The Meaning of Entropy

Entropy is a word that we see a lot in various forms. It’s classical use comes from thermodynamics: e.g. “the entropy in the universe is always increasing.” With the recent boom in statistics and machine learning, the word has also seen a surge in use in information-theoretic contexts: e.g. “minimize the cross-entropy of the validation set.”

It’s been an ongoing investigation for me, trying to figure out just what the hell this information-theoretic entropy is all about, and how it connects to the notion I’m familiar with from statistical mechanics. Reading through the wonderful book Data Analysis: a Bayesian Tutorial by D. S. Sivia, I found the first connection between these two notions that really clicked for me. I’m going to run through the basic argument here, in the hope that reframing it in my own words will help me understand it more thoroughly.

Entropy in Thermodynamics

Let’s start with the more intuitive notion, which is that of thermodynamic entropy. This notion, when poorly explained, can seem opaque or quixotic; however, when viewed through the right lens, it is straightforward, and the law of increasing entropy becomes a highly intuitive result.

Counting Microstates

Imagine, if you will, the bedroom of a teenager. We want to talk about the entropy of two different states: the state of being “messy” and the state of being “clean.” We will call these macrostates; they describe the macroscopic (large-scale) view of the room. However, there are also many different microstates. One can resolve these on a variety of scales, but let’s just say they correspond to the location/position of each individual object in the room. To review:

Type Definition Example
Macrostate Overall Description “Messy”
Microstate Fine-Scale Description “Underwear on lamp, shoes in bed, etc.”

The Boltzmann Entropy

One might notice an interesting fact: that there are many more possible microstates that correspond to “messy” than there are microstates that correspond to “clean.” This is exactly what we mean when we say that a messy room has higher entropy. In particular, the entropy of a macrostate is the log of the number of microstates that correspond to that macrostate. We call this the Boltzmann entropy, and denote it by \(S_B\). If there are \(\Omega\) possible microstates that correspond to the macrostate of being “messy,” then we define the entropy of this state as1

\[S_B(\text{messy}) = \log(\Omega).\]

This is essentiall all we need to know here.2 The entropy tells us how many different ways there are to get a certian state. A pyramid of oranges in a supermarket has lower entropy than the oranges fallen all over the floor, because there are many configurations of oranges that we would call “oranges all over the floor,” but very few that we would call “a nicely organized pyramid of oranges.”

In this context, the law of increasing entropy becomes almost tautological. If things are moving around in our bedroom at random, and we call most of those configurations “messy,” then the room will tend towards messyness rather than cleanliness. We sometimes use the terms “order” and “disorder” to refer to states of relatively low and high entropy, respectively.

Entropy in Information Theory

One also frequently encounters a notion of entropy in statistics and information theory. This is called the Shannon entropy, and the motivation for this post is my persistent puzzlement over the connection between Boltzmann’s notion of entropy and Shannon’s. Previous to reading D. Sivia’s manual, I only knew the definition of Shannon entropy, but his work presented such a clear exposition of the connection to Boltzmann’s ideas that I felt compelled to share it.

Permutations and Probabilities

We’ll work with a thought experiment.3 Suppose we have \(N\) subjects we organize into \(M\) groups, with \(N\gg M\). Let \(n_i\) indicate the number of subjects that are in the \(i^\text{th}\) group, for \(i=1,\ldots,M\). Of course,

\[\sum_{i=1}^M n_i = N,\]

and if we choose a person at random the probability that they are in group \(i\) is

\[p_i = \frac{n_i}{N}.\]

The Shannon entropy of such a discrete distribution is defined as

\[S = -\sum_{i=1}^M p_i\log(p_i)\]

But why? Why \(p\log(p)\)? Let’s look and see.

A macrostate of this system is defined by the size of the groups \(n_i\); equivalently, it is defined as the probability distribution. A microstate of this system is specifying the group of each subject: the specification that subject number \(j\) is in group \(i\) for each \(j=1,\ldots,N\). How many microstates correspond to a given macrostate? For the first group, we can fill it with any of the \(N\) participants, and we must choose \(n_1\) members of the group, so the number of ways of assigning participants to this group is

\[{N\choose n_1} = \frac{N!}{n_1!(N-n_1)!}\]

For the second group, there are \(N - n_1\) remaining subjects, and we must assign \(n_2\) of them, and so on. Thus, the total number of ways of arranging the \(N\) balls into the groups of size \(n_i\) is

\[\Omega = {N\choose n_1}{N-n_1 \choose n_2}\ldots {N-n_1-\ldots-n_{M-1}\choose n_M}.\]

This horrendous list of binomial coefficients can be simplified down to just

\[\Omega = \frac{N!}{n_1!n_2!\ldots n_M!}.\]

The Boltzmann entropy of this macrostate is then

\[S_B = \log(\Omega) = \log(N!) - \sum_{i=1}^M \log(n_i!)\]

From Boltzmann to Shannon

We will now show that the Boltzmann entropy is (approimxately) a scaling of the Shannon entropy; in particular, \(S_B \approx N\,S\). Things are going to get slightly complicated in the algebra, but hang on. If you’d prefer, you can take my word for it, and skip to the next section.

We will use the Stirling approximation \(\log(n!)\approx n\log(n)\)4 to simplify:

\[S_B \approx N\log(N) - \sum_{i=1}^M n_i\log(n_i)\]

Since the probability \(p_i=n_i/N\), we can re-express \(S_b\) in terms of \(p_i\) via

\[S_B \approx N\log(N)-N\sum_{i=1}^M p_i\log(Np_i)\]

Since \(\sum_ip_i=1\), we have

\[S_B \approx -N\sum_{i=1}^M p_i\log(p_i) = N \, S.\]

Phew! So, the Boltzmann entropy \(S_b\) of having \(N\) students in \(M\) groups with sized \(n_i\) is (approximately) \(N\) times the Shannon entropy.

Who Cares?

Admittedly, this kind of theoretical revalation will probably not change the way you deploy cross-entropy in your machine learning projects. It is primarily used because its gradients behave well, which is important in the stochastic gradient-descent algorithms favored by modern deep-learning architectures. However, I personally have a strong dislike of using tools that I don’t have both a theoretical understanding of; hopefully you now have a better grip on the theoretical underpinnings of cross entropy, and its relationship to statistical mechanics.

  1. Often a constant will be included in this definition, so that \(S=k_B \log(\Omega)\). This constant is arbitrary, as it simply rescales the units of our entropy, and it will only serve to get in the way of our analysis, so we omit it. 

  2. All we need to know for the purpose of establishing a connection between thermodynamic and information-theoretic entropy; of course there is much more to know, and there are many alternative ways of conceptualizing entropy. However, none of these have ever been intuitive to me in the way that Boltzmann’s definition of entropy is. 

  3. We have slightly rephrased Sivia’s presentation to fit our purposes here. 

  4. The most commonly used form of Stirling’s approximation is the more precise \(\log(n!)\approx n\log(n)-n\), but we use a coarser form here.