactment— of performative gender actions. The specific claim being made is that one's gender identity is an artifact whose ontological existence arises from particular deeds, in the exact same way that a marriage is an artifact arising from nuptial ceremony, that a promise is an artifact arising from the swearing of a vow, that a state of war is an artifact arising from the declaration of its existence, and so on. The performative theory of gender is often paraphrased as "gender is something we do"— but this paraphrase is grossly misleading. The paraphrase elides the entire specific content of the thesis! Sure, gender is something we do, but it's something we do in very specific ways and it is in virtue of doing those things in those ways that we come to identify with our gender. That's the thesis.
As discussed before, there are some crucial issues with performativity as a theory of gender. (Though these issues can be corrected by changing the focus without giving up the crucial insight.) But the issue with performativity has nothing whatsoever to do with the fact that performances are artificial, that performances are interruptible, that performances can be altered on whimsy, that performances can be disingenuous, that performances are "only" art, etc. Those latter complaints are why performance-based theories of gender are flat out wrong. And they're evidence of why claiming that performance-based theories were built upon performative theories grossly misconceptualizes performativity.
 Don't take my word for it, Butler herself has continually argued that performance-based theories are a gross misinterpretation of her work (Gender Trouble, xxii–xxiv; Bodies That Matter, 125–126; "Gender as Performance: An interview with Judith Butler", 32–39; Judith Butler (by Sara Salih), 62–71).Twitter Facebook Google+ Tumblr
A lot of ink has been spilt over trigger warnings lately. And I don't want to write about it because I feel like I don't have much to add to the conversation. But when I stop, that feeling nags at me. You can't think with your mouth open; and as someone who always had issues keeping her damn mouth shut, it took me a long time to learn that to listen you must be silent. ... And yet. ... And yet, when someone experiences strong emotions about her own marginalization, but feels compelled to self-silence: that's when you need to listen harder.
Because there are a lot of voices I know full well, and I don't hear them talking.
I know them because they're the voices of my friends, and among friends we talk about things we don't talk about. In the workaday world we put on our faces and never hint at the behemoths raging through our china cabinets. And when we let down our hair, those faces stay on, because you always know who might be listening. And behind closed doors, still, we keep them on because elsewise love would be too tragic. But in secret spaces, we talk. We are, every one of us, damaged. I may not know who hurt you yet, I may not know your story of pain, but I never assume there isn't one; because every single person I've known, when we get close enough, they tell me things we don't talk about. Sometimes it takes years before they feel safe enough, sometimes they never feel safe enough, but if they've ever lowered their guard to me, they've told me. Every. single. person.
We are born and raised and live in a world drenched in abuse. And that abuse doesn't leave scars, it leaves open wounds waiting to have dirt rubbed in them. The first rule of what doesn't happen is that it cannot be spoken of. So healing only happens in those secret spaces, one-on-one, in the dark of night, far far from friends and strangers alike. This privatization of healing only compounds the problem of abuse. When we cannot see past others' faces, when we cannot see the living wounds they bear, when we do not hear their daily resistance against reiterations of violence, we come to think that somehow maybe they haven't been hurt as badly as we. When we see our own people succeed, or see leaders of resistance and "survivors" and "healed" voices speaking up against the injustice of the world, we think that somehow maybe they must be stronger than us, more resilient than us, more determined than us. When we cannot witness their struggle, we think that somehow maybe when they go to bed at nights they need not take the time to scrub out that daily dirt from their wounds. And when we cannot bear that witness, we see ourselves as lesser, broken, impostors.
These are the voices I do not hear speaking out, or hear speaking in only roundabout whispers. These are the voices for whom trigger warnings are writ. As so precisely put by Aoife,
Here's something I need you to understand: the vast majority of students when 'triggered' don't write howlers to department heads or flip laptops over in crowded classrooms for YouTube counts.
On the contrary, they most often shut down and collapse into numbness.
