As other speakers have said, it's a rather daunting experience -- a particularly daunting experience -- to be speaking in front of this audience. But unlike the other speakers, I'm not going to tell you about the mysteries of the universe, or the wonders of evolution, or the really clever, innovative ways people are attacking the major inequalities in our world. Or even the challenges of nation-states in the modern global economy. My brief, as you've just heard, is to tell you about statistics -- and, to be more precise, to tell you some exciting things about statistics. And that's -- (Laughter) -- that's rather more challenging than all the speakers before me and all the ones coming after me. (Laughter) One of my senior colleagues told me, when I was a youngster in this profession, rather proudly, that statisticians were people who liked figures but didn't have the personality skills to become accountants. (Laughter) And there's another in-joke among statisticians, and that's, "How do you tell the introverted statistician from the extroverted statistician?" To which the answer is, "The extroverted statistician's the one who looks at the other person's shoes." (Laughter) But I want to tell you something useful -- and here it is, so concentrate now. This evening, there's a reception in the University's Museum of Natural History. And it's a wonderful setting, as I hope you'll find, and a great icon to the best of the Victorian tradition. It's very unlikely -- in this special setting, and this collection of people -- but you might just find yourself talking to someone you'd rather wish that you weren't. So here's what you do. When they say to you, "What do you do?" -- you say, "I'm a statistician." (Laughter) Well, except they've been pre-warned now, and they'll know you're making it up. And then one of two things will happen. They'll either discover their long-lost cousin in the other corner of the room and run over and talk to them. Or they'll suddenly become parched and/or hungry -- and often both -- and sprint off for a drink and some food. And you'll be left in peace to talk to the person you really want to talk to.
Beste hizlariek esan duten moduan, nahiko esperientzia beldulgarria da - esperientzia bereziki beldulgarria da - entzuleria honen aurrean hitz egitea. Baina besteek ez bezala, nik unibertsoko misterioei edo eboluzioaren edertasunari edo gure munduko desberdintasun handienei aurre egiteko erabiltzen ari diren modu berritzaileei buruz hitz egingo dizuet. Edo ekonomia global modernoan nazioek dituzten erronkei buruz. Nire lana, estatistikaz hitz egitea da -- hobe esanda, estatistikaren gauza liluragarriak kontatzea. Eta hori... (barreak) hori nire aurrekoek egindakoa, eta ondorendoek egingo dutena baino zailagoa da. (barreak) Lanbide honetan berria nintzenean, lankide batek esan zidan estatistikariak zenbakiak maite zituzten, baina kontable izateko pertsonalitaterik ez zuten pertsonak zirela. (barreak) Estatistikoen arteko beste txiste batek dio: "Nola ezberdindu estatistikari introbertitu bat estatistikari extrobertitu batengandik?" "Estatistikari extrobertitua beste pertsonaren zapatetara begiratzen duena da" (barreak) Baina gauza bat esan nahi dizuet - eta orain doa, beraz adi. Gaur harrera bat dago Unibertsitateko Natur Zientzien Museoan. Ikusiko duzuen bezala, toki zoragarri bat da, tradizio victoriar hoberenaren ikono handi bat. Nekez gertatuko da, toki berezi horretan, hainbeste jende artean, baina gerta daiteke, nahi ez duzuen norbaitekin hitz egiten amaitzea. Hau da egin behar duzuena. "Zein da zure lanbidea?" galdetzean, "Estatistikaria naiz" erantzun. (barreak) Beno, orain abisatuta zaudete, eta asmatzen ari zaretela jakingo du, baina bestela, bi gauza pasa daitezke. Gelaren beste puntan lehengusu bat aurkituko du, eta harekin hitz egitera joango da, edo bapatean goseak eta egarriak ipiniko da -askotan biak- eta edateko eta jateko zerbaiten bila joango da. Eta zu lasai geratuko zara, benetan hitz egin nahi duzunarengana joateko.
It's one of the challenges in our profession to try and explain what we do. We're not top on people's lists for dinner party guests and conversations and so on. And it's something I've never really found a good way of doing. But my wife -- who was then my girlfriend -- managed it much better than I've ever been able to. Many years ago, when we first started going out, she was working for the BBC in Britain, and I was, at that stage, working in America. I was coming back to visit her. She told this to one of her colleagues, who said, "Well, what does your boyfriend do?" Sarah thought quite hard about the things I'd explained -- and she concentrated, in those days, on listening. (Laughter) Don't tell her I said that. And she was thinking about the work I did developing mathematical models for understanding evolution and modern genetics. So when her colleague said, "What does he do?" She paused and said, "He models things." (Laughter) Well, her colleague suddenly got much more interested than I had any right to expect and went on and said, "What does he model?" Well, Sarah thought a little bit more about my work and said, "Genes." (Laughter) "He models genes."
