to introduce one of the stars not only of our department but of the entire University of Wisconsin.  Elliot Sober's widely considered the preeminent philosopher of biology in the world and among the top handful of all philosophers of science, a fact attested to by his recently being given the Carl Gustav Hempel Award by the Philosophy of Science Association, an award given only once every two years for career achievement.  He served as president of the Union of the History of Philosophy of Science and Technology as well as the chair of the History and Philosophy of Science Division of the American Association for the Advancement of Science.  He's been president of the Philosophy of Science Association and of the APA Central Division.  I could be here all night if I listed all of Elliot's awards and honors, but he's additionally a named WARF professor here at UW.  He's been named a Vilas research professor and he has also received the Hilldale Award, which is given annually to only one professor from each of the four divisions of the university.   These accolades have accrued to Elliot in large part because of his broad and often groundbreaking work in central issues in philosophy of science and philosophy of biology in particular.  It would hardly be an overstatement to say that Elliot has invented contemporary philosophy of biology, taking the tools of probability theory and applying them to central theoretical questions about biology and evolution.  In Unto Others, Elliot with coauthor David Wilson explored how and under what conditions altruism could emerge from natural selection, which seems to be a fundamentally egoistic process.  In evidence and evolution, he discussed how to understand evidence, in particular, in connection with evolution where there's nothing like direct experimentation with which to confirm and test theories and show how such evidence can be used to test the theory of natural selection and claims of common ancestry, though not so much for intelligent design. 

 In all, Elliot has written or cowritten 11 books and edited two others.  He has published, wait for it, over 250 articles, producing highly influential work about evolution, adaptation, explanation, probability, evidence, parsimony, Pascal's wager and much more.  His work has been translated into many languages, including Japanese, Chinese, Korean and Turkish and Pig Latin.  (laughter) Given all this research and the hundreds of talks around the world he's been invited to give and all of his administrative work as president of this and that, what we'd expect that he'd have little time for teaching, mentoring, an aspect of what most people consider normal life.  But Elliot has had dozen of advisees, many of whom he's written papers with and students flock to his wonderful classes.  He's an avid sailor and makes his wine and cider.  He's a devoted and busy father and grandfather.  Goes to lots of operas and shows.  He's an excellent cook and spends lots of time with his friends.  When you look at ll of these involvements to fill up his days, it can seem like the dumbfounding question isn't how does he find time to do anything other than research and give presentations, but how does he find time for that. 

 I'm suspicious there may be some cloning involved.  (laughter) Both as a person and as an academic, Elliot is nothing short of astounding.  We're extremely fortunate to have him here at UW and I consider myself even more fortunate to count him as a treasured friend.  Tonight he'll discuss some themes from his new book, Ockham's Razors: A User's Manual, which will be available for purchase after the talk, signed if you wish by the author.  And the talk tonight is entitled Ockham's Razors: When is a Simpler Theory Better.  Would you help me in welcoming Elliot Sober? (applause)  - Thanks, Alan.  You make me blush.   So it's great to be here, but I'm puzzled as to why.  This is a series the philosophy department puts on called UW Philosophers Tackle Contemporary Issues and if you look at the history up there, this was over a long time ago.  But I'm kidding of course. 

 The principle of parsimony is with us still in contemporary science and in everyday life also.  It's named for Ockham, a medieval philosopher, but contemporary science uses this tool in reasoning.  Just a little vocabulary.  Parsimony, I have to tell this to my intro students so I hope you won't mind me telling it to you.  The word parsimony means an unwillingness to spend money or to use resources so an example would be the cooks were parsimonious in their use of butter in the recipe.  And the principle of parsimony applied to theories is the theories are better when they postulate fewer causes or processes or entities and the goal tonight is to try to understand what makes this theory tick.  So there's Einstein saying what I just said before.  He thinks that Ockham's razor is an indispensable tool in doing science and the puzzle as I say, is to understand why this much used principle makes sense.  And here's what I'll do.  I'll give you just a tiny bit of history. 

