Alex Alemi's Blog

Why KL?

Alexander A. Alemi.

The Kullback-Liebler divergence, or KL divergence, or relative entropy, or relative information, or information gain, or expected weight of evidence, or information divergence (it goes by a lot of different names) is unique among the ways to measure the difference between two probability distributions. It holds a special and privileged place, being used to define all of the core concepts in information theory, such as mutual information.

Why is the relative information so special and where does it come from? How should you interpret it? What is a nat anyway? In this note, I'll try to give a better understanding and set of intuitions about what KL is, why it's interesting, where it comes from and what its good for.

Information Gain

Let's see if we can motivate the form of the KL axiomatically.

Imagine we have some prior set of beliefs summarized as a probability distribution $q$. In light of some kind of evidence, we update our beliefs to a new distribution $p$. How much did we update our beliefs? How do we quantify the magnitude of that update? What are some properties we might want this hypothetical function to have? Let $I[p; q]$ denote the function that measures how much we moved beliefs when we switch from beliefs $q$ to beliefs $p$. We'll call this amount of update the information gain when we move from $q$ to $p$. 1

We want our information function to satisfy the following properties:

  1. It's continuous. A small change in the distributions makes a small change in the amount of information in the move.
  2. It's permutation or reparameterization independent. It doesn't matter if we change the units we've specified our distributions in or if we relabel the sides of our dice, the answer shouldn't change.
  3. We want it to be non-negative and have the value $I = 0$ if and only if $p = q$. If $p=q$ we haven't updated our beliefs and so have no information gain.
  4. We want it to be monotonic in a natural sense. If we, for instance start with some uniform distribution over the 24 people in a game of Guess Who? and then update to only 5 remaining suspects, $I$ should be larger than if there were still 12 remaining suspects.
  5. Finally we want our information function to decompose in a natural and linear way.2 In particular, we want to be able to relate the information between two joint distributions in terms of the information between their marginal and conditional distributions.

These are all very natural properties for our information function to have. That last point about composition needs to be elaborated. The point is that we have alternative ways we might express a probability distribution. Apropos of nothing, imagine we are concerned that we might have been exposed to a disease and are thinking about getting a test done. There are two random variables under consideration, we will label them $\mathcal{D}$ for whether we actually had the disease or not, and $\mathcal{T}$ for whether the test result is positive. Each of these random variables can take on two possible states, we'll denote them as $\mathcal{D} \in \{ D, \overline D \}, \mathcal{T} \in \{ T, \overline T \}$. $D$ represents the state of our having-had-the-disease random variable $\mathcal{D}$ being positive, meaning we actually did have the disease. $\overline D$ denotes we actually didn't. With two binary random variables, there are 4 possible outcomes $(\{ DT, D\overline T, \overline D T, \overline D \overline T\})$ and fully specifying our set of beliefs requires 3 independent probabilities.

What are our prior beliefs? Let's imagine while we are concerned we might have had the disease, but if we are being honest, we almost certainly didn't,3 so we'll put our prior belief in having had the disease at 7%. $(q(D) = 0.07)$. How do we expect the antibody test to go if we have it done? You do a bit of research and discover that if you had had the disease, the sensitivity or true positive rate of the test you're about to take is 93.8% $(q(T|D) = 0.938)$. The specificity or true negative rate of that same test is 95.6% $(q(\overline T | \overline D) = 0.956)$. 4

Conditional characterization of distribution. Joint characterization of distribution.
Figure 1. Two equivalent ways to express the joint distribution $q(\mathcal{D}\mathcal{T})$.
We've just specified our prior beliefs with 3 numbers, imagining our process as having two steps, first we either had the disease or not $(q(\mathcal{D}))$ and then, conditioned on that we get the result of our test $(q(\mathcal{T}|\mathcal{D}))$. Equivalently, we could have just given the joint probability distribution, as shown in Figure 1.

