This course describes Bayesian statistics, in which one's inferences about parameters or hypotheses are updated as evidence accumulates. You will learn to use Bayes’ rule to transform prior probabilities into posterior probabilities, and be introduced to the underlying theory and perspective of the Bayesian paradigm. The course will apply Bayesian methods to several practical problems, to show end-to-end Bayesian analyses that move from framing the question to building models to eliciting prior probabilities to implementing in R (free statistical software) the final posterior distribution. Additionally, the course will introduce credible regions, Bayesian comparisons of means and proportions, Bayesian regression and inference using multiple models, and discussion of Bayesian prediction. We assume learners in this course have background knowledge equivalent to what is covered in the earlier three courses in this specialization: "Introduction to Probability and Data," "Inferential Statistics," and "Linear Regression and Modeling."
Bayesian methods and big data: a talk with David Dunson
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Do curso por Duke University
Bayesian Statistics
415 classificações
Na lição
Perspectives on Bayesian Applications
This week consists of interviews with statisticians on how they use Bayesian statistics in their work, as well as the final project in the course.
Conheça os instrutores
Mine Çetinkaya-RundelAssociate Professor of the Practice
Department of Statistical Science
David BanksProfessor of the Practice
Statistical Science
Colin RundelAssistant Professor of the Practice
Statistical Science
Merlise A ClydeProfessor
Department of Statistical Science
Hi, I'm Kyle Burris.
I'm a first-year statistics PhD student here at Duke.
>> Hi, I'm Callie Mao, and I'm an undergraduate statistics student at Duke.
This is Dr. David Dunson, Professor of Statistical Science.
>> Hi I'm David.
I work a lot in Bayesian statistical methods and so I've been doing that for
about 20 or 25 years now and a lot of it is motivated by high
dimensional interesting biomedical applications.
>> Could you tell us about some of the application in
Bayesian methods in the health data that you're working with?
>> Yeah, we have a whole enormous variety of high dimensional health data that
we're motivated by.
I think that there's been this enormous profusion of high dimensional really
complicated data in a variety of fields, and so I just give one example
that we're really excited about lately which is neurosciences.
And so there's a lot of new imaging technology that allows you to actually
measure the connection structure in an individual's brain and so
you guys can go in and do a brain scan called a diffuse intenser imaging scan.
And also get structural imaging, and based on that imaging scan, you can actually
recreate the fibers connecting different regions of your brain.
And so you guys might be wired slightly differently, and
that might be related to your personality traits,
tendency to have depression, creativity, intelligence in different domains.
And so we'd like to kind of figure out those relationships and so
we've been working on that a lot.
Lately using Bayesian methods and so the data are really intriguing.
They're kind of curves linking up different regions of the brain.
They have a network type structure and we're trying to build models for
discovering low dimensional structure
in the brain related to phenotypes like intelligence, creative reasoning, etc.
>> And so how can Bayesian statistics in particular tell us more about the brain?
>> It's interesting that most of the literature recently
has been dominated by optimization methods,
a really simple methods that take the data and look at tiny pieces of it separately.
And so imagine that we have a big network in your brain, okay?
And so everyone in the study has a slightly different network,
and we'd like to know how are the features of the network related to
traits of that individual.
And so can we do that using non Bayesian methods?
Well, we could take little pieces of the network and just do a test separately for
the relationship between, say, IQ and this link in the network, get a p-value and
do that for every possible link in the network.
And then maybe do some sort of adjustment to avoid having too many false
discoveries.
So that turns out to do extremely badly, if you do tests.
The other thing people do in kind of modern statistics is to do an optimization
type of approach.
And so you try to take that brain network and
then maybe do some sort of singular value decomposition or
matrix factorization to learn some low-dimensional structure.
Then you can do that really fast, but the problem with that type of approach,
which has really dominated a lot of literature is that
it just gives you a point estimate, okay?
And so, let's say I want to study the relationship between
some mental health disorder or aging and brain structure, okay?