 It didn't start with Ockham, the medieval philosopher.  It started much earlier, maybe in Aristotle's own work.  I'm going to give you just a tiny snapshot of one episode in this long history and then for the rest, I'm going to talk about two frameworks within which the principle of parsimony, Ockaham's razor can be justified.  So in this time of the scientific revolution, we have great philosophers, scientists like Descartes, Leibniz and Newton and in those days the distinction between philosophy as we now call it and science as we now call it was much blurrier and continuous so all three of these figures did work that we now look on as being both philosophical and scientific.  So I'm going to say a little bit about one of these guys and give you a picture of how he understood Ockham's razor, which was not unique to him and then I'm going to fast forward to now and talk about a very different way of thinking about this idea.  So here's Newton.  In his book, The Principia: The Mathematical Principles of Natural Philosophy, he called it and he lays down four rules of reasoning there.  I'm going to talk about two of them just to give you a flavor of how Ockham's razor is part of the scientific revolution.  So Newton says we're to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.  Here's his justification in The Principia, to this purpose the philosophers say that nature does nothing in vain. 

 That's a little catchphrase of Aristotle.  And more is in vain when less will serve.  And here's some beautiful poetry.  For nature is pleased with simplicity and effects not the pomp of superfluous causes.  Isn't that great? The second rule is related.  It says therefore to the same natural effects, we must, as far as possible, assign the same causes and the examples he gives are respiration in man and beast, the descent of stones in Europe and America, the light in our culinary fires and of the sun and the reflection of light in the Earth and on the planets.  And what he's saying is that when you have two systems that are observationally similar, you should try to explain them in terms of the same causes.  Newton was famous for unifying the theory of motion and Aristotle and after, there were separate theories of motion, one for terrestrial motion, the other for celestial motion.  Totally different laws and Newton's great acheivement was to unify the theory of motion so it would both the motion of the moon and the falling of an apple.  So unification is rule two and you can think of unification as part of the idea that simpler theories are better. 

 A unified theory of two or more phenomena is simpler than a disunifying theory that gives a separate explanation for each little piece.   So what's the justification? Newton says nature does nothin in vain, but how dow e know that.  And he says so what's the justification of thinking that nature does nothing in vain.  Well, he wrote a book never published about interpreting the Bible, in particular the book in the Christian Bible called the Book of Revelations and it tells you how he really thought of Ockham's razor in science and also in biblical interpretation.  For Newton, these were not separate problems.  He wrote more about alchemy and religion than he wrote about physics as we would now call physics.  And for him, the rules of reasoning are the same, both for biblical interpretation and observations as interpreted by science.  So this is what he says about Revelations, but you'll see science coming into this.  He says to chose those interpretations which without straining reduce things to the greatest simplicity, truth is ever to be found in simplicity and not in the multiplicity and confusion of things.  As the world, which to the naked eye exhibits the greatest variety of objects, appears very simple in its internal constitution when surveyed by a philosophic understanding so it is in these visions, the visions reported in the Book of Revelation, so we should interpret the visions in the same way. 

 And how do we do the two tasks, interpreting the Bible and interpreting observations in science? Here's the key.  It's the perfection of God's works that they are all done with the greatest simplicity.  He is the god of order and not of confusion and therefore as they do understand the frame of the world, must endeavor to reduce their knowledge to all possible simplicy so it must be in seeking to understand these visions.  So here's the picture that Newton and many other of these 17th century geniuses who created the scientific revolution, this is how they saw the world, that God created the laws of nature.  He also built us to have the minds we have.  He made the laws of nature simple and he gave each of us a taste for simple theories.  We prefer simple theories over complex things because God gave us that preference and the result is that our thinking and the way the world is, the simplicity of nature fit together.  So this is Newton's justification for Ockham's razor.  The justification is nature is simple and how do you know that.  Because God made nature, that's why it's simple. 