Gure lanbidearen erronketako bat egiten duguna azaltzea da. Afari, hitzaldi eta horrelakoetara ez gaituzte gonbidatzen. Sekula ez dut asmatu hori nola lortu. Baina nire emazteak - orduan nire neskalagunak - nik sekula lortu ez dudana lortu zuen. Duela urte asko, elkarrekin hasi ginenean, berak BBCrako lan egiten zuen, Britainia Handian, eta ni une horretan Estatu Batuetan nengoen lanean. Bera bisitatzera joan nintzen batean, bere laneko batek "eta zure mutil lagunak zer egiten du?" galdetu zion Sarah-k nik azaldu nizkion gauzei buruz pentsatu, egun haietan entzuten arreta jarri zuen, (barreak) Ez esan halakorik esan dudanik. eboluzioa eta genetika ulertzeko eredu matematikoak garatzen burutu nuen lanean pentsatu zuen eta bere lankideak "zer egiten du?" galdetzean Sarah-k etenaldi bat egin eta esan zion "gauzak modelatzen ditu". (barreak) Bere lankidea, bapatean, espero zitekeena baina gehiago interesatu zen eta jarraitu zuen "zer modelatzen du?" Sarh-k nire lanean pixka bat gehiago pentsatu eta "geneak" esan zion (barreak) "geneak modelatzen ditu".
That is my first love, and that's what I'll tell you a little bit about. What I want to do more generally is to get you thinking about the place of uncertainty and randomness and chance in our world, and how we react to that, and how well we do or don't think about it. So you've had a pretty easy time up till now -- a few laughs, and all that kind of thing -- in the talks to date. You've got to think, and I'm going to ask you some questions. So here's the scene for the first question I'm going to ask you. Can you imagine tossing a coin successively? And for some reason -- which shall remain rather vague -- we're interested in a particular pattern. Here's one -- a head, followed by a tail, followed by a tail.
Hau da nire bizitzako amodioa, eta hortaz pixka bat hitz egingo dut. Gure munduan zoriak eta probabilitateak duten lekuan pentsatzea nahi dut, eta horren aurrean nola jokatzen dugun. Orain arte nahiko erraza izan da, orain arteko hitzaldietan barre batzuk egin dituzue. Orain pentsatu egin behar duzue, galderak egingo dizkizuet. Beraz, hau da lehen galderaren eszenatokia: Imaginatu zaitezte txanpon bat behin eta berriz airera botatzen eta arrazoi batengatik, ez dugu zehaztuko zergatik, patroi zehatz batetan interesa dugu. Esaterako: aurpegia, gurutzea, gurutzea.
So suppose we toss a coin repeatedly. Then the pattern, head-tail-tail, that we've suddenly become fixated with happens here. And you can count: one, two, three, four, five, six, seven, eight, nine, 10 -- it happens after the 10th toss. So you might think there are more interesting things to do, but humor me for the moment. Imagine this half of the audience each get out coins, and they toss them until they first see the pattern head-tail-tail. The first time they do it, maybe it happens after the 10th toss, as here. The second time, maybe it's after the fourth toss. The next time, after the 15th toss. So you do that lots and lots of times, and you average those numbers. That's what I want this side to think about.
Beraz txanpon bat behin eta berriz jaurtitzen dugu. Eta... aurpegia-gurutzea-gurutzea, gure patroia agertzen da. Kontatu eta bat, bi, hiru, lau, bost, sei, zazpi, zortzi, bederatzi, hamar, hamargarren jaurtiketaren ostean gertatu da. Gauza interesgarriagoak egin daitezkeela pentsatuko duzue, baina jarrai iezaidazue une batez. Imajinatu entzuleriaren alde honetako bakoitzak txanpon bat atera eta aurpegi-gurutze-gurutze patroia atera arte jaurtitzen duela. Egiten duten lehen aldian agian hamargarren jaurtiketan gertatzen da. Bigarrenean agian laugarrenean. Eta ondoren hamabosgarrenean. Beraz txanpona askotan botatzen duzue, eta zenbaki horien bataz bestekoa kalkulatzen duzue. Horretan pentsatzea nahi dut.
The other half of the audience doesn't like head-tail-tail -- they think, for deep cultural reasons, that's boring -- and they're much more interested in a different pattern -- head-tail-head. So, on this side, you get out your coins, and you toss and toss and toss. And you count the number of times until the pattern head-tail-head appears and you average them. OK? So on this side, you've got a number -- you've done it lots of times, so you get it accurately -- which is the average number of tosses until head-tail-tail. On this side, you've got a number -- the average number of tosses until head-tail-head.
Entzuleriaren beste aldeak ez du aurpegi-gurutze-gurutze nahi, arrazoi kulturalengatik aspergarria dela uste dute, eta gehiago gustatzen zaie aurpegi-gurutze-aurpegi patroia. Beraz hemen ere, txanponak atera eta jaurti eta jaurti hasten dira. Jaurtiketak kontatzen dituzte aurpegi-gurutze-aurpegi patroia atera arte. Eta bataz bestekoa ateratzen dute, ados? Beraz alde honetan zenbaki bat dute, askotan egin dute beraz zenbakia zehatza da, aurpegi-gurutze-gurutze lortu arte behar diren jaurtiketen bataz bestekoa da. Hemen beste zenbaki bat dute, aurpegi-gurutze-aurpegi lortu harteko bataz besteko jaurtiketa kopurua.
So here's a deep mathematical fact -- if you've got two numbers, one of three things must be true. Either they're the same, or this one's bigger than this one, or this one's bigger than that one. So what's going on here? So you've all got to think about this, and you've all got to vote -- and we're not moving on. And I don't want to end up in the two-minute silence to give you more time to think about it, until everyone's expressed a view. OK. So what you want to do is compare the average number of tosses until we first see head-tail-head with the average number of tosses until we first see head-tail-tail.