That numbness, that collapse, is the last tool our minds have to keep our faces in place when some sudden shock reopens sore wounds. The second rule of what we do not talk about is that wounds never heal, not entirely. We —some of us— can manage not flinching when someone raises their hand. We —some of us— learn to laugh along when someone touches our back. We —some of us— learn to feel safe in a room alone with a man. We —some of us— learn to turn blind to the "tranny" jokes, to the blackface, to the jibes about trailer parks and country living, to the "sex" scene where she lay sleeping, the scene where he takes the other man 'round back, the man who slaps his wife, the mother who cuffs her child, being told to go pick a switch, to the child starving on the street, to the college kids playing "tricks" on the homeless. We —some of us— learn to live as stone. But stone don't heal, and we all have our rituals of self-care we won't talk about. But when everywhere all you ever see is stone, you know your flesh will never make it if the light still shines in your eyes.
And I too am guilty of this silence culture. Because the fact of the matter is, in this day and age, to speak is to jeopardize my career. I can talk about being trans or being a dyke, and I can at least pretend that the laws on the books will mean a damn. But if I talk about my childhood, I won't be seen as an adult. If I talk about my abuse, I won't be seen as stable. If I bring up my mental life, I won't be seen as professional. If I talk about spoons, I won't be seen as reliable. And so I stuff it down and self-silence and hide what it's like, that daily living with depression and PTSD, til some trigger sets it off and out comes that rage which grows on silence. Some full-force punch to the gut, some words like "I'm not sure suicide is ever the answer" and my eyes go black, and words come out, and they sound nice enough, but every one means "I hate you".
Not to be rude, but sometimes suicide is the answer. It may not be the best answer, but it is an answer. And, unfortunately, sometimes that is all that's required. Sometimes a terrible fucking answer is the only answer to be found.
I say this as someone who's spent more of her life being suicidal than not, as someone who's survived multiple attempts, as someone whose friends have almost invariably spent years being suicidal. Yes, it sucks. And no, it doesn't "solve" anything. But think of the suffering of the victim. It is incredibly difficult to overcome the self-preservation instinct. Profoundly difficult. Imagine the volume of suffering it takes, the depths and duration of misery required to actively overcome the single most powerful compulsion any living creature can experience. There comes a point, long after endurance has already given out, when the full weight of that volume cannot be borne.
Whenever this happens, my thoughts are always with the victim. I cannot help but empathize with that terrible terrible suffering
Because the fact of the matter is, I'm too scared to talk. We live in a culture where suicide is "the easy way" and you're supposed to "take it like a man", but the fact of the matter is noone can take it. We are, every one of us, damaged. We privatize our healing because the first rule of abuse is that it must never be mentioned, must never never be discussed. The learning of silence is the first abuse: it is how we are taught to abuse ourselves, to never never hear that we're not alone.
This isn't about suicide and depression. Isn't about rape and racism. Isn't about violence and neglect. This is about silence. About the words we don't use to not say what you can't talk about. This is about learning to speak using words. About how we must open our mouths in order to listen.Twitter Facebook Google+ Tumblr
We are having another PLATEAU workshop at SPLASH 2014. We have a new category for "Hypotheses Papers" and thought this would be particularly appealing to the LTU community.
Programming languages exist to enable programmers to develop software effectively. But how efficiently programmers can write software depends on the usability of the languages and tools that they develop with. The aim of this workshop is to discuss methods, metrics and techniques for evaluating the usability of languages and language tools. The supposed benefits of such languages and tools cover a large space, including making programs easier to read, write, and maintain; allowing programmers to write more flexible and powerful programs; and restricting programs to make them more safe and secure.
PLATEAU gathers the intersection of researchers in the programming language, programming tool, and human-computer interaction communities to share their research and discuss the future of evaluation and usability of programming languages and tools.
