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Table of contents
The performance of this campaign seems extremely high given how our other campaigns have done historically. Let's overlay this likelihood function with the distribution of click-through rates from our previous campaigns:. Clearly, the maximum likelihood method is giving us a value that is outside what we would normally see. Perhaps our analysts are right to be skeptical; as the campaign continues to run, its click-through rate could decrease.
Alternatively, this campaign could be truly outperforming all previous campaigns. We can't be sure. Ideally, we would rely on other campaigns' history if we had no data from our new campaign. And as we got more and more data, we would allow the new campaign data to speak for itself. This skepticism corresponds to prior probability in Bayesian inference. Generally, prior distributions can be chosen with many goals in mind:.
For our example, because we have related data and limited data on the new campaign, we will use an informative, empirical prior. Below, we fit the beta distribution and compare the estimated prior distribution with previous click-through rates to ensure the two are properly aligned:. The beta distribution with these parameters does a good job capturing the click-through rates from our previous campaigns, so we will use it as our prior.
We will now update our prior beliefs with the data from the facebook-yellow-dress campaign to form our posterior distribution. From the earlier section introducing Bayes' Theorem, our posterior distribution is given by the product of our likelihood function and our prior distribution:.
Usually, the true posterior must be approximated with numerical methods. To see why, let's return to the definition of the posterior distribution:. A more descriptive representation of this quantity is given by:. This integral usually does not have a closed-form solution, so we need an approximation. One method of approximating our posterior is by using Markov Chain Monte Carlo MCMC , which generates samples in a way that mimics the unknown distribution.
We begin at a particular value, and "propose" another value as a sample according to a stochastic process. We may reject the sample if the proposed value seems unlikely and propose another. If we accept the proposal, we move to the new value and propose another. PyMC is a python package for building arbitrary probability models and obtaining samples from the posterior distributions of unknown variables given the model.
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In our example, we'll use MCMC to obtain the samples. Let's now obtain samples from the posterior. We select our prior as a Beta Let's see how observing 7 clicks from 10 impressions updates our beliefs:. Model creates a PyMC model object.
Bayesian inference - Wikipedia
All PyMC objects created within the context manager are added to the model object. This random variable is generated from a beta distribution pm. Beta ; we name this random variable "prior" and hardcode parameter values We could have set the values of these parameters as random variables as well, but we hardcode them here as they are known. This statement represents the likelihood of the data under the model. Again we define the variable name and set parameter values with n and p. Note that for this variable, the parameter p is assigned to a random variable, indicating that we are trying to model that variable.
Lastly, we provide observed instances of the variable i. Because we have said this variable is observed, the model will not try to change its values. These three lines define how we are going to sample values from the posterior. The sampling algorithm defines how we propose new samples given our current state. The proposals can be done completely randomly, in which case we'll reject samples a lot, or we can propose samples more intelligently. Other choices include Metropolis Hastings, Gibbs, and Slice sampling. Lastly, pm. The data has caused us to believe that the true click-through rate is higher than we originally thought, but far lower than the 0.
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Why is this the case? 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. Very good introduction to Bayesian Statistics. Very interactive with Labs in Rmarkdown.
The course is compact that I've learnt a lot of new concepts in a week of coursework. A good sampler of topics related to Bayesian Statistics. It prints out a bunch of descriptive statistics and a reminder of what the null and alternative hypotheses are, before finally getting to the test results. I wrote it that way deliberately, in order to help make things a little clearer for people who are new to statistics.
Notice that format of this command is pretty standard. As usual we have a formula argument in which we specify the outcome variable on the left hand side and the grouping variable on the right. The data argument is used to specify the data frame containing the variables. This is because the BayesFactor package does not include an analog of the Welch test, only the Student test.
So what does all this mean? Just as we saw with the contingencyTableBF function, the output is pretty dense. But you already knew that. So the only part that really matters is this line here:. This is the Bayes factor: the evidence provided by these data are about 1. According to the orthodox test, we obtained a significant result, though only barely.
Bayesian methods usually require more evidence before rejecting the null. Back in Section The easiest way to do it with this data set is to use the x argument to specify one variable and the y argument to specify the other.
Introduction to bayesian statistics
At this point, I hope you can read this output without any difficulty. The data provide evidence of about in favour of the alternative. We could probably reject the null with some confidence! In Chapter 15 I used the parenthood data to illustrate the basic ideas behind regression. Back in Chapter 15 I proposed a theory in which my grumpiness dan. We tested this using a regression model. In order to estimate the regression model we used the lm function, like so:.
The hypothesis tests for each of the terms in the regression model were extracted using the summary function as shown below:.
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When interpreting the results, each row in this table corresponds to one of the possible predictors. The important thing for our purposes is the fact that dan. Okay, so how do we do the same thing using the BayesFactor package? The easiest way is to use the regressionBF function instead of lm. As before, we use formula to indicate what the full regression model looks like, and the data argument to specify the data frame. So the command is:. The output, however, is a little different from what you get from lm.
The format of this is pretty familiar. At the bottom we have some techical rubbish, and at the top we have some information about the Bayes factors.
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One possibility is the intercept only model , in which none of the three variables have an effect. At the other end of the spectrum is the full model in which all three variables matter. So what regressionBF does is treat the intercept only model as the null hypothesis, and print out the Bayes factors for all other models when compared against that null. What I find helpful is to start out by working out which model is the best one, and then seeing how well all the alternatives compare to it.
First, we have to go back and save the Bayes factor information to a variable:. This is telling us that the model in line 1 i.