The point now is that if we were to update our beliefs, in the diagram on the right there is just a single distribution $q(\mathcal{D},\mathcal{T})$, in the one on the left there are essentially three different distributions $(q(\mathcal{D}), q(\mathcal{T}|D), q(\mathcal{T}| \overline D))$ and we want some sort of structural consistency between the two sides: $$ I[p(\mathcal{D},\mathcal{T}); q(\mathcal{D},\mathcal{T})] \quad \textrm{versus} \quad I[p(\mathcal{D}); q(\mathcal{D})], I[p(\mathcal{T}|D); q(\mathcal{T}|D)], I[p(\mathcal{T}|\overline D), q(\mathcal{T}|\overline D)] . $$

The consistency we will require is that our information measure decomposes linearly between these two different descriptions. The information between the joints should be a weighted linear combination of the informations of three constituent distributions. In this particular case we will require: $$ I[p(\mathcal{D},\mathcal{T}); q(\mathcal{D},\mathcal{T})] = I[p(\mathcal{D}); q(\mathcal{D})] + p(D) I[p(\mathcal{T}|D); q(\mathcal{T}|D)] + p(\overline D) I[p(\mathcal{T}|\overline D), q(\mathcal{T}|\overline D)] . $$ In words: The information in the full joint update is the information update for your belief in whether or not you had the disease $(q(\mathcal D))$ plus the informations in the two conditional distributions, but weighted by how often we find ourselves in each of those branches, as measured by our updated beliefs $(p(\mathcal{D}))$.

More generally we are requiring that our information function satisfies a natural chain rule: $$ I[ p(X,Y); q(X,Y) ] = I[ p(X); q(X) ] + \mathbb{E}_{p(X)} \left[ I[ p(Y|X); q(Y|X) ] \right] $$

Notice that it is here, in this sort of structural independence that we make our information function manifestly asymmetric. Here our $p$ distribution becomes distinguished over our $q$ as it is the one we use to weight the child contributions. This makes sense if we imagine or if $p$ is the actual distribution that events are drawn from, for it means that this will correspond to the informations we would observe in expectation.

The interesting thing is that if you want your information function to satisfy all of these seemingly reasonable properties, that is enough to determine it uniquely. The only function satisfying all of these properties is the relative entropy, or KL divergence we all know and love: $$ I[p;q] = \int \mathrm dx\, p(x) \log \frac{p(x)}{q(x)} $$

See A New Theorem of Information Theory by Arthur Hobson for a complete proof, but here I'll offer a more colloquial argument like the one given by Ariel Caticha.5

We will start with and focus on the continuous setting, where we have two probability distributions $p$ and $q$. We seek a functional that takes our two distributions and gives back our information gain and we seek one that is local in the physics sense, meaning that our functional can be written as the integral of a function depending only on the values the probability densities take at each point: $$ I[p;q] = \int \mathrm dx\, \mathcal{A}(x, p(x), q(x)). $$

Our requirement that our information gain be reparameterization independent means it has to be invariant to any remapping of our coordinates, or in other words, it has to be dimensionless. Imagine $x$ has units of a length, here our integral measure $\mathrm dx$ has units of a length, and the densities $p(x), q(x)$ would have units of an inverse length. In order to be dimensionally consistent our functional must take the form:6 $$ I[p;q] = \int \mathrm dx\, p(x) f\left( \frac{p(x)}{q(x)} \right). $$

Finally, our decomposability requirement above when written out in terms of continuous densities takes the form: $$ I[ p(x,y); q(x,y) ] = I[ p(x); q(x) ] + \int \mathrm dx\, p(x) I[p(y|x) ; q(y|x)] $$