 And we seek out simple theories because God built our minds that way so that is the theological setting that so many people in the 17th century had of the justification of the principle of parsimony.  Well, here we are the 21st century and what I'm going to talk about tonight is what you ought to think about Ockham's razor and its justification if you set aside this theological story that was so influential in the 17th century.  How could you justify a preference for simple theories without bringing God, the designer of the universe, into your story? In the 20th century, a lot of progress was made on this problem and I'm going to describe two lines of inquiry that I think help us really understand this in a totally different way from the way Newton, Leibniz and Descartes thought about it.  So here's the basic question which we want to make precise.  Why should you think that simpler theories are better than theories that are more complex? And one obvious answer is well, we should ask better, well, simple theories are easier to understand, they're easier to remember, they're more beautiful.  That's the sense in which simpler theories are better.  It's correct, but it really misses the hard problem that is involved in thinking about the justification of Ockman's razor.  The hard problem, there's several, are like this.  So okay, so what? Simpler theories are beautiful and easy to understand.  Tell me why I should think that simpler theories of having a higher probability of being true. 

 Tell me why simpler theories are better supported by the evidence we have or possibly, tell me why simple theories are more predictively accurate than theories that are more complex.  So the hard problems about Ockham's razor have to do with the relevance of simplicity to thinking about the way the world is and you're not addressing that question by talking about yourself and the fact that you like simple theories, that you can remember them well, that they're beautiful to you.  That is not on topic.  So these three problems are the hard problems and I'm going to be talking about ways of solving these two of the three.   Now I'm going to get a little mathy and I bet a lot of you are math phobic, but I'm going to try to make it without numbers and calculations, give you a feeling for some mathematical ideas from the theory of probability that will help you get a grip on ways of thinking about Ockham's razor and connecting it with those two questions, about it's justification.  So bear with me, take a deep breath if you are math phobic.  I hope I can communicate the basics here without making it a blur.   Okay, so this is the only picture in Darwin's Origin of Species.  It's a picture that biologists would now call a picture of a phylogenetic tree.  At the top of the page are species that exist now. 

 As you go down the page, you see present day species tracing back to common ancestors and those common ancestors tracing back to still older common ancestors.  And Darwin thought that everything alive on our planet traces back to one or a few common acestors.  Current biology doesn't hedge and say one or a few.  The standard picture in evolutionary biology now is that all life traces back to a single common ancestor.  And so I want to start thinking about Ockham's razor in the context of this idea of common ancestry.  So let's think about what kind of evidence would be relevant to justifying the hypothesis of common ancestry that Darwin endorsed and that contemporary biology continues to endorse.  So here's an example.  Let's work with an example.  So human beings and monkeys both have tailbones.  And is that evidence that they have a common ancestor? Is that similarity evidence for common ancestor, common ancestry? Well, first I want to get you, have a feeling of how that question is connected to a question about parsimony and simplicity. 

 Well, if human beings and monkeys had a common ancestor, then it's possible that tailbones evolved exactly once in history of those two lineages they go back in the past and was retained thereafter.  On the other hand, if human beings and monkeys had separate ancestors, then tailbones would have to evolve separately, once in each of those lineages so there seems to be a difference in how many changes need to occur in the history of these lineages if you're going to get what you observe now, that present day monkeys and human beings have tailbones.  So its in that sense that Darwin and biologists now especially, think of the common ancestry hypothesis as simpler, as more parsimonious.  The common ancestry hypothesis requires fewer originations of new characteristics than the separate ancestry hypothesis.  But our question, which will be repeated several times is okay, so what, it's simpler, it's more parsimonious, but why is that a reason to think the common ancestry hypothesis has a better claim on being of the world we live in.  What does parsimony have to do with the world we live in as opposed to our subjective preferences?  Okay, here's some terminology.  It's a terminology about probability, but nothing hard.  So if we're going to use O for observation and H for hypothesis and this symbolism here with O vertical slash H, the probability of O/H means the probability that O has on the assumption that H is true and the terminology out there in probability and statistics is really horrible because the terminology is the probability of O given H is called H's likelihood, whereas the probability of H given O is called H's probability.  It's the worst possible terminology because in ordinary English, for something to be likely and for something to be probable, mean exactly the same thing, but the mathematical genius RA Fisher chose this terminology 100 years ago about and it has stuck so unfortunately we have to use this.  I'm going to be especially interested in likelihoods and I'll be using that word 25 times, but I want you to make sure you understand what I do and what the world of probability and statistics understands by likelihood. 