Well, if I just get a point estimate of that, that's just one guess or one, maybe,
best guess of what's going on.
It doesn't tell me how uncertain I am in that and so I can't really publish
an article on that or feel like I am confident about that result.
It might be that there's 10,000 other different relationships that are equally
consistent with the data, and I've just estimated one of them.
And so, the really distinct characteristic of Bayesian methods
is their ability to characterize uncertainty.
Uncertainty in scientific inferences, in this case.
And so I can say, what's the post to your probability that there's any
relationship between IQ and brain structure, say.
I can also say, well, what's the post to your probability in particular region
that there's a relationship between IQ and brain structure.
The other thing that people do is they will just take the data in the brain
network and they'll extract certain features.
They're called topological features of the network.
And they might take three, or four, or five of these different features and
then just do a statistical analysis based on that.
But that also fixates to some particular features, and
if you collect too many you'll end up having false positives.
In a Bayesian approach you can holistically model
the entire brain structure flexibly while allowing uncertainty.
So I think it's been quite exciting.
We've already found very intriguing relationships between brain structure and
Alzheimer's disease.
And also big differences between individuals having low creative
reasoning scores and high creative reasoning scores in terms of
connections in the frontal lobe across hemispheres and so it's been a lot of fun.
>> You just talked a little bit about the advances that Bayesian statistics is
making in a variety of applications such as neuroscience and brain imaging.
But what advances are being made in the field of Bayesian statistics in general?
>> Yeah, it's been really exciting, I think.
In general, Bayesian statistics if you kind of back up in time,
there's been these different kind of revolutions essentially.
And so prior to the onset of Markov Chain Monte Carlo methods in the late 80s,
early 90s,
it was more of a small philosophical field where people would work on toy problems.
And then once we had more computational tools,
it sort of exploded into being a very applied driven, applied motivated
field that could have a substantial impact on lot of application areas.
But that's kind of changing a little recently, as with the data becomes so
big, that these traditional algorithms that people were using in the 90s,
early 2000s, no longer usable.
And so one of the really exciting things in Bayesian statistics is trying to design
completely new algorithms, new ways to do inferences, new types of models for
really hugely high dimensional complicated data while allowing uncertainty.
Can we talk to people in computer science and machine learning, and figure out ways
to kind of scale up algorithms using big computing systems, distributed computing.
And that's been really interesting and intriguing,
while also maintaining theoretical guarantees that these methods do well.
So that's been a big part of our work and
there's been an explosion in this area in general, I think.
>> Can you tell us a bit more about the application of Bayesian
methods to big data?
>> Yeah, so it's interesting because what people mean by big data exactly.
And so, a lot of the people in the literature are working on
Bayesian methods have been focused on big data meaning really large sample sizes.
And so that we might have a 100 million subjects or something.
But, if you don't have a really large sample size, but
the number of variables you're collecting isn't that big, and
you're doing something like say, a logistic regression, which is a toy
example people almost always use in motivating these types of methods,
then actually, there's often not that much motivation for
doing a Bayesian approach relative to some fast optimization approach.
Because, I have 50 parameters and I have a 100 million sample size,
well, my posterior distribution might be essentially,
really super concentrated right around something that we would get
by just doing say maximum likely estimation.
And so the really interesting problems in big data are when you actually
have really large numbers of variables or you're trying to fit a really flexible
model like a non-parametrical model or you have something like rare events.
So, for example, in computational advertising, we might be looking at people
going from a large number of websites to a small set of websites, client websites.
And so there might be hundreds of thousands websites and
then a hundred client websites, and those transitions can be really rare.
And so, even though you have millions of people, you have rare events.
And so, then we found that the uncertainty in those transitions is super important.
And so, if you just do an optimization approach, you might be quite misled.
And so this problem of kind of scaling up to characterize the complexity of big data
while allowing uncertainty, is really important in most of these settings where
you have really high dimensional data, but also the sample size is enormous, I think.
>> All right, well thank you so much for your time.
>> Thank you, it's been fun.
>> Pleasure speaking with you.
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