Combining this linear decomposition requirement with our requirement for the form required and pushing some equations around gives us: $$ \begin{align} I[ p(x,y); q(x,y) ] &= I[p(x); q(x)] + \int \mathrm dx\, p(x) I[p(y|x); q(y|x)] \\ \int \mathrm dx\, \mathrm dy\, p(x,y) f\left(\frac{p(x,y)}{q(x,y)} \right)&= \int \mathrm dx\, p(x) f\left(\frac{p(x)}{q(x)} \right) + \int \mathrm dx\, p(x) \int dy\, p(y|x) f\left(\frac{p(y|x)}{q(y|x)} \right) \\ \int \mathrm dx\, \mathrm dy\, p(x) p(y|x) f\left(\frac{p(x)p(y|x)}{q(x)q(y|x)} \right)&= \int dx\, dy\, p(x) p(y|x) \left[ f\left(\frac{p(x)}{q(x)} \right) + f\left(\frac{p(y|x)}{q(y|x)} \right)\right] . \end{align} $$ Notice that this demonstrates that our function $f$ must satisfy the property: $$ f(ab) = f(a) + f(b). $$ This well known functional equation has a unique (up to a multiplicative constant) continuous solution: $$ f(x) = c \log x. $$ We can roll the choice of multiplicative constant into our choice of basis for the logarithm and arrive at our final form for our information gain: $$ I[p;q] = \int \mathrm dx\, p(x) \log \frac{p(x)}{q(x)}. $$

As for the non-negativity, our final form satisfies that property. Because we have that $\log x \leq x -1$: $$ I[p;q] = \int \mathrm dx\, p(x) \log \frac{p(x)}{q(x)} = -\int \mathrm dx \, p(x) \log \frac{q(x)}{p(x)} \geq -\int \mathrm dx\, p(x) \left( \frac{q(x)}{p(x)} - 1 \right) = 0. $$

Bayes Rule

Having identified the right way to measure how much information is gained when we update a distribution from $q$ to $p$, why don't we put this to practical use and try to figure out how we ought to update our beliefs in light of evidence or observations.7

Returning to our disease testing example, let's say you get the test done and receive a positive result $(\mathcal T = T)$. What should your new distribution of beliefs be? Well, first off if we've observed the results of the test we should probably have our updated beliefs reflect the observation we made, making it consistent with our observation, setting $p(T) = 1$, but this doesn't fully specify $p$; we need two more numbers. How should we set those?

Why don't we aim to be conservative and try to find the new set of beliefs that are as close as possible to our prior beliefs while still being consistent with the observation that we've made.
Namely, let's look now for a joint distribution $p(\mathcal T, \mathcal D)$ that is as close as possible to $q(\mathcal T, \mathcal D)$ but for which we have that $p(T)=1$. $$ \DeclareMathOperator{\argmin}{arg\,min} $$ $$ \argmin_{p(\mathcal D, \mathcal T)} I[p(\mathcal D, \mathcal T); q(\mathcal D, \mathcal T)] \quad \text{ s.t. }\quad p(T) = 1 $$ Now that we know how to measure how much information is gained in updating our beliefs, we will find the $p$ that minimizes this update while still being true to the observation we made. Writing $p(\mathcal D,\mathcal T) = p(\mathcal T)p(\mathcal D|\mathcal T)$ and using our linear decomposition rule from above (the other way around), we have: $$ I[p(\mathcal D,\mathcal T); q(\mathcal D,\mathcal T)] = I[p(\mathcal T);q(\mathcal T)] + I[p(\mathcal D|T);q(\mathcal D|T)]. $$ Because we've decided to fix $p(T)=1$ in order to be consistent with our observation, the way to minimize the information between the joints is to set $p(\mathcal D|T)=q(\mathcal D|T)$ so that our second term vanishes. In this particular case this means: $$ p(T)=1 $$ $$ p(D|T) = q(D|T) = \frac{q(T|D)q(D)}{q(T|D)q(D) + q(T|\overline D)q(\overline D)} = 0.616 $$

Furthermore, the marginal distribution of our updated beliefs about our disease status is: $$ p(D) = p(D|T)p(T) = q(D|T) = 0.616$$ In this particular case our updated belief is only 3 to 2 on that we actually had the disease, despite our positive test result. In Figure 2 we show both our prior in this factorization as well as our new beliefs.

