Frequentist, Likelihood, and Bayesian Approaches to Statistical Inferences by Daniel Lakens

[Music] in this lecture we'll talk about three different ways in which you can draw two tist achill 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 towards this truth there are many different ways to try to do this one useful metaphor that I think that help can help you think about these differences is if we look at Hinduism where there are three ways to reach enlightenment the karmayoga the Yan'an yoga and the bhakti yoga now if you see what the Hindu his book bhagavad-gita says about this this is 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 quiz questions that you can ask if you do statistics now Royalle 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 now this approach uses B values and alpha levels to either make a decision to 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 hypothesis given the data that you have collected let's take a look at this situation where we flip a coin 10 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 curves tells us all the likelihoods of different hypothesis given the data that we have now we've observed six heads so you can see that this according to the likelihood functions 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 10 times we saw that it came up heads 6 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 absolutely I've observed this this one time there were six heads but I don't really believe that this is what I'll see if I continue flipping the coin I think still that 50% probability of heads is what's gonna 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 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 alternative hypothesis Bayesian statistics where we talk about the degree of belief in an hypothesis and likelihood switch 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 is 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-pearson 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 ha ha after all these years and this intense discussion that we have 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 to said 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 Fischer 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 Moe 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 very high prior that this is ever really going to happen which of course is a slight 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 a sudden of course these to 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 likely hoots underlie Bayesian statistics and the difference between likelihoods and bayesian statistics so 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 Royale and he might say something 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 c'mon don't be such a nuisance nobody even knows what likelihood paradigm is and at this moment this might be true but we'll see that 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 that's these two different approaches for you it's very important that you can use whatever answers your question and that's really the main points 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]

Channel: Daniel Lakens

Views: 7,341

Keywords: statistics, probability, frequentist, likelihood, bayesian, statistical inference

Id: 7-Ud4nyHO_Q

Length: 9min 26sec (566 seconds)

Published: Sun Sep 15 2019