Hemen gauza matematiko sakon bat topatuko dugu, bi zenbaki badituzu, hiru gauza gerta daitezke. Edo berdinak dira, bat bestea baino handiagoa da, edo alderantziz. Beraz, hemen zer gertatzen da? Guztiok pentsatu behar duzue, eta guztiok erantzun behar duzue, bestela ez dugu jarraituko. Eta ez dut bi minutuko isilunearekin amaitu nahi guztioi erantzuteko denbora emateko. aurpegi-gurutze-aurpegi patroia lortu arte behar ditugun bataz besteko jaurtiketa kopurua aurpegi-gurutze-gurutze patroia lortu arte behar ditugunekin konparatu behar duzue.
Who thinks that A is true -- that, on average, it'll take longer to see head-tail-head than head-tail-tail? Who thinks that B is true -- that on average, they're the same? Who thinks that C is true -- that, on average, it'll take less time to see head-tail-head than head-tail-tail? OK, who hasn't voted yet? Because that's really naughty -- I said you had to. (Laughter) OK. So most people think B is true. And you might be relieved to know even rather distinguished mathematicians think that. It's not. A is true here. It takes longer, on average. In fact, the average number of tosses till head-tail-head is 10 and the average number of tosses until head-tail-tail is eight. How could that be? Anything different about the two patterns? There is. Head-tail-head overlaps itself. If you went head-tail-head-tail-head, you can cunningly get two occurrences of the pattern in only five tosses. You can't do that with head-tail-tail. That turns out to be important.
Zeintzuk uste dute A egia dela, bataz beste denbora gehiago beharko dela aurpegi-gurutze-aurpegi lortzeko aurpegi-gurutze-gurutze baino? Nork uste du B egia dela, batez bestekoa berdina dela? Nork uste du C egia dela, bataz beste denbora gutxiago beharko dela aurpegi-gurutze-gurutze lortzeko aurpegi-gurutze-gurutze lortzeko baino? Ados, nor falta da erantzuteko? Hori bihurrikeria bat da, erantzun egin behar zela esan dut. (barreak) Ados, gehiengoak uste du B dela egia. Eta lasai, matematikari ezagun batzuek ere hori pentsatzen dute eta. Baina ez, A da egia. Bataz beste denbora gehiago behar du. Izatez, aurpegi-gurutze-aurpegi lortzeko bataz besteko jaurtiketa kopurua 10 da eta aurpegi-gurutze-gurutze lortzeko 8. Nola da posible hau? Patroietan desberdintasunen bat dago? Bai. aurpegi-gurutze-aurpegi gainjarri egiten da. Aurpegi-gurutze-aurpegi bilatzen baduzu, zortearekin patroiaren bi sekuentzia lor ditzakezu bost jaurtiketatan. Hori ezin duzu aurpegi-gurutze-gurutze patroiarekin lortu. Eta hori garrantzitsua da.
There are two ways of thinking about this. I'll give you one of them. So imagine -- let's suppose we're doing it. On this side -- remember, you're excited about head-tail-tail; you're excited about head-tail-head. We start tossing a coin, and we get a head -- and you start sitting on the edge of your seat because something great and wonderful, or awesome, might be about to happen. The next toss is a tail -- you get really excited. The champagne's on ice just next to you; you've got the glasses chilled to celebrate. You're waiting with bated breath for the final toss. And if it comes down a head, that's great. You're done, and you celebrate. If it's a tail -- well, rather disappointedly, you put the glasses away and put the champagne back. And you keep tossing, to wait for the next head, to get excited.
Bi modu daude honen inguruan pentsatzeko. Bat erakutsiko dizuet. Imajinatu, demagun egiten ari garela. Alde honetan, gogoratu aurpegi-gurutze-gurutze eta zuek aurpegi-gurutze-aurpegi. Txanpona jaurti eta aurpegia ateratzen da, zuen eserlekuaren iskinan zaudete, zerbait handia ederra edo sinesgaitza gerta daitekeelako. Bigarren jaurtiketa gurutzea ateratzen da, benetan gustora zaudete. Txanpaina izotzetan sartuta dago, eta kopak ospatzeko prest daude. Bihotza abiada bizian duzue azken jaurtiketan. Aurpegia ateratzen bada izugarria izango da. Lortu eta ospatu egingo duzue. Gurutzea ateratzen bada, beno etsigarria da, kopak gorde eta txanpaina bere lekuan uzten duzue. Eta jaurtitzen jarraitzen duzue, hurrengo aurpegiaren zain.
On this side, there's a different experience. It's the same for the first two parts of the sequence. You're a little bit excited with the first head -- you get rather more excited with the next tail. Then you toss the coin. If it's a tail, you crack open the champagne. If it's a head you're disappointed, but you're still a third of the way to your pattern again. And that's an informal way of presenting it -- that's why there's a difference. Another way of thinking about it -- if we tossed a coin eight million times, then we'd expect a million head-tail-heads and a million head-tail-tails -- but the head-tail-heads could occur in clumps. So if you want to put a million things down amongst eight million positions and you can have some of them overlapping, the clumps will be further apart. It's another way of getting the intuition.
Alde honetan esperientzia ezberdina da. Berdina da sekuentziaren lehen bi zatitan. Pixka bat gustora zaudete lehen aurpegiarekin, eta oso gustura hurrengo gurutzearekin. Orduan txanpona jaurtitzen duzue. Gurutzea bada txanpaina irekitzen duzue. Aurpegia bada, etsigarria da, baina zuen patroiaren herena badaukazue jada. Eta hori aurkezteko modu ez formala litzateke, baina hori da desberdintasuna. Ikusteko beste modu bat, txanpona 8 milioi aldiz botatzen badugu, milioi bat aurpegi-gurutze-aurpegi esperoko genituzke eta milioi bat aurpegi-gurutze-gurutze, baina aurpegi-gurutze-gurutzeak multzoka ager daitezke. Beraz, milioi bat gauza zortzi milioi posiziotan ipintzen badituzue eta gainjartze apur bat onartzen baduzue, multzoak elkarrengandik hurrunago egongo dira. Hau ulertzeko beste modu bat da.