Some particular areas of interest are:
- empirical studies of programming languages
- methodologies and philosophies behind language and tool evaluation
- software design metrics and their relations to the underlying language
- user studies of language features and software engineering tools
- visual techniques for understanding programming languages
- critical comparisons of programming paradigms
- tools to support evaluating programming languages
- psychology of programming
PLATEAU encourages submissions of three types of papers:
Research and position papers: We encourage papers that describe work-in-progress or recently completed work based on the themes and goals of the workshop or related topics, report on experiences gained, question accepted wisdom, raise challenging open problems, or propose speculative new approaches. We will accept two types of papers: research papers up to 8 pages in length; and position papers up to 2 pages in length.
Hypotheses papers: Hypotheses papers explicitly identify beliefs of the research community or software industry about how a programming language, programming language feature, or programming language tool affects programming practice. Hypotheses can be collected from mailing lists, blog posts, paper introductions, developer forums, or interviews. Papers should clearly document the source(s) of each hypothesis and discuss the importance, use, and relevance of the hypotheses on research or practice. Papers may also, but are not required to, review evidence for or against the hypotheses identified. Hypotheses papers can be up to 4 pages in length.
Papers will be published in the ACM Digital Library at the authors’ discretion.
Workshop paper submission due - 1 August, 2014
Notification to authors - 22 August, 2014
Early registration deadline - 19 September, 2014
Josh Bloch, former Chief Java Architect at Google and Distinguished Engineer at Sun Microsystems.
I want to archive this solution if anyone else hits this issue.
On x64 Windows installing 2013.2.0.0
Close all running programs, for me this included but was not limited to eclipse, skype and firefox. I'm not sure which was the offender. Now re-run the installer and you should be good.
This post doesn't need upvotes but i want to store this solution for if someone searches for it.submitted by IndexPlusPlus
[link] [1 comment]
Sorry for the newbie question, but Google wasn't too helpful (googlebot doesn't parse punctuation).
Is there a Monadic equivalent to <* that I've overlooked? Something like << ? I find myself doing things like:mobitFoo >>= (\x -> bar >> return x)
ordo x <- mobitFoo bar return x
Also, I've only been Haskelling for a little while, and I'm pretty sure this is reason #97 I've found why Applicative should be a superclass of Monad.submitted by Considered_Harmful
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Hi! I've started reading Learn you a haskell for great good! as I suppose most of you guys have read. I'm at the data type section where the author is making a 3D vector type. He goes on saying
Let’s implement a 3D vector type and add some operations for it. We’ll make it a parameterized type, because even though it will usually contain numeric types, it will still support several of them, like Int, Integer, and Double, to name a few.
data Vector a = Vector a a a
But what I truly want is "data Vector = Vector Num Num Num" right? Why is this not possible (or if it is, how to do it?). Num takes a parameter so I tried something like "data Vector a = Vector (Num a) (Num a) (Num a)" but this did not work. So what's the deal?
Thanks for the help!submitted by frerom
[link] [36 comments]
-- Sample test data
type Sales = (String, String, Int)
-- First String is title of track, Second String is artist name, Int is number of tracks sold
testData :: [Sales]
testData :: [("Me and My Broken Heart","Rixton",5),("It’s My Birthday","will.i.am",4),("Problem","Ariana Grande",3)]
-- record a sale of a track
sellTrack :: String String Int -> [Sales]
sellTrack :: trackName artistName numberSold = [Sales] ++ [(trackName, artistName, numberSold)]
I get the error "illegal literal in type : "Me and My Broken Heart"
Thanks in advance.submitted by Rifaz1
Just a quick note on the relationship between ESI scores and allele frequencies. Allele frequency differences is of course related to – perhaps even the definition of – diversification, but the information we gain from observing an allele also depends on the specific allele frequencies involved. The graph below shows how this is related.
Each line represents a fixed allele difference, from 0.05 at the bottom, to 0.95 at the top, and the x-axis is the average allele frequency between populations. We see that for small differences, the actual frequencies matter little, but for moderate to large allele differences, allele frequencies near the extremes have a large effect.
Note that this is information per allele, and thus not ESI (which is the expected information from observing the site, in other words a weighted average over all alleles).