 It means the probability of an observation given a hypothesis.  How likely the hypothesis is, is not how probable it is.  Rather the likelihood of the hypothesis has to do with how probable the hypothesis makes your observations.  So just to crystalize that distinction and maybe this will be the only thing you'll remember in an hour or two, I hope not, is here's a simple way to keep the difference between the probability of O given H and the probability of H given O really clear in your mind.  Here's the example.  So let O be the observation that there are noises coming from your attic and let H be the hypothesis that says that there are noisy gremlins bowling in your attic.  I hope you see that the probability of O given H is high, but the probability of H given O, I hope none of us thinks that's high.  (laughter) So this is an example which sort of crystalizes the idea that to talk about the likelihood of a hypothesis is a whole different question from talking about the probability of the hypothesis and I'm going to be talking about likelihoods now.   So here's an idea about the interpretation of evidence that is not totally uncontroversial.  What is in philosophy, but I like it. 

 And I'm going to be using it.  I'm going to try to jolly you into thinking that it's a sensible principle and then I'm going to use it to think about Ockham's razor in connection with the example of human beings and monkeys both having tailbones.  So here's the principle I want to use.  It says that we have some observations in two possible hypotheses and the idea is that the observations favors the first hypothesis over the second exactly when the observations are more probable if the first hypothesis is true than the observations would be if the second hypothesis were true.  And here is the example that's supposed to make you all think, "Oh yeah, that's a good principle." So we have an urn. It's filled with balls.  Each ball has one color.  And we're interested in testing two different hypothesis about all these balls that are in the urn.  The first hypothesis says that 80% of the balls in the urn are green, H2 says that only 50% are and we're going to test these two hypothesis by drawing some balls out of the urn.  We're going to draw 100 out and suppose it turns out that 90 of the balls drawn are green.  Now, that doesn't prove that one of those hypotheses has got to be true and the other has to be false, but I hope you'll agree with me that the observation is pointing in the direction of one of those and away from the other, that the observation favors one over the other and I hope you'll furthermore agree that the observation favors H1 over H2. 

 If H1 were true, it's not that surprising that you would get these observations.  If H2 were true, it's rather more surprising, much less probably that you'd get those observations.  So this is supposed to be an illustration that gets you to think yeah, the law of likelihood, that's a cool way to understand how you interpret observations as they bear on competing hypotheses, but what is this going to have to do with the problem we started with, the problem of understanding why common ancestry and separate ancestry will not only differ in parsimony in their explanation of why human beings and monkeys have tailbones, but why that's a significant fact about them, why there's something better in some serious sense of better about the common ancestry hypothesis versus the separate ancestry hypothesis.  And to answer that-- So here's what we want.  First of all, there's the VF that we want to get clearly in mind.  If I can show that the observed similarities of human beings and monkeys is more probable under the common ancestry hypothesis than it is under the separate ancestry hypothesis, then the law of likelihood is going to take that inequality, that greater than thing that we have in the yellow box and conclude oh, that means the observed similarity of human beings and monkeys favors common ancestry over separate ancestry.  So that's got an if in it and we have two questions we want to think about.  First of all, why should I think that the observed similarity of monkeys and human beings is more probable if they have a common ancestor than it would be if they only had separate ancestry? Like why is that first inequality is true? Why is it true? And what we're looking for is assumptions that seem reasonable and plausible that will tell us that inequality is true and then we'll have our answer, but what are those assumptions.  And then separately, maybe there's some other assumptions which will have the opposite significance.  It will tell us that no, under these weird assumptions, these different assumptions, the observed similarity is more probable under the separate ancestry hypothesis than it is under the common ancestry hypothesis. 

