Learning Bayesian Models with R: Become an expert in Bayesian Machine Learning methods using R and apply them to solve real-world big data problems

In both classical and Bayesian approaches, a probability distribution function is the central quantity, which captures all of the information about the relationship between variables in the presence of uncertainty. A probability distribution assigns a probability value to each measurable subset of outcomes of a random experiment. The variable involved could be discrete or continuous, and univariate or multivariate. Although people use slightly different terminologies, the commonly used probability distributions for the different types of random variables are as follows:

One of the well-known distribution functions is the normal or Gaussian distribution, which is named after Carl Friedrich Gauss, a famous German mathematician and physicist. It is also known by the name bell curve because of its shape. The mathematical form of this distribution is given by:

Here, is the mean or location parameter and is the standard deviation or scale parameter ( is called variance). The following graphs show what the distribution looks like for different values of location and scale parameters:

One can see that as the mean changes, the location of the peak of the distribution changes. Similarly, when the standard deviation changes, the width of the distribution also changes.

Many natural datasets follow normal distribution because, according to the central limit theorem, any random variable that can be composed as a mean of independent random variables will have a normal distribution. This is irrespective of the form of the distribution of this random variable, as long as they have finite mean and variance and all are drawn from the same original distribution. A normal distribution is also very popular among data scientists because in many statistical inferences, theoretical results can be derived if the underlying distribution is normal.

Now, let us look at the multidimensional version of normal distribution. If the random variable is an N-dimensional vector, x is denoted by:

Then, the corresponding normal distribution is given by:

Here, corresponds to the mean (also called location) and is an N x N covariance matrix (also called scale).

To get a better understanding of the multidimensional normal distribution, let us take the case of two dimensions. In this case, and the covariance matrix is given by:

Here, and are the variances along and directions, and is the correlation between and . A plot of two-dimensional normal distribution for , , and is shown in the following image:

If , then the two-dimensional normal distribution will be reduced to the product of two one-dimensional normal distributions, since would become diagonal in this case. The following 2D projections of normal distribution for the same values of and but with and illustrate this case:

The high correlation between x and y in the first case forces most of the data points along the 45 degree line and makes the distribution more anisotropic; whereas, in the second case, when the correlation is zero, the distribution is more isotropic.

We will briefly review some of the other well-known distributions used in Bayesian inference here.

Often, one would be interested in finding the probability of the occurrence of a set of random variables when other random variables in the problem are held fixed. As an example of population health study, one would be interested in finding what is the probability of a person, in the age range 40-50, developing heart disease with high blood pressure and diabetes. Questions such as these can be modeled using conditional probability, which is defined as the probability of an event, given that another event has happened. More formally, if we take the variables A and B, this definition can be rewritten as follows:

Similarly:

The following Venn diagram explains the concept more clearly:

In Bayesian inference, we are interested in conditional probabilities corresponding to multivariate distributions. If denotes the entire random variable set, then the conditional probability of , given that is fixed at some value, is given by the ratio of joint probability of and joint probability of :

In the case of two-dimensional normal distribution, the conditional probability of interest is as follows:

It can be shown that (exercise 2 in the Exercises section of this chapter) the RHS can be simplified, resulting in an expression for in the form of a normal distribution again with the mean and variance .

From the definition of the conditional probabilities and , it is easy to show the following:

Rev. Thomas Bayes (1701–1761) used this rule and formulated his famous Bayes theorem that can be interpreted if represents the initial degree of belief (or prior probability) in the value of a random variable A before observing B; then, its posterior probability or degree of belief after accounted for B will get updated according to the preceding equation. So, the Bayesian inference essentially corresponds to updating beliefs about an uncertain system after having made some observations about it. In the sense, this is also how we human beings learn about the world. For example, before we visit a new city, we will have certain prior knowledge about the place after reading from books or on the Web.

However, soon after we reach the place, this belief will get updated based on our initial experience of the place. We continuously update the belief as we explore the new city more and more. We will describe Bayesian inference more in detail in Chapter 3, Introducing Bayesian Inference.

In many situations, we are interested only in the probability distribution of a subset of random variables. For example, in the heart disease problem mentioned in the previous section, if we want to infer the probability of people in a population having a heart disease as a function of their age only, we need to integrate out the effect of other random variables such as blood pressure and diabetes. This is called marginalization:

Or:

Note that marginal distribution is very different from conditional distribution. In conditional probability, we are finding the probability of a subset of random variables with values of other random variables fixed (conditioned) at a given value. In the case of marginal distribution, we are eliminating the effect of a subset of random variables by integrating them out (in the sense averaging their effect) from the joint distribution. For example, in the case of two-dimensional normal distribution, marginalization with respect to one variable will result in a one-dimensional normal distribution of the other variable, as follows:

The details of this integration is given as an exercise (exercise 3 in the Exercises section of this chapter).

