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Frequentism, Likelihoods, Bayesian statistics

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Improving your statistical inferences

アイントホーフェン工科大学（Eindhoven University of Technology）

4.9 (433 件の評価) | 29K在籍する学生

This course aims to help you to draw better statistical inferences from empirical research. First, we will discuss how to correctly interpret p-values, effect sizes, confidence intervals, Bayes Factors, and likelihood ratios, and how these statistics answer different questions you might be interested in. Then, you will learn how to design experiments where the false positive rate is controlled, and how to decide upon the sample size for your study, for example in order to achieve high statistical power. Subsequently, you will learn how to interpret evidence in the scientific literature given widespread publication bias, for example by learning about p-curve analysis. Finally, we will talk about how to do philosophy of science, theory construction, and cumulative science, including how to perform replication studies, why and how to pre-register your experiment, and how to share your results following Open Science principles. In practical, hands on assignments, you will learn how to simulate t-tests to learn which p-values you can expect, calculate likelihood ratio's and get an introduction the binomial Bayesian statistics, and learn about the positive predictive value which expresses the probability published research findings are true. We will experience the problems with optional stopping and learn how to prevent these problems by using sequential analyses. You will calculate effect sizes, see how confidence intervals work through simulations, and practice doing a-priori power analyses. Finally, you will learn how to examine whether the null hypothesis is true using equivalence testing and Bayesian statistics, and how to pre-register a study, and share your data on the Open Science Framework. All videos now have Chinese subtitles. More than 10.000 learners have enrolled so far!

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学習するスキル

Likelihood Function, Bayesian Statistics, P-Value, Statistical Inference

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4.9 (433 件の評価)

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BH

Oct 06, 2017

This is a top-notch course. The ground (especially pitfalls) is very well covered, and useful free tools are engaged (R, G*Power, prof's own spreadsheets for calculating effect size).

TZ

Jun 19, 2018

This course changed my concepts not only about statistics but about research and science. Daniel Lakens is a fantastic lecturer and scientist. I can't recommend this course enough.

レッスンから

Introduction + Frequentist Statistics

Frequentism, Likelihoods, Bayesian statistics9:25

講師

Daniel Lakens
Associate Professor

字幕

[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 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.

The Karma yoga, the Jnana yoga, and the Bhakti yoga.

Now if you see what the Hinduist book [UNKNOWN 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.

Now this approach uses p-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 hypotheses given the data that you have collected.

Let's take a look at the situation where you flip a coin ten times.

This is an old Dutch from when I was young.

And we see that there are six heads and four tales.

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 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 likelihood 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's supposed to be 1,

if there's no difference between different hypothesis, 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 hypothesis.

And likelihoods which tell us something about relative evidence between different

hypothesis.

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, Jerzey Neyman, and the person on the right, Ronald Fisher.

Now, Jerzey 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, 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, gentleman, Quit 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 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 underlay 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 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, come on, don't be such a nuisance.

Nobody even knows what 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 all

these different approaches, so

you can ask the question that will give you the answer that you want.

[MUSIC]

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