 So notice how the inequality flips here.  Greater than, less than.  So we're looking for assumptions that will justify the first inequality, but we want to be alert to the possibility there might be other assumptions that justify the second one.   Well, one of my heroes in philosophy, Hans Reichenbach, created this principle called the principle of the common cause.  He didn't talk about common ancestry, but we can adapt, we can adopt what he's doing, adapt what he's doing to the issue of common ancestry and separate ancestry because after all, when you say that two things have a common ancestor, when you say that human beings and monkeys have a common ancestor, you're talking about some organisms or organism who's a common cause of the characteristics that human beings and monkeys have and similarly the hypotheses of separate ancestry, being a hypothesis of separate causes.  And it turns out that Reichenbach described assumptions that suffice for the common cause hypothesis to have the higher likelihood and these assumptions are often very plausible.  They're used pervasively in evolutionary biology and I'm not going to tell you what the assumptions are.  If you want, in the Q&A I'm happy to describe what they say, but all I want you to get so far is there are these assumptions that seem reasonable to lots of people in at least a lot of settings, which have the consequence that that top inequality is true, that under those assumptions the common ancestry hypothesis makes the observed similarity more probable than the separate ancestry hypothesis does so we've got the common ancestry hypothesis having a higher likelihood in the technical sense of likelihood that we're using.  Okay, that's nice, isn't it? (laughter) But the assumptions aren't written in granite.  They have to be true in every possible situation. 

 You can tinker with Reichenbach's assumptions and create a situation in which you get this flip happening so that under these different kinds of assumptions, it turns out that a similarity has a higher probability under the separate cause, the separate ancestry hypothesis, than it has under the common ancestry hypothesis so let me describe a couple of those weirdo cases.  So before I do that, here's the picture.  Nice Reichenbach assumptions that tell you that the common ancestry hypothesis makes the similarity of human beings and monkeys more probable than the separate ancestry hypothesis has.  That's nice, but now change those assumptions.  You can get the reverse kind of result and I'll describe some examples, too.  Okay, suppose there is a gene that doesn't make you sterile, but it makes you have very low fertility.  Your chance of having a kid is extremely low.  Now imagine the two individuals in some population both have that gene.  I think that's evidence that favors...  That's evidence against their being siblings because if they were siblings, how did their parents produce two kids. 

 Whereas if they're not siblings, then it's more probable that each of them would have it so this is a case violating Reichenbach's assumptions where it turns out that a similarity can favor separate cause over common cause, unlike the Reichenbachian assumptions which tell you that common cause is more likely than separate cause.  That's the one example. Here's a second one.  Not about biology at all.  Suppose that each of several businesses has one boss, has a single boss and multiple employees and suppose in this in this bunch of businesses, you look at two employees, X and Y, and it turns out they're similar in a certain way.  They're both unhappy with their jobs.  Is that shared characteristic evidence that the two employees have the same boss? That would be the common cause hypothesis.  Well, maybe sometimes, but not under some assumptions.  Here is an assumption that wrecks that conclusion.  Suppose that if X's boss is strict, this raises the probability that X will be unhappy, whereas if Y's boss is strict, this lowers the probability that Y will be unhappy.  It turns out that with that assumption in place, the observed similarity X and Y are both unhappy with their jobs favors separate bosses, not the same boss. 

  I hope this is a little intuitive.  It's not obvious why that works out in the mathy background that I'm not really giving you, but it's true.  Okay, so let me sum up what I just said about parsimonium likelihood.  So we had these two hypotheses, a common cause and a separate cause hypothesis, and we have an observation that two objects, they could be two species, two employees, whatever, have a trait in common and in the common cause, in the common ancestry example, you see how the common cause hypothesis is more parsimonious and given Reichenbachian assumptions, it'll turn out that the common cause hypothesis has a higher likelihood and I claim that this answers at least part of one of the hard problems.  One of the hard problems I had up there before was why think that simpler theories are better supported by the evidence.  Well, the simpler theory, the common cause hypothesis, is better supported.  It's favored by the observations with these assumptions in place, but it's not always true that when X and Y share a trait, that that always favors common cause over separate cause.  There are ways of violating the Reichenbachian assumptions that give you that weird flip of the inequality side.  Okay, so that's one way of justifying Ockham's razor.  What's great about simple theories is not just that they're beautiful, not just that they're easy to understand, but in a rather general setting, simpler theories are better supported by the evidence than more complex theories. 