Prior distribution of beliefs. Posterior distribution of beliefs.
Figure 2. Our prior (left, blue, notice that we've swapped the order of the conditioning) and updated (right, orange) beliefs after observing that the test was positive.

Notice what just happened. If we look for a new distribution that is as close as possible to our previous distribution of beliefs (as measured by $I[p;q]$) which is also consistent with our observations, we end up with an updated, or posterior set of beliefs given by Bayes' Rule. Imagine we had some observable $x$ and some parameters $\theta$. Our prior set of beliefs are described by the joint distribution $q(\theta,x) = q(x|\theta)q(\theta)$: a likelihood $q(x|\theta)$ of how we expect the data to be distributed given the parameter values and some prior $q(\theta)$ set of beliefs about what values those parameters can take. If we make an observation and see some value for our observable $x=X$, what ought our new beliefs be? If we search for the joint distribution $p(x,\theta)$ that is as close as possible to our previous beliefs $q(x,\theta)$ but that no longer has any uncertainty about the value the observable will take $(p(x) = \delta(x-X))$ we see that minimizing the information gain: $$ I[p;q] = I[p(x);q(x)] + \int \mathrm dx\, p(x) \, I[p(\theta|x); q(\theta|x)], $$ is accomplished if we set $p(\theta|x) = q(\theta|x)$, yielding the updated joint: $$ p(x,\theta) = p(x)p(\theta|x) = \delta(x-X) q(\theta|x) $$ and the marginal beliefs about the parameters to be: $$ p(\theta) = \int \mathrm dx\, p(x,\theta) = \int \mathrm dx\, \delta(x-X) q(\theta|x) = q(\theta|X), $$ or precisely what you probably thought it should have been anyway if you've heard of Bayesian inference.

Although, if you stop to think about it, even though many of us know of and have used Bayes Theorem for a long time, the way its normally presented, it is just a trivial statement about how joint distributions factor. $$ q(\theta, D) = q(\theta) q(D|\theta) = q(D) q(\theta|D) \implies q(\theta|D) = \frac{q(D|\theta) q(\theta)}{q(D)}. $$ But, this is just a statement about distribution $q$, our prior beliefs. It tells us nothing about how we should update those beliefs in light of observations. However, the previous argument demonstrates that if you want to set your updated beliefs such that they are as close as possible to your prior beliefs while being consistent with your observations, you should set your updated beliefs according to Bayes rule run on the prior beliefs.

Expected Weight of Evidence

Traditionally, KL is interpreted from a coding perspective, a view I've included in an appendix below, but here I offer a different perspective from the viewpoint of model selection.

Above we saw that we can motivate Bayesian inference as choosing a posterior belief distribution that has the minimal information gain over our prior distribution of beliefs while being consistent with our observations. This guides us towards forming better belief distributions, but what if we just have two different belief distributions and wish to decide between them?

Really what we want to know is what is the probability that our beliefs are correct in light of evidence? Symbolically you might write this as $p(P|E)$ where $P$ is some belief distribution and $E$ is some evidence, data or observations. If we run Bayes Theorem we can see that: $$ p(P|E) = \frac{p(E|P) p(P)}{p(E)}. $$ We can update our belief in our beliefs being correct by setting our updated weight in the belief $p(P|E)$ to be proportional to our initial weight $p(P)$ times the likelihood that the evidence we observed would have been generated if our belief was true $(p(E|P))$. The probability of the evidence given the belief $P$ is just the likelihood $P(E)$. Proportional because we would need to know how likely the evidence would be $p(E)$ amongst all possible beliefs. This last part, the marginal likelihood is notoriously difficult to compute. In principle it is asking us to evaluate how likely the evidence would be from all possible models.