Having known the distribution of a set of random variables , what one would be typically interested in for real-life applications is to be able to estimate the average values of these random variables and the correlations between them. These are computed formally using the following expressions:

For example, in the case of two-dimensional normal distribution, if we are interested in finding the correlation between the variables and , it can be formally computed from the joint distribution using the following formula:

Binomial distribution

A binomial distribution is a discrete distribution that gives the probability of heads in n independent trials where each trial has one of two possible outcomes, heads or tails, with the probability of heads being p. Each of the trials is called a Bernoulli trial. The functional form of the binomial distribution is given by:

Here, denotes the probability of having k heads in n trials. The mean of the binomial distribution is given by np and variance is given by np(1-p). Have a look at the following graphs:

The preceding graphs show the binomial distribution for two values of n; 100 and 1000 for p = 0.7. As you can see, when n becomes large, the Binomial distribution becomes sharply peaked. It can be shown that, in the large n limit, a binomial distribution can be approximated using a normal distribution with mean np and variance np(1-p). This is a characteristic shared by many discrete distributions that, in the large n limit, they can be approximated by some continuous distributions.

Beta distribution

The Beta distribution denoted by is a function of the power of , and its reflection is given by:

Here, are parameters that determine the shape of the distribution function and is the Beta function given by the ratio of Gamma functions: .

The Beta distribution is a very important distribution in Bayesian inference. It is the conjugate prior probability distribution (which will be defined more precisely in the next chapter) for binomial, Bernoulli, negative binomial, and geometric distributions. It is used for modeling the random behavior of percentages and proportions. For example, the Beta distribution has been used for modeling allele frequencies in population genetics, time allocation in project management, the proportion of minerals in rocks, and heterogeneity in the probability of HIV transmission.

Gamma distribution

The Gamma distribution denoted by is another common distribution used in Bayesian inference. It is used for modeling the waiting times such as survival rates. Special cases of the Gamma distribution are the well-known Exponential and Chi-Square distributions.

In Bayesian inference, the Gamma distribution is used as a conjugate prior for the inverse of variance of a one-dimensional normal distribution or parameters such as the rate () of an exponential or Poisson distribution.

The mathematical form of a Gamma distribution is given by:

Here, and are the shape and rate parameters, respectively (both take values greater than zero). There is also a form in terms of the scale parameter , which is common in econometrics. Another related distribution is the Inverse-Gamma distribution that is the distribution of the reciprocal of a variable that is distributed according to the Gamma distribution. It's mainly used in Bayesian inference as the conjugate prior distribution for the variance of a one-dimensional normal distribution.

Dirichlet distribution

The Dirichlet distribution is a multivariate analogue of the Beta distribution. It is commonly used in Bayesian inference as the conjugate prior distribution for multinomial distribution and categorical distribution. The main reason for this is that it is easy to implement inference techniques, such as Gibbs sampling, on the Dirichlet-multinomial distribution.

The Dirichlet distribution of order is defined over an open dimensional simplex as follows:

Here, , , and .

Wishart distribution

The Wishart distribution is a multivariate generalization of the Gamma distribution. It is defined over symmetric non-negative matrix-valued random variables. In Bayesian inference, it is used as the conjugate prior to estimate the distribution of inverse of the covariance matrix (or precision matrix) of the normal distribution. When we discussed Gamma distribution, we said it is used as a conjugate distribution for the inverse of the variance of the one-dimensional normal distribution.

The mathematical definition of the Wishart distribution is as follows:

Here, denotes the determinant of the matrix of dimension and is the degrees of freedom.

A special case of the Wishart distribution is when corresponds to the well-known Chi-Square distribution function with degrees of freedom.

Wikipedia gives a list of more than 100 useful distributions that are commonly used by statisticians (reference 1 in the Reference section of this chapter). Interested readers should refer to this article.

To summarize this chapter, we discussed elements of probability theory; particularly those aspects required for learning Bayesian inference. Due to lack of space, we have not covered many elementary aspects of this subject. There are some excellent books on this subject, for example, books by William Feller (reference 2 in the References section of this chapter), E. T. Jaynes (reference 3 in the References section of this chapter), and M. Radziwill (reference 4 in the References section of this chapter). Readers are encouraged to read these to get a more in-depth understanding of probability theory and how it can be applied in real-life situations.

In the next chapter, we will introduce the R programming language that is the most popular open source framework for data analysis and Bayesian inference in particular.