 So that's a reason to take parsimony and simplicity seriously here, I claim.  Here's a different problem, which has to be looked at, I think, in a totally different way.  We're going to talk about is there ever a general thing you can say about simple theories being more predictively accurate.  That would be a reason to care about them.  If you have a stake in making accurate predictions about the future and if simpler theories were more predictively accurate, that would be worth knowing and that's, as I keep saying, not a different kind of thought than saying oh, simpler theories are easier to remember and more beautiful.  So what about predictive accuracy? Here's the example I want to use to explain the sort of second paradigm for thinking about parsimony that's out there, thanks to work in the 20th century.  So you drive south from Madison, you stop your car, you get out and you see that on one side of the road is a giant field of corn plants and on the other side of the road is a second giant population of corn plants.  Each of those populations has its own average height.  Thousands and thousands of corn plants in each field.   So you being an empirical scientist, you sample 100 corn plants from population one, 100 from population two and you get, in your samples, you get the mean height of 56 inches and 52 inches. 

 That's your data and you're going to use the data you just collected from these 100 corn plants from here, 100 corn plants from there, to try to evaluate two models.  The null model says that the mean height in the first population is identical with the mean height in the second.  That's why it's called a null hypothesis.  It's saying the difference between the two population is zero.  And then there's this model call diff, which just says that there exists a number beta such that M1 minus M2 is equal to beta.  Beta could be positive, it could be negative, it could be zero.  So diff doesn't commit to very much here.  Null is a more assertive kind of hypothesis.  As I say, if null is true, diff has to be true.  And here's some terminology from statistics. 

 Null has no adjustable parameters, whereas diff has a single adjustable parameter.  That's what beta is.  Beta is like this blank.  You could replace beta with 35, you could replace it with four and so on.  So what we want to talk about next is what it is to fit models to data.  What that means is that you replace the adjustable parameters in a model with estimates of their value.  So diff says that beta is M1 minus M2.  What's the best estimate of the value of beta, given your observation that in the samples, you got 56 inches in one sample, 52 inches in the other sample? Well the maximum likelihood estimate, the estimate that makes your observations most probable is you should guess that it's four.  Notice that if that were true, if four inches were the real difference between the populations, it's not that surprising that you get these observations, whereas if you say that beta is equal to 100 inches, it's kind of weirdly improbable that you actually got this data.  So four seems like a reasonable estimate. 

 So we've now fitted the two models to data.  We didn't have to do anything to null because it has no adjustable parameters.  Diff had one adjustable parameters.  We've now adjusted it. It's gone.  It's replaced by its best estimate.  Okay, that's nice. So what? Now here's the prediction problem.  You're going to go back to these fields and gather 100 corn plants from each of them.  That'll be your new data.  And I want you to think about which of these two models will do a better job of predicting what the new data looks like.  You know which of them fits the old data better. 

 You've got the data right in front of you, but you haven't looked at the new data yet so how are you supposed to go about thinking about which of these models will be more predictively accurate.  That is do a better job of predicting data you do not have in front of you yet, but you will.   You know from before that diff is fitting the data of the old data a lot better than null does.  On the other hand, null is simpler because it doesn't have any adjustable parameters.  Diff has a single adjustable parameter so now we're going to along with this idea that's so common in science that simplicity in a model has to do with paucity of parameters.  The smaller the number of adjustable parameters, the simpler the model.  And again we're going to ask that relevance question.  Okay, null is a simpler model and has fewer adjustable parameters because zero is smaller than one, but so what.  Why is that relevant to the question of figuring out which of these models is going to be more predictively accurate when I look at new data?  Before I give you the surprise, the mathy surprise about how to think about this, I want to describe an experience that scientists have all the time when they're working with models of data.  Here's the experience. 

 A really complicated model which has lots of adjustable parameters will do a great job of fitting the data you have already, but when you use the model fitted to the old data to predict new data, it often does poorly.  So fitting old data, yeah, it's good to be complicated, but predicting new data, ah, not so much.  On the other hand, simpler models usually fit the old data less well than complicated models do.  That was true in our example of null and diff.  Yet when those simpler models are used to predict new data, they often do a bit better than their complicated competitor models.  So that's just an experience that scientists have.  Who knows why they're having that experience, but at least it suggests the idea that when you're thinking about evaluating models for their predictive accuracy, it's not just fit to the old data that you should be thinking about.  Parsimony seems to be relevant.  But why should it be relevant? Maybe it's this brute inexplicable mystery.  No. 

 We have this guy, genius statistician from Japan named Akaike...   who invented...  He invented ideas that led others to invent this idea called AIC, the Akaike Information Criterion.  The reason the AIC was invented is that Akaike proved a theorem.  This is the mathy part.  He proved, based on some assumptions, you always need assumptions to prove stuff, that an unbiased estimate of the predictive accuracy of a model is equal to, forget about the log thing, the probability the data given the fitted model minus K where K is the number of adjustable parameters so what the theorem tells you is an unbiased estimate of predictive accuracy should attend to how well the model fits the data.  That's what that red term over there is all about.  But it also should attend to how complicated the model is, how many adjustable parameters it has.  And you see that minus sign there.  There's a penalty. 

 Complicated models are penalized when you estimate their predictive accuracy because K is big.  They have a lot of adjustable parameters.  Simpler models pay a smaller penalty.  And K, wasn't K what we said we mean by simplicity of a model? The parsimony of a model is how complicated.  What does that mean? How many adjustable parameters it contains.  That's why null is more parsimonious, simpler than diff.  And here's a thing from mathematical statistics that tells us that parsimony matters in estimating predictive accuracy.  It's not just an aesthetic thrill.  No, it's actually to judging which models will do better at predicting and which worse.  Now, the AIC framework and other ideas in this part of statistics called model selection theory don't say the simpler theory is always better because of course, notice that there are two things that are relevant to evaluating them. 

 Not just how simple it is, but how well it fits the data and there's a kind of conflict or tradeoff between those two considerations.  Complicated theories have better fit the data, but simpler theories have lower values of K.   Okay.  But what is true in the AIC framework is that if you have two models that fit the data about equally well, then AIC is going to tell you that the simpler model will be more predictively accurate.   Here is one more wrinkle about AIC and I'm done telling you about it.  By the way, the terms of the "math-y" hard part, you're well over the hump now.  Okay.  Here's a weird thing about this example and it's not unique to this example.  Before you even look at one corn plant drawn from either of these populations, I think you know with about as much certainty as you know anything, that null is false and that diff is true.  Remember what diff says. 

 It just says there is a number, which is the mean height in the first population minus the mean height in the second.  How could that be wrong? Whereas diff says-- but null says something very-- The difference is exactly zero.  Thousands of corn plants over here, thousands over there and null is saying that the difference in the mean height, it's not like within 10% of zero.  It says it's zero.  So why do scientists bother testing null against diff in examples like that if we know from the get-go that null is false and diff is true? Well the answer is, if they care about predictive accuracy, the fact that null is false and diff is true doesn't settle the question that's being asked.  It doesn't tell which of those will be more predictively accurate.  And one of the philosophically interesting things about this area of statistics is that it often turns out that simpler false models are more predictively accurate than more realistic true models.  So that's why it's not crazy to test these two things.  If your interest is in predictive accuracy rather than truth, then by all means you got to do it.  AIC if you got to do it. 

 Okay now I talked about two frameworks for justifying Ockham's razor, for justifying the principle of parsimony.  One was the Reichenbach story about likelihoods, common ancestry versus separate ancestry.  When will the common ancestry hypothesis have a higher likelihood than the separate acestry hypothesis given that humans and monkeys both have tailbones.  First story.  Second story was null and diff and I was describing how Akaike and the AIC show you why the number of adjustable parameters is relevant to estimating the predictive accuracy of a model.  So they're really very different ways of thinking about Ockham's razor, very different frameworks, but they have a couple things in common.  First of all, neither of them says, "Oh, parsimony.  "There's no justification for parsimony.  "It's a rock bottom thing.  "We love it for its own sake. 