What's the point I want to make? It's a very, very simple example, an easily stated question in probability, which every -- you're in good company -- everybody gets wrong. This is my little diversion into my real passion, which is genetics. There's a connection between head-tail-heads and head-tail-tails in genetics, and it's the following. When you toss a coin, you get a sequence of heads and tails. When you look at DNA, there's a sequence of not two things -- heads and tails -- but four letters -- As, Gs, Cs and Ts. And there are little chemical scissors, called restriction enzymes which cut DNA whenever they see particular patterns. And they're an enormously useful tool in modern molecular biology. And instead of asking the question, "How long until I see a head-tail-head?" -- you can ask, "How big will the chunks be when I use a restriction enzyme which cuts whenever it sees G-A-A-G, for example? How long will those chunks be?"
Zer esan nahi dut? Oso adibide sinplea da, probabilitate galdera xume bat, eta guztiek, eta lagunarte onean zaudete, gaizki erantzuten dute. Hau da nire pasioarekin, genetikarekin, lotuta nire dibertimentu txikia. Bada erlazio bat, aurpegi-gurutze-aurpegi eta aurpegi-gurutze-gurutzeren artean. Eta hau da. Txanpona jaurtitzean, aurpegi eta gurutzeen sekuentzia bat lortzen duzu. DNA ikustean sekuentzia bat dago, baina ez bi gauzena soilik, lau hizkiena baizik, A, G, C eta T. Eta guraize kimiko txiki batzuk daude, errestrikzio entzimak, patroi jakin bat ikustean DNA mozten dutenak. Biologia molekular modernoan oso erabilgarriak diren tresna bat dira. Eta "aurpegi-gurutze-aurpegi lortzeko zenbat jaurtiketa behar dira?" galdetu beharrean, "zein tamaina izango dute adibidez G-A-A-G patroia ikustean mozten duten errestrikzio entzimak erabiltzen baditut?" galdetu dezakegu. Ze tamainako pusketak izango ditut?
That's a rather trivial connection between probability and genetics. There's a much deeper connection, which I don't have time to go into and that is that modern genetics is a really exciting area of science. And we'll hear some talks later in the conference specifically about that. But it turns out that unlocking the secrets in the information generated by modern experimental technologies, a key part of that has to do with fairly sophisticated -- you'll be relieved to know that I do something useful in my day job, rather more sophisticated than the head-tail-head story -- but quite sophisticated computer modelings and mathematical modelings and modern statistical techniques. And I will give you two little snippets -- two examples -- of projects we're involved in in my group in Oxford, both of which I think are rather exciting. You know about the Human Genome Project. That was a project which aimed to read one copy of the human genome. The natural thing to do after you've done that -- and that's what this project, the International HapMap Project, which is a collaboration between labs in five or six different countries. Think of the Human Genome Project as learning what we've got in common, and the HapMap Project is trying to understand where there are differences between different people.
Hau probabilitatearen eta genetikaren arteko azaleko lotura bat da. Lotura sakonago bat ere badago, baina ez daukat hura aztertzeko denborarik, genetika modernoa zientziaren esparru oso kitzikagarri bat baita benetan. Eta beranduago honen inguruko hitzaldiak entzungo ditugu. Baina teknologia experimental modernoek sortzen duten informazioaren sekretu batzuk jakiteko gakoetako batzuk teknika sofistikatuetatik etortzen dira, jakin ezazute nire eguneroko lanean zerbait erabilgarria egiten dudala, aurpegi-gurutze-gurutzeren istorioa baino sofistikatuagoa, eredu konputazional eta matematiko nahiko konplexuekin, eta teknika estatistiko modernoekin. Nire taldeak Oxford-en daramatzan bi proiekturen zati txiki batzuk, adibide batzuk, azalduko ditut interesgarriak direla uste dut eta. Giza genomaren proiektuari buruz entzun duzue. Giza genomaren kopia bat deszifratzea helburu zuen proiektu bat zen. Hau lortu ostean, noski, beste proiektu bat doa, HapMap nazioarteko proiektua, 5 edo 6 herrialdeetako laborategiek kolaborazioan garatzen duten proiektua. Giza genomaren proiektuak zer dugun amankomunean aurkitu nahi du, HapMap proiektuak, pertsona desberdinen arteko diferentziak non dauden ulertu nahi du.
Why do we care about that? Well, there are lots of reasons. The most pressing one is that we want to understand how some differences make some people susceptible to one disease -- type-2 diabetes, for example -- and other differences make people more susceptible to heart disease, or stroke, or autism and so on. That's one big project. There's a second big project, recently funded by the Wellcome Trust in this country, involving very large studies -- thousands of individuals, with each of eight different diseases, common diseases like type-1 and type-2 diabetes, and coronary heart disease, bipolar disease and so on -- to try and understand the genetics. To try and understand what it is about genetic differences that causes the diseases. Why do we want to do that? Because we understand very little about most human diseases. We don't know what causes them. And if we can get in at the bottom and understand the genetics, we'll have a window on the way the disease works, and a whole new way about thinking about disease therapies and preventative treatment and so on. So that's, as I said, the little diversion on my main love.
