This is the second of a two-course sequence introducing the fundamentals of Bayesian statistics. It builds on the course Bayesian Statistics: From Concept to Data Analysis, which introduces Bayesian methods through use of simple conjugate models. Real-world data often require more sophisticated models to reach realistic conclusions. This course aims to expand our “Bayesian toolbox” with more general models, and computational techniques to fit them. In particular, we will introduce Markov chain Monte Carlo (MCMC) methods, which allow sampling from posterior distributions that have no analytical solution. We will use the open-source, freely available software R (some experience is assumed, e.g., completing the previous course in R) and JAGS (no experience required). We will learn how to construct, fit, assess, and compare Bayesian statistical models to answer scientific questions involving continuous, binary, and count data. This course combines lecture videos, computer demonstrations, readings, exercises, and discussion boards to create an active learning experience. The lectures provide some of the basic mathematical development, explanations of the statistical modeling process, and a few basic modeling techniques commonly used by statisticians. Computer demonstrations provide concrete, practical walkthroughs. Completion of this course will give you access to a wide range of Bayesian analytical tools, customizable to your data.
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From the course by University of California, Santa Cruz
Bayesian Statistics: Techniques and Models
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From the lesson
Count data and hierarchical modeling
Poisson regression, hierarchical modeling
Meet the Instructors
Matthew HeinerDoctoral Student
Statistics
As we just saw, the Poisson model is lacking for these data.
However, assuming the model fit is adequate, we could interpret the results.
Let's look at a summary from these model simulations.
In this model, the intercept is not necessarily interpretable
because it corresponds to a healthy 0 year old.
Whereas in this dataset the youngest person is 20 years old.
It is still an important parameter we just won't interpret it.
For healthy individuals it appears that age
has a positive association with the number of doctor visits.
Clearly bad health is associated with an increase in expected number of visits.
The interaction term is interpreted as an adjustment
to the age coefficient for people in bad health.
So in this case it looks like, for
a person with bad health, the age coefficient would switch
from being positive to being negative because we would subtract this off.
Remember, here the interpretation is if a coefficient is positive,
then there is a positive association between that variable and
the expected number of doctor visits.
If the coefficient is negative, then it's a negative association between that
variable and the number of doctor visits.
Everything we've done so far, could have easily been computed
using the Default Reference Analysis from the functions in R.
So now, let's do something where it really pays off
to have an Bayesian Analysis with posterior samples.
Let's say, we have two people age 35.
One person is in good health and the other person is in poor health.
What is the posterior probability
that the individual with poor health will have more doctor visits?
This goes beyond the posterior probabilities we've calculated
comparing expected responses in the previous lessons.
Here, we need to create Monte Carlo samples for
the actual responses themselves.
This is done by taking the Monte Carlo samples of the model parameters and
for each of those drawing a sample from the likelihood.
Let's walk through this.
First, we need the x values.
The predictor variable values for each individual.
We'll say that the healthy person is person 1,
for person 1, their bad health indicator will be 0 because they are in good health.
Their age is 35, and
the interaction term which is the product of the first two is also 0.
For the second person who is in poor health, we have 1 to indicate bad health.
They are also age 35 and the product of those two for
the interaction term of course is 35.
Let's run these two, okay.
Remember that all the posterior samples for
the model parameters are stored In mod_csim.
So let's look at the head of that, mod_csim.
For these two individuals, we need to compute the linear part of the predictor.
So let's do the log of lambda for person 1.
Which will be mod_csim.
And we want to grab first the intercept term for all of the rows.
And we want the intercept column of that matrix.
Then to that we'll add mod_csim.
We want all rows again.
And now we want columns 2, 1, and
3 in that order because remember, we fit the model.
Because we fit the model first with the bad health indicator, followed by age,
followed by the interaction.
So column 2 is the bad health indicator.
Then we need the age, which is column 1 and
then we need the interaction term which is column three.
We will matrix multiply this with the values of the predictors for person one.
We also need to do this for person two.
And the only thing we need to change is that we're using the predictor values for
x for person two.
We'll call that logLan2.
We run both of these and
this gives us a Monte Carlo sample
of the linear part of the model.
So we have one for each sample from the posterior.
Now that we have samples from the posterior distribution of the log of
lambda for these individuals, we need to get the posterior distribution of lambda.
So we'll say lam1 will be an exponentiated version of log lambda 1.
And we'll do the same for person two.
We now have Monte Carlo samples for the Poisson mean for these two individuals.
Let's take a look at the posterior distribution for Lambda 1 for person 1.
Let's plot a density estimate of lam1.
For person 1, we would expect on average for
them to have between 1.8 and 2 doctor visits.
Remember, our question was what is the posterior probability that
person two will have more doctor visits?
That's different than the number of doctor visits on average,
we want actual doctor visits.
So now, with the posterior distribution for Lambda, we now need to
simulate a Monte Carlo sample of actual number of visits for each of them.
First we need to know how many simulations we're calculating.
This would be the length of lambda 1.
Which is also the length of lambda 2.
Its 15,000 because we ran three chains each with 5,000 iterations.
Simulating from the likelihood is very straightforward here.
For person1 we'll draw from a Poisson distribution, rpois.
And we're going to create n_sim,
simulations from that Poisson distribution, where the mean is lambda 1.
So for each draw of the Poisson we're going to use a different value
from our Monte Carlo sample of lambda 1.
The same goes for person 2.
Their doctor visits that we're going to simulate
will come from a Poisson distribution.
Where the means will come from our Monte Carlo sample for the lambda twos.
We'll run that.
And let's plot the posterior distribution for these number of visits for
the two people.
First we want to plot for person 1, we want to plot a table
of y1 as a factor variable.
So we need to say factor(y1) and
let's plot this for up to 18 visits.
We want this table to output probabilities instead of counts, so
let's divide by the number of simulations.
Let's run this plot.
So along the x axis we have the number of doctor visits and
along the y axis the probability of those number of doctor visits.
There is also a possibility that they have 0 doctor visits so
let's include 0 as a level.
Right there's a 15% probability that this person has zero doctor visits.
Now let's add the posterior distribution for person two.
I'm going to shift them just slightly here so that we can see both sets of points.
We'll make this one colored red.
And we need to change this from plots to points.
We can also get rid of the factor, variable here.
And now let's run it.
We now have the posterior distribution of the number of visits for
the person in poor health and the posterior distribution for
the number of visits of the person in good health.
Finally, we can answer our original question.
What is the probability that the person with poor health
will have more doctor visits than the person with good health?
We can take take the Monte Carlo mean of that indicator variable,
y2 is greater than y1.
For each Monte Carlo sample,
it evaluates whether person two had more doctor visits than person one.
And the average of that is the posterior probability.
Let's run that.
We get about 92%.
That is, the posterior probability that person2
will have more doctor visits than person1 is 92%.
Because we used our posterior samples for the model parameters in our simulation.
This posterior predictive distribution on the number of visits for these two new
individuals naturally takes into account our uncertainty in these model estimates.
That creates for us a more honest or
realistic distribution than we would get if we had just fixed the model
parameters at their maximum likelihood estimates or some other point estimate.
It's a nice example of the advantages that a Bayesian model can give us here.
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