 "It doesn't have to be connected with anything more fundamental." Neither of them is saying that.  They're both saying the justification for Ockham's razor is that simpler theories deliver something that we care about that's more fundamental.  Greater likelihood in the case of Reichenbach, greater predictive accuracy in the case of Akaike.  And the second similarity between them is the justifications, the Reichenbachian justification of common cause being better than separate cause and AIC and Akaike's theorem, they all depend on assumptions and if you change the assumptions, you get a difference in the outcome so both of them involve conditional justifications of Ockham's razor.  It's not like you can say no matter what the world is like, simplicity is a good thing.  That's not what's going on in these two frameworks.  Okay, that's how they're similar.  How do they differ? Well, we have likelihood being the story that's going on in the Reichenbach common cause, separate cause thing, whereas in the AIC thing, we've got the complexity of a model being relevant to estimating predictive accuracy, but here's the way to sort of crystalize how they're really about different things.  Reichenbach's framework is all about which hypothesis, the common cause or the separate cause hypothesis, fits the data you actually have in front of you.  It's about fitting old data. 

 That's what this likelihood comparison is about, whereas AIC is all about predicting new data so the whole problem that's being addressed is different between these two frameworks.   Now these are really different and no one in statistics or probability or philosophy, I think has seen a way of unifying these two justifications of Ockham's razor so as of 2015, I think we're in a situation where Ockham's razor is broken into pieces and understood in this way, it has that kind of justification, understood in this way, it has a different justification.  That could all change tomorrow, but I think that's how things are right now.  And the other thing I want you to nice about this is I never said that these are the only two possible justifications for parsimony, Maybe there are other ways to justify Ockham's razor beyond the Reichenbach likelihood story and the Akaike story about predictive accuracy.  And that leads me-- So I want to sort of speculate a little bit about this last possibility.  Maybe there are examples of good parsimony arguments in science which cannot be captured either in the first way or the second way.  So in closing, I want to describe an example that might be of that sort.  So I went to a lecture a couple weeks ago about why the dinosaurs went extinct and it was a rather surprising lecture.  The lecturer started off by describing the conventional explanation of the extinction of the dinosaurs, but then he said I have got a far more parsimonious explanation of the extinction.  It's better because it's more parsimonious. 

 So let me describe each of the theories in turn and how the lecture developed.  So he started off by describing the conventional theory.  The conventional theory is a giant meteor crashes into the Yucatan Peninsula, which sends up a dust cloud, which blocks the sun, which makes all the grass and vegetation die so the dinosaurs had nothing to eat and so they went extinct because they starved to death.  That's the first explanation.  And the lecturer then says that's what everybody else thinks, but I have a simpler theory.   And his theory was this.  In a galaxy far from us, there was an advanced civilization which built giant spacecrafts, which they sent to Earth in large numbers and when the aliens got to Earth, they walked out of their spaceships and they noticed these beautiful animals, dinosaurs walking around and they thought these are so fantastic we should take them back to our home planet.  So the aliens, they had big spaceships and there were a lot of them, they loaded all the dinosaurs on Earth into these spaceships.  This is a theory. (laughter) And they were about to take off when one alien said to the other you know, we don't need to take all these dinosaurs back to our planet.  We'll just take a small breeding population back. 

 That'll work.  So they offloaded most of the dinosaurs and then they took off and went back to their planet.  The lecturers paused there and looked around and someone in the back row raised her hand and said, "But wait a minute, why did the dinosaurs go extinct then?" And the lecturer said, "Oh, I forgot a detail in my theory.  "After the aliens went back to their planet, "a giant meteor hit." (laughter) So that wasn't my story.  That was from Saturday Night Live as told by Guido Sarducci.  (laughter) I didn't have the nerve to give you the story with Guido's accent, but anyhow that's where it came from.  So here are the people in Madison who helped me put these ideas together.  Thanks.  (applause)