Zergatik axola digu? Beno, arrazoi asko daude. Premiazkoena, zera ulertzea da, diferentzia batzuk nola egiten duten batzuk gaixotasun batzuk izateko joera izatea, 2 motako diabetesa izatera adibidez, edo gaixotasun kardiakoak izateko joera izatea, edo aplopejiak, autismoa... Hori proiektu handi bat da. Badago beste proiektu handi bat, Wellcome Trust-ek berriki finantziatua, genetika ulertzeko, ikerketa handiak, milaka pertsona 8 gaixotasun desberdin, 1 eta 2 motako diabetesa, gaixotasun koronarioak eta nahaste bipolarra adibidez, barne hartzen dituena. Ze desberdintasun genetikok sortzen dituzten gaixotasunak eta zergatik ulertzeko. Zergatik egin nahi dugu? Giza gaixotasun gehienen inguruan ezer gutxi dakigulako. Ez dakigu zerk sortzen dituen. Eta oinarrira iritsi eta genetika ulertu ahalko bagenu, gaixotasunak nola jokatzen duen jakingo genukeelako. Eta terapiak eta tratamentu prebentiboak ikusteko modu berri bat izango genukeelako. Beraz hori da, nire benetako maitasunaren barnean daukadan dibertimentu txikia.
Back to some of the more mundane issues of thinking about uncertainty. Here's another quiz for you -- now suppose we've got a test for a disease which isn't infallible, but it's pretty good. It gets it right 99 percent of the time. And I take one of you, or I take someone off the street, and I test them for the disease in question. Let's suppose there's a test for HIV -- the virus that causes AIDS -- and the test says the person has the disease. What's the chance that they do? The test gets it right 99 percent of the time. So a natural answer is 99 percent. Who likes that answer? Come on -- everyone's got to get involved. Don't think you don't trust me anymore. (Laughter) Well, you're right to be a bit skeptical, because that's not the answer. That's what you might think. It's not the answer, and it's not because it's only part of the story. It actually depends on how common or how rare the disease is. So let me try and illustrate that. Here's a little caricature of a million individuals. So let's think about a disease that affects -- it's pretty rare, it affects one person in 10,000. Amongst these million individuals, most of them are healthy and some of them will have the disease. And in fact, if this is the prevalence of the disease, about 100 will have the disease and the rest won't. So now suppose we test them all. What happens? Well, amongst the 100 who do have the disease, the test will get it right 99 percent of the time, and 99 will test positive. Amongst all these other people who don't have the disease, the test will get it right 99 percent of the time. It'll only get it wrong one percent of the time. But there are so many of them that there'll be an enormous number of false positives. Put that another way -- of all of them who test positive -- so here they are, the individuals involved -- less than one in 100 actually have the disease. So even though we think the test is accurate, the important part of the story is there's another bit of information we need.
Ziurtasunik ezaren inguruan egiten ditugun arrazoiketetara itzuliz, hona zuentzat beste askmakizun bat: Imajinatu gaixotasun bat detektatzeko proba bat daukagula. Ez da hutsezina, baino nahiko ona da. kasuen %99an asmatzen du. Eta zuetako bat, edo kaleko norbait hartzen dut, zoriz, eta proba hori egiten diot. Demagun GIB-rako proba dela, IHESA sortzen duen birusa, eta probak pertsona gaixo dagoela esaten duela. Zein da gaixotasuna izateko probabilitatea? Probak kasuen %99an asmatzen du. Beraz erantzun azkarra %99 da. Nori gustatzen zaio erantzun hori? Benga, guztiok parte hartu behar dugu. Ez pentsatu jada ezin duzuenik nigan konfidantza izan. (barreak) Ongi dago pixka bat eszeptiko egotea, hori ez baita erantzun zuzena. Hori pentsa dezakezue. Baina ez da erantzun zuzena, historiaren zati bakarra baita. Berez, gaixotasunaren hedapenaren araberakoa izango da probabilitatea. Utzi iezaidazue erakusten. Milioi bat pertsonaz osatutako lagina dugu. Pentsa dezagun gaixotasun bitxi batean, 10.000tik bati eragiten dion batean. Milioi horretan, gehienak osasuntsu daude, eta batzuk gaixotasun hori izango dute. Berez, hori bada gaixotasunaren maiztasuna, gutxi gora behera 100 gaixo izango genituzke. Demagun proba guztiei egiten diegula. Zer gertatzen da? Gaixotasuna duten 100 pertsonetan, Frogak %99tan asmatuko du, eta 99 gaixo detektatuko ditu. Gaixotasuna ez duten pertsonetan, frogak %99tan asmatuko du. Kasuen %1ean erratuko da. Baina hainbeste osasuntsu daude, positibo faltsu asko egongo direla. Beste era batera esanda, frogak gaixo dagoela esaten duen horietan, ehunetik batek baino gutxiagok izango du gaixotasuna benetan. Beraz froga zehatza dela uste badugu ere, garrantzitsua zera da, beharrezko den beste informazio bat falta dela.
Here's the key intuition. What we have to do, once we know the test is positive, is to weigh up the plausibility, or the likelihood, of two competing explanations. Each of those explanations has a likely bit and an unlikely bit. One explanation is that the person doesn't have the disease -- that's overwhelmingly likely, if you pick someone at random -- but the test gets it wrong, which is unlikely. The other explanation is that the person does have the disease -- that's unlikely -- but the test gets it right, which is likely. And the number we end up with -- that number which is a little bit less than one in 100 -- is to do with how likely one of those explanations is relative to the other. Each of them taken together is unlikely.