However, we can make further progress if we content ourselves to not necessarily knowing the absolute probability our model or beliefs are correct, but instead just its probability relative to some other model. If we consider the ratio of two different models $P$ and $Q$ we have: $$ \frac{p(P|E)}{p(Q|E)} = \frac{p(E|P)}{p(E|Q)} \frac{p(P)}{p(Q)}. $$ Notice that the marginal likelihoods cancel out. This is saying that whatever prior relative odds for the two models being correct, if we compute the Bayes factor $\left( \frac{p(E|P)}{p(E|Q)} \right)$, it tells us how the relative probabilities of the two beliefs should update in light of the evidence. Taking a log on both sides: $$ \log \frac{p(P|E)}{p(Q|E)} = \log \frac{p(E|P)}{p(E|Q)} + \log \frac{p(P)}{p(Q)},$$ turns this multiplicative factor into an additive one.

If what we are deciding between is two different probability distributions, you may recognize that this additive weight of evidence for $p$ over $q$ when we observe $x$ is precisely the integrand in our information gain: $$ w[x; p,q] = \log \frac{p(x)}{q(x)}. $$ The log ratio of two probability distributions measures by how much you should update your prior log odds between the two distributions being correct. The KL divergence is just then the expected weight of evidence if we draw samples from $p(x)$ itself: $$ I[p;q] = \mathbb{E}_p\left[ \log \frac{p(x)}{q(x)} \right] = \mathbb{E}_p \left[ w[x; p,q] \right]$$

So, one way to interpret the relative entropy is that if our data was actually coming from distribution $p$ and we had some other hypothesis $q$, the $I[p;q]$ measures on average how much we should believe $p$ over $q$ on each observation. In order to make that statement more precise, we need a better language to talk about the magnitudes of these quantities.

How loud is the Evidence?

Our measurement of the amount of information was only unique up to a choice of multiplicative constant. This is equivalent to our choice of base for the logarithm. We can think of this as the units we use to measure our information. The traditional choices would be to use the base-2 logarithm and measure the information in bits,8 or to use the more mathematically convenient natural logarithm and measure the information in nats. Another option is to measure the information in decibans or decibels or Hartley's, wherein we use ten times the base-10 logarithm.

$$ I[p;q] = 10 \int \mathrm dx\, p(x) \log_{10} \frac{p(x)}{q(x)}\, \textrm{dB} $$

The nice thing about measuring information in decibans or decibels is the people already have some familiarity with the unit, such as for measuring the loudness of sounds. Its always a comparative measurement, for sound taking $10 \log_{10} \frac{P}{P_0}$ of the power to some reference or baseline power. In the same way we could besides just measuring the KL between two distributions, measure the comparative difference between any two probabilities on the log scale: $$ 10 \log_{10} \frac{p(x)}{q(x)} \textrm{ dB}. $$

In particular, we could get some feeling for these quantities by comparing the probability something happens to the probability it doesn't. Consider a simple binary outcome and taking $q=1-p$, in this case the weight of evidence that the thing happens versus it doesn't upon observing it happen once is: $$ 10 \log_{10} \frac{p}{1-p} \text{ dB}. $$ This essentially gives us a new scale to measure probabilities on. Instead of expressing probabilities as a number between 0 and 1, here we are computing the log odds of an event happening on the decibel scale.

Below in Table 1 is a summary of the correspondence between decibans and odds or probabilities, and in Figure 3 is a large visual representation you can play with.

dbodds~oddsprobabilityspinner
01.001:1 50%
11.265:456%
21.58π:261%
32.002:167%
42.515:271.5%
53.16π:176%
63.984:180%
75.015:183%
86.312π:186%
97.948:189%
101010:191%
1112.64π:192.6%
1215.816:194%
132020:195%
Table 1: A table of the correspondence between decibans/decibels and odds or probabilities.


Figure 3: A larger visual representation of decibels as a probability that you can play with. Here the set value of decibels measure the weight of evidence between the spinner giving a blue versus a white outcome.

Another nice property of measuring evidence and probabilities in decibels is that it seems like 1 dB roughly corresponds the smallest detectable value that people notice in terms of a change in underlying distribution, being the difference between even chance and 5 to 4 odds, moderate probability or better than even chance.

