[MUSIC] In this lecture we'll talk about three different ways in which you can draw statistical inferences from your data. We can see that there are three main questions that you can ask which are all related to different statistical approaches. So let's take a look at each of these in turn. Now in statistics, we'll see that there's basically one truth that we all try to figure out. But the paths (to follow) toward this truth, there are many different ways to try to do this. One useful metaphor that I think can help you think about these differences is if we look at Hinduism, where there are three ways to reach enlightenment: The Karma yoga, the Jnana yoga, and the Bhakti yoga. Now if you see what the Hinduist book Bhagavad Gita says about this, there's basically a path of action, a path of devotion and a path of knowledge. And I think there's a nice relation to make between these three paths and three questions that you can ask if you do statistics. Now, Royall talks about these three different questions that one might ask. And he differentiates between, "what should I do?", "what should I believe?" and "what's the relative evidence?" And we see that these are three different statistical approaches that we can use to draw inferences from data. The first is the path of action. The path of action uses rules to govern our behavior such that, in the long run, we won't make a fool out of ourselves too often
(by saying there's something there that isn't. Now this approach uses p-values and alpha levels to make a decision to either reject the null hypothesis, or to accept the null hypothesis, or - if you want - to remain in doubt about whether the alternative hypothesis is true or not. This is a rule to govern our behavior in the long run. It's important to keep in mind that this doesn't tell you anything about one single test that you're performing. So any current test might be either true or false. We don't know, but what we know is that in the long run, there's a certain percentage of the time that will be correct. So this is the main goal of the path of action. Making decisions about what you should do. The path of knowledge, the second way, focuses on likelihoods. And it tries to answer the question what the likelihood is of different hypotheses given the data that you have collected. Let's take a look at the situation where we flip a coin ten times. This is an old Dutch guilder from when I was young. And we see that there are six heads and four tails. So if you flip this, if you flip a coin ten times and this is the data that you have observed, you can ask yourself the question, is this coin biased or not? So what is the likelihood that this is a fair coin where every option comes up 50% of the time? Or what's the likelihood that this is a biased coin? Now we can plot the likelihood function given the data that we have observed. Let's take a look at the likelihood function. So this curve tells us all the likelihoods of different hypotheses given the data that we have. Now we've observed six heads, so you can see that this, according to the likelihood function, is the most likely possibility, but we can also calculate the likelihood ratio. How much more likely is the data that we have given a fair coin? And we see that this is not very impressive. The likelihood ratio is supposed to be 1, if there's no difference between different hypotheses, and this is pretty close to 1. Later on in the course, we'll talk about how you can really calculate these things. The last option is the path of belief. Now we have flipped a coin ten times, and we saw that it came up heads six times. Now if you see this, do you really believe that the coin in the long run will come up heads 60% of the time? Now that seems rather unlikely. You have previous beliefs about coins, and you have a very strong belief that most coins are fair. So in the long run you might say, well, I've observed this. This one time there were six heads, but I don't really believe that this is what I will see if I continue flipping the coin. I think, still, that 50% probability of heads is what's going to happen if I do this over and over again. So you see that in this case, the data did not really change your prior beliefs. And this path is known as Bayesian statistics, which allows you to express evidence in terms of the degrees of belief. So how much do you believe in a certain hypothesis? So these three different paths, the path of action, the path of devotion, and the path of knowledge. We can sort of make a relationship to Neyman-Pearson statistics, which is the path of action using alpha levels to decide between the null hypothesis and the alternate hypothesis. Bayesian statistics, where we talk about the degree of belief in hypotheses. And likelihoods which tell us something about relative evidence between different hypotheses. Now, in the history of statistics, we see that there's a lot of discussion going on between different sides in this debate. And if you want to know how nasty science can get, then I invite you to take a look at the discussion in the scientific literature between this person on the left, Jerzy Neyman, and the person on the right, Ronald Fisher. Now, Jerzy Neyman is about the path of action. And Ronald Fisher uses p-values in a slightly different way. P-values as a measure of evidence. Now, Ronald Fisher is a genius, one of the few geniuses that we have in science. He's not only a godfather of statistics, he worked a lot on introducing analysis of variance, for example. But he's also a very respected scholar in the field of biology, where he did groundbreaking work as well. So this is no doubt a very smart individual, but there's a lot of debate about the way that he uses p-values to draw inferences from your data. So if there's a discussion between these two individuals, you can say, well, the Neyman-Pierson approach to inference is definitely the best way, the most coherent way, to draw inferences from data using alpha levels and the p-value. So Neyman would be really happy, say, "Haha, after all these years and this intense discussion that we had, I've won. My way of doing statistics is the only logical approach to draw statistical inferences". On the other hand, Fisher might not be too sad. He might say, "Oh forget it! No one knows who you are", which is in general true. I think not a lot of people have heard of the Neyman-Pearson approach of statistics. And he would be pretty happy that everybody uses p-values in, well, arguably, not the optimal way, but at least the way that he proposed. And, well, he gets a lot of respect for this. You might not know this, but the F-distribution, the F-value that you calculate in an ANOVA, is actually named after Fisher in his honor. All right, so these two sides are debating, but now we see that Bayesian statistics is on the rise in recent years. This is Reverend Thomas Bayes, or actually probably not. This is a picture that's circulating that might be him, but it's doubtful that it's actually him. Nevertheless, we'll use it in this course to illustrate the Bayesian perspective. And he would say, "Gentlemen, stop fighting! Who cares about these frequentist approaches to statistics that you think are important because everybody in the future will use Bayesian statistics anyway". We can see whether that's true or not. And maybe Neyman would respond, "Well, I don't have a really high prior that this is ever really going to happen. Which of course, is a sly joke, because he's using prior information to draw an inference in this case. And you see that when there's a discussion between Bayesian statistics and frequentist statistics, all of sudden, of course, these two become perfect friends. They say, "Of course, very good joke!" And they will be in agreement about the way to do statistics. Now the third approach is the likelihood approach. It's not very popular at the moment, but likelihoods underlie Bayesian statistics. And the difference between likelihoods and Bayesian statistics, is that likelihoods do use the relative evidence that's present in the data, as Bayesian statisticians do, but it doesn't rely on this subjective prior. One of the proponents of this approach is Rich Royall. And he might say that nobody really cares about your subjective opinion when you draw inferences from data. So you should ignore this subjective prior and only rely on the likelihood. To which Thomas Bayes might say, "Oh come on, don't be such a nuisance! Nobody even knows what the likelihood paradigm is!" At this moment this might be true, but we'll see you can use it for certain useful things later on in these lectures. Now, it's very important to realize that in this debate, which sometimes feels a little bit like Microsoft versus Apple, there's always this clash between two sides. And people will start to argue vehemently between these two different approaches. For you, it's very important that you can use whatever answers your question. And that's really the main point. It's not either/or. You can even combine these approaches if you want to. So the important take-home message here is that there are three different approaches in how you can draw inferences from your data. These all answer a question you might be interested in. And the main thing is that you realize that there are these different approaches, so you can ask the question that will give you the answer that you want. [MUSIC]