Hau da ideia garrantzitsua. Froga positiboa dela jakin ostean egin behar duguna zera da, lehian dauden bi azalpenen probabilitatea aztertu. Azalpen bakoitzak zati probable eta zati inprobable bat ditu. Azalpen bat pertsona gaixo ez egotea da, hau oso probablea da, norbait zoriz hautatzen baduzu, baina froga erratu egiten da, eta hau inprobablea da. Beste azalpena, pertsona gaixo egotea da, inprobablea dena, eta froga zuzen egotea, probablea dena. Eta topatu nahi dugun zenbaki horrek, ehuneko bat baino txikiagoa den horrek, azalpen batek bestearekiko duen probabilitatearekin du zerikusia. Multzo bakoitza banaka inprobablea da.
Here's a more topical example of exactly the same thing. Those of you in Britain will know about what's become rather a celebrated case of a woman called Sally Clark, who had two babies who died suddenly. And initially, it was thought that they died of what's known informally as "cot death," and more formally as "Sudden Infant Death Syndrome." For various reasons, she was later charged with murder. And at the trial, her trial, a very distinguished pediatrician gave evidence that the chance of two cot deaths, innocent deaths, in a family like hers -- which was professional and non-smoking -- was one in 73 million. To cut a long story short, she was convicted at the time. Later, and fairly recently, acquitted on appeal -- in fact, on the second appeal. And just to set it in context, you can imagine how awful it is for someone to have lost one child, and then two, if they're innocent, to be convicted of murdering them. To be put through the stress of the trial, convicted of murdering them -- and to spend time in a women's prison, where all the other prisoners think you killed your children -- is a really awful thing to happen to someone. And it happened in large part here because the expert got the statistics horribly wrong, in two different ways.
Hona hemen gai bera jorratzen duen adibide berriago bat. Britainia Handikoak zateretenak jakingo duzue, kasua nahiko famatua egin baita. Sally Clark izeneko emakume batek bapatean hil ziren bi haur izan zituen. Hasieran informalki "sehaskako heriotza" deritzonarengatik, formalki, haurren bapateko heriotzagatik hil zirela uste zen. Hainbat arrazoirengatik, erahilketa leporatu zitzaion. Bere epaiketan, pediatra oso ezagun batek ebidentzia bezala esan zuen bi horrelako heriotza izateko probabilitatea, berea bezalako familia batean, profesionala eta ez erretzailea, 73 milioiren artean batekoa zela. Laburtuz kondenatu egin zuten. Geroago, bere bigarren apelazioan, errugabea zela erabaki zen. Pentsa ezazue zer izan daitekeen norbaitentzat haur bat galtzea, gero bestea galtzea, eta errugabea izanik erahilketa leporatuta kondenatua izatea. Epaiketaren eta seme-alabak galtzearen estresa jasatea eta emakumezkoen gartzelan denbora pasatzea, beste preso guztiek zure seme-alabak hil zenituela uste duten bitartean. Horrez izugarria izan behar du. Eta gertatu egin zen. Adituak estatistikak ereabiltzerakoan bi akats egin zituelako.
So where did he get the one in 73 million number? He looked at some research, which said the chance of one cot death in a family like Sally Clark's is about one in 8,500. So he said, "I'll assume that if you have one cot death in a family, the chance of a second child dying from cot death aren't changed." So that's what statisticians would call an assumption of independence. It's like saying, "If you toss a coin and get a head the first time, that won't affect the chance of getting a head the second time." So if you toss a coin twice, the chance of getting a head twice are a half -- that's the chance the first time -- times a half -- the chance a second time. So he said, "Here, I'll assume that these events are independent. When you multiply 8,500 together twice, you get about 73 million." And none of this was stated to the court as an assumption or presented to the jury that way. Unfortunately here -- and, really, regrettably -- first of all, in a situation like this you'd have to verify it empirically. And secondly, it's palpably false. There are lots and lots of things that we don't know about sudden infant deaths. It might well be that there are environmental factors that we're not aware of, and it's pretty likely to be the case that there are genetic factors we're not aware of. So if a family suffers from one cot death, you'd put them in a high-risk group. They've probably got these environmental risk factors and/or genetic risk factors we don't know about. And to argue, then, that the chance of a second death is as if you didn't know that information is really silly. It's worse than silly -- it's really bad science. Nonetheless, that's how it was presented, and at trial nobody even argued it. That's the first problem. The second problem is, what does the number of one in 73 million mean? So after Sally Clark was convicted -- you can imagine, it made rather a splash in the press -- one of the journalists from one of Britain's more reputable newspapers wrote that what the expert had said was, "The chance that she was innocent was one in 73 million." Now, that's a logical error. It's exactly the same logical error as the logical error of thinking that after the disease test, which is 99 percent accurate, the chance of having the disease is 99 percent. In the disease example, we had to bear in mind two things, one of which was the possibility that the test got it right or not. And the other one was the chance, a priori, that the person had the disease or not. It's exactly the same in this context. There are two things involved -- two parts to the explanation. We want to know how likely, or relatively how likely, two different explanations are. One of them is that Sally Clark was innocent -- which is, a priori, overwhelmingly likely -- most mothers don't kill their children. And the second part of the explanation is that she suffered an incredibly unlikely event. Not as unlikely as one in 73 million, but nonetheless rather unlikely. The other explanation is that she was guilty. Now, we probably think a priori that's unlikely. And we certainly should think in the context of a criminal trial that that's unlikely, because of the presumption of innocence. And then if she were trying to kill the children, she succeeded. So the chance that she's innocent isn't one in 73 million. We don't know what it is. It has to do with weighing up the strength of the other evidence against her and the statistical evidence. We know the children died. What matters is how likely or unlikely, relative to each other, the two explanations are. And they're both implausible. There's a situation where errors in statistics had really profound and really unfortunate consequences. In fact, there are two other women who were convicted on the basis of the evidence of this pediatrician, who have subsequently been released on appeal. Many cases were reviewed. And it's particularly topical because he's currently facing a disrepute charge at Britain's General Medical Council.
Beraz, nondik atera zuen "73 milioitik bat" zenbakia? Bapateko heriotzaren estatistika batzuk kontsultatu zituen, eta Sally Clark-ena bezelako familia batean hori gertatzeko probabilitatea 8500etik batekoa zela ikusi zuen. Gero pentsatu zuen "familian horrelako heriotza bat izateak ez du beste bat izateko probabilitatea aldatuko". Horri estatistikoek independentziaren aurretikoa deitzen diote. "Txanpon bat airera bota eta aurpegia ateratzeak, ez du bigarren aldiz aurpegia ateratzeko probabilitatea aldatuko" esatea bezala da. Beraz txanpon bat airera bi aldiz botatzen baduzu, bi aldiz aurpegia ateratzeko probabilitatea erdia, lehen aldiz aurpegia ateratzeko probabilitatea, bider erdia, bigarreneko probabilitatea, da. Beraz esan zuen, "demagun bi gertaerak independenteak direla pentsatuko dut, 8500 bider 8500, 73 milioi inguru da". Eta guzti hau ez zitzaion zinpekoei suposizio gisa, edo modu honetara azaldu. Zoritxarrez, lehenik eta behin egoera horretan enpirikoki egiaztatu beharko litzateke. Eta bigarrenik, erabat faltsua da. Gauza asko dira bapateko haurren heriotzari buruz ez dakizkigunak. Posible da ezagutzen ez ditugun faktore anbientalak egotea, eta oso litekeena da ere, ezagutzen ez ditugun faktore genetikoak egotea. Beraz, familia batek horrelako heriotza jasaten badu, arrisku altuko taldean sartuko zenuke. Ziurrenik, ezagutzen ez ditugun arrisku faktore anbiental eta genetikoak izango dituzte. Bigarren heriotzaren probabilitatea, informazio hori ezagutuko ez bazenukeenaren berdina izango dela argudiatzea benetan inozoa da. Inozoa baino okerrago, benetan zientzia txarra da. Hala ere, hala aurkeztu zen, eta epaiketan inork ez zuen eztabaidatu. Hori da lehen arazoa. Bigarren arazoa zera da, zer esan nahi du 73 miliotik batek? Beraz Sally Clark kondenatua izan ostean, imajina dezakezue prentsan izan zuen oihartzuna, Britainia Handiko egunkari errespatuenetako kazetari batek adituak zera esan zuela idatzi zuen: "Errugabea izateko aukera 73 miliotik batekoa zela" Akats logiko izan zen %99ko zehaztasuna duen gaixotasunaren froga eta gero gaixotasuna izateko aukera %99koa dela pentsatzearen errore bera. Gaixotasunaren adibidean bi gauza izan behar genituen kontuan, bat froga erratu zitekeela, eta bestea a priori, pertsonak gaixo egoteko edo ez egoteko zuen probabilitatea. Testuinguru honetan gauza bera gertatzen da. Bi gauza daude, azalpenaren bi zati. Bi azalpenek duten probabilitatea jakin nahi dugu. Bat Sally Clark errugabea dela, a priori oso posible dena, ama gehienek ez dituzte beren seme-alabak hiltzen. Eta azalpenaren bigarren zatia oso inprobablea zen zerbait pasa zela. Ez 73 miliotik behin bezain inprobablea, baina inprobablea hala ere. Beste azalpena erruduna zela da. A priori inprobablea dela pentsa dezakegu. Eta horrela pentsatu beharko genuke epaiketa kriminal baten testuinguruan, inprobablea dela, inozentziaren aurretikoari esker. Beraz, seme-alabak hil nahi bazituen, lortu zuen. Beraz errugabea izateko aukera ez da bat 73 miliotik. Ez dakigu zenbatekoa den. Bere aurkako ebidentzia eta ebidentzia estatistikoa aztertu behar dira. Badakigu haurrak hil zirela. Beraz jakin behar dena bi azalpenen probabilitate erlatiboa da. Biak sinesgaitzak dira. Estatistikako akatsak ondorio lazgarriak izan zituzten egoeretako bat da hau. Izatez, beste bi emakume ere kondenatuak izan ziren pediatra honen ebidentziagatik, eta gero aske geratu dira, apelatu ere egin gabe. Kasu asko errebisatu ziren. Eta orain Britainia Handiko Kontseilu Mediku Orokorrean ospea galdu du.
So just to conclude -- what are the take-home messages from this? Well, we know that randomness and uncertainty and chance are very much a part of our everyday life. It's also true -- and, although, you, as a collective, are very special in many ways, you're completely typical in not getting the examples I gave right. It's very well documented that people get things wrong. They make errors of logic in reasoning with uncertainty. We can cope with the subtleties of language brilliantly -- and there are interesting evolutionary questions about how we got here. We are not good at reasoning with uncertainty. That's an issue in our everyday lives. As you've heard from many of the talks, statistics underpins an enormous amount of research in science -- in social science, in medicine and indeed, quite a lot of industry. All of quality control, which has had a major impact on industrial processing, is underpinned by statistics. It's something we're bad at doing. At the very least, we should recognize that, and we tend not to. To go back to the legal context, at the Sally Clark trial all of the lawyers just accepted what the expert said. So if a pediatrician had come out and said to a jury, "I know how to build bridges. I've built one down the road. Please drive your car home over it," they would have said, "Well, pediatricians don't know how to build bridges. That's what engineers do." On the other hand, he came out and effectively said, or implied, "I know how to reason with uncertainty. I know how to do statistics." And everyone said, "Well, that's fine. He's an expert." So we need to understand where our competence is and isn't. Exactly the same kinds of issues arose in the early days of DNA profiling, when scientists, and lawyers and in some cases judges, routinely misrepresented evidence. Usually -- one hopes -- innocently, but misrepresented evidence. Forensic scientists said, "The chance that this guy's innocent is one in three million." Even if you believe the number, just like the 73 million to one, that's not what it meant. And there have been celebrated appeal cases in Britain and elsewhere because of that.
Beraz, amaitzeko, zein da guzti honen mezua? Badakigu zoria, probabilitatea eta ziurtasunik eza gure bizitzako zati direla. Egia da ere, nahiz eta zuek oso publiko berezia izan, oso tipikoa dela jarri ditudan adibide horiek ez asmatzea. Oso ongi dokumentatuta dago jendea gauza hauetan erratu egiten dela. Logikako akatsak egiten dira ziurtasunik ezaren inguruan arrazoitzean. Hizkuntzaren txikikeriekin oso ongilan egin dezakegu, eta hori nola lortu dugunaren inguruan oso galdera interesgarriak daude. Baina ez gara onak ziurtasunik ezaren inguruan arrazoitzen. Hori gure eguneroko bizitzan arazo bat da. Hitzaldi askotan entzun duzuen bezala, estatistika ikerketa zientifiko askoren, gizarte zientzietan, medikuntzan... eta industriaren zati handi baten oinbarrian dago. Industriaren prozesuan inpaktu handia duen kalitate kontrol hori guztia, estatistikan oinarritzen da. Gaizki egiten dugun zerbait da. Gutxienez onartu egin beharko genuke, baina ez onartzeko joera dugu. Testuinguru legalera bueltatuz, Sally Clark-en epaiketan, abokatu guztiek adituaren hitzak onartu zituzten, besterik gabe. Beraz pediatra batek zinpekoei zera esan bazien: "Badakit zubiak eraikitzen. Kale horretan bat eraiki dut. Mesedez, pasa bertatik zure autoarekin", zinpekoek erantzungo zuten: "pediatrek ez dakite zubiak eraikitzen. Hori injeniarei dagokie." Baina horren ordez iritsi eta esan zuen, edo aditzera eman zuen: "Badakit nola arrazoitu ziurtasunik gabeko egoeretan, badakit estatistikarekin lan egiten." Eta guztiek esan zuten, ados, aditu bat da. Beraz gure konpetentziak non amaitzen diren jakin behar dugu. Horrelakoxe gauzak atera ziren DNA sekuentziatzen hasi zirenean, zientzialariak, abokatuak eta epaileak ere sistematikoki frogak desitxuratu zituztenean. Orokorrean, uste dugu, maliziarik gabe, baina frogak desitxuratu zituzten. Zientzialari forenseek esan zuten "hau errugabea izateko probabilitatea 3 miliotik batekoa da". Zenbakia sinistuta ere, 73 miliotik bat bezala, ez du hori esan nahi. Eta apelazio famatuak egon dira horregatik Britainia Handian, eta beste lekuetan.
And just to finish in the context of the legal system. It's all very well to say, "Let's do our best to present the evidence." But more and more, in cases of DNA profiling -- this is another one -- we expect juries, who are ordinary people -- and it's documented they're very bad at this -- we expect juries to be able to cope with the sorts of reasoning that goes on. In other spheres of life, if people argued -- well, except possibly for politics -- but in other spheres of life, if people argued illogically, we'd say that's not a good thing. We sort of expect it of politicians and don't hope for much more. In the case of uncertainty, we get it wrong all the time -- and at the very least, we should be aware of that, and ideally, we might try and do something about it. Thanks very much.
Eta lege-sistemaren testuinguruarekin amaitzeko. Oso ongi dago "ahalik eta ongien froga aurkeztea". Baina gero eta gehiago, batez ere DNA-ren azterketen kasuetan, zinpekoak, pertsona normalak direnak, eta jakina den arren horretan oso txarrak direla, arrazoitze modu horrekin lan egiteko gai direla uste dugu. Bizitzaren beste esparruetan, jendeak ilogikoki argudiatuko balu, beno politikoak kenduta, beste esparruetan jendeak ilogikoki argudiatuko balu, ez dela ona esango genuke. Politikoengandik espero dugu, baina ez beste inorrengandik. Ziurtasunik ezaren kasuan beti erratuta gaude, eta gutxienez kontziente izan beharko genuke. Eta idealki horen inguruan zerbait egin beharko genuke. Mila esker.