Additionally, $10 \textrm{ dB}$ corresponds to 10 to 1 odds, or 91% probability, which people associate with events being almost certain or happening almost always. 9.

The traditional statistical threshold for reported results is a p-value of 0.05, which is often misinterpreted to mean that the probability the null hypothesis is less than 5%. While this isn't what the p-value measures, if we obtain more than 13 dB of evidence against some null hypothesis, this does mean that the relative odds that it is correct have decreased by a factor of 20, taking us below 20 to 1 against if we started with even odds.

We have the conversions: $$ 1 \textrm{ nat} = \frac{10}{\log 10} \textrm{ dB} = 4.34 \textrm{ dB} $$ $$ 1 \textrm{ bit} = \frac{10}{\log_2 10} \textrm{ dB} = 3.01 \textrm{ dB} $$

Examples and Magnitudes

Double headed Coin

Let's say I have two coins in my pocket, the first is an ordinary unbiased coin, the second is doubled headed. I give you one of them and you start flipping the coin. You get a heads, then another heads, then another. How many heads would you need to see in a row until you're sure you've been given the doubled headed coin? Let's work out the relative entropy between these two distributions. On the one hand we have $p(H)=1, p(\overline H) =0$, and the other $q(H) = q(\overline H)= 0.5$.

$$ I[p;q] = 10 \sum_i p_i \log_{10} \frac{p_i}{q_i} = -10 \log_{10} 2 = 3.01 \text{ dB} $$

The relative entropy of a sure thing and a coin flip is 3 decibels. This means that if we want to be more sure than 20 to 1 that we have the doubled headed coin we'd need to observe 5 heads in a row, giving us 15 dB of evidence.

Births

Perhaps the first hypothesis test to be resolved with modern statistics was the question of whether more male or female babies are born. Using data from 1745 to 1770, Laplace found that in those 26 years, 251,527 boys and 241,945 girls were born. This gives a fraction of male births of $\sim 51\%$. Is this just a statistical fluke, or are boys more common than girls at birth? What Laplace did was to analytically work out the Bayesian posterior distribution for the probability that a male baby was born using a uniform prior, obtaining a $\operatorname{Beta}(251528, 241946)$ distribution, for which the probability that the probability a male is born is less than or equal to $1/2$ is $$ \int_0^{1/2} \mathrm dx \, \operatorname{Beta}(x; 251528, 241946) \sim 10^{-42}$$ enough for Laplace to declare that he was morally certain that males are born more frequently than females.

Let's work out the weight of evidence in this case, let's say we were comparing two hypotheses, the first that males are born 51% of the time, and the second that they are born 50% of the time. With Laplace's data, the total weight of evidence in this case is:

$$ 2515270 \log_{10} \frac{0.51}{0.50} + 2419450 \log_{10} \frac{0.49}{0.50} = 404 \text{ dB} $$ a whopping 400 decibels of evidence for males being born 51% of the time rather than 50%.
At the same time, I'm not sure most people are aware that males are born with a higher proportion and it doesn't seem to effect most people's lives. Why is that? Well, let's evaluate the relative entropy between a 51% Bernoulli and a 50% Bernoulli: $$ I = 5.1 \log_{10}\frac{0.51}{0.50} + 4.9 \log_{10} \frac{0.49}{0.50} = 8.7 \times 10^{-4} \text{ dB}. $$ Notice that the relative entropy is quite small. On average, if the true distribution was 51%, the evidence we accumulate on each observed birth is less than 8 microbels. This means that on average in order to be reasonably sure that the 51% hypothesis is true, we'd have to observe $\sim \frac{13}{8.7 \times 10^{-4}} \sim 15,000$ births. This makes clear how with enough data we could both be very sure that males are born with a higher frequency than females, but at the same time this could have very little impact on our individual lives.

Likelihoods and Learning

What we would really like to do is learn a model of some real life distribution. If the true distribution of data is $p(x)$, and we have some kind of parametric model $q(x;\theta)$, we would like to set our model parameters $\theta$ so that we get as close as possible to the true distribution. In other words, we want to minimize the relative entropy from the real world to our model: $$\min I[p;q] = \int \mathrm dx\, p(x) \log \frac{p(x)}{q(x;\theta)}. $$ The biggest complication is that we don't actually know what the true distribution of the data is. We can however, sample data. Luckily for us, as far as this as an objective for $\theta$ goes, we can treat the entropy of $p(x)$ as a constant. This motivates the traditional maximum likelihood objective: $$ \max \int \mathrm dx \, \log q(x;\theta). $$

If we had an infinite dataset, maximum likelihood is the same as minimizing the relative entropy between the real world and our model. Unfortunately, we don't often have infinite datasets.10 On finite datasets, maximum likelihood can still be interpreted as minimizing a KL divergence, but now the KL divergence between the *empirical distribution* $\hat p(x) = \sum_i \delta(x - x_i) $ and our model $q(x;\theta)$.

Unfortunately, the cross entropy is no longer reparameterization invariant a point I elaborate in an appendix below, and so is difficult to interpret directly, but if we take the difference of any two cross entropies, we can still interpret that as the weight of evidence for one model with regards to the other. Because of the lack of reparameterization independence, care must be taken to ensure that the likelihoods of the two models are evaluated using the same measure, but provided they are:

$$ L_1 - L_2 = \mathbb{E}\left[ \log q_1(x) \right] - \mathbb{E}\left[ \log q_2(x) \right] = \mathbb{E}\left[ \log \frac{q_1(x)}{q_2(x)} \right] $$

Given the size of test sets we have for modern image datasets, this means that very small changes in likelihood can be interpreted as large confidences in the superiorities of models. Take for instance something as simple as binary static MNIST.11 Here, with 10,000 test set images, a difference in likelihoods of 0.0013 dB or 0.0004 nats corresponds to 13 dB of evidence for the one model over the second.

Appendix A: Whither Continuous Entropy

The relative entropy really is the proper way to define entropy. For all of the things that Shannon got right, he flubbed a bit when he defined the entropy of a distribution as: $$ H(P) = -\sum_i p_i \log p_i $$

Why do I say he flubbed? Because this notion of entropy doesn't generalize to continuous distributions. The continuous analog: $$ H(P) = -\int \mathrm dx\, p(x) \log p(x) $$ isn't reparameterization independent. Consider for instance the distribution of adult human heights: 12

Distribution of adult heights.
Figure 1. Distribution of adult heights. 13

If you measure the continuous entropy of this distribution measured in centimeters you get 5.4 bits. If you instead measure the entropy of the same distribution in feet you get 0.43 bits. If you instead were to measure heights in meters it would be -1.3 bits! 14

Appendix B: Coding Interpretation

The traditional interpretation offered for the KL is from the coding perspective. Imagine we have a simple 4-letter alphabet that we want to communicate over the wire. If the four letters occurred with different probabilities: $p(A)=1/2, p(B)=1/4, p(C)=p(D)=1/8$, with an optimally designed Huffman Code we could encode our letters with a variable length code: $A:0, B:10, C:110, D:111$, and on average we'd only be spending $1/2 + 2/4 + 3/8 + 3/8 = 7/4$ bits per letter.

ABCD
$p$1/21/41/81/8
p-code010110111
$q$1/41/41/41/4
q-code00011011
Table 2: A simple example of two different distributions over a 4 letter alphabet.

Imagine however we didn't know what the true distribution of letters was and instead designed an optimal code using a different distribution $q$. If we believed each of the 4 letters were equally likely $(q(A)=q(B)=q(C)=q(D)=1/4)$, the optimal way to encode messages would just assign a two bit code to each letter $(A : 00, B:01, C:10, D:11)$. If we used this suboptimal code to send messages that were actually distributed as $p$ it would cost $2/2 + 2/4 + 2/8 + 2/8 = 2$ bits per letter. Our incorrect belief leads to a $2 - 7/4 = 1/4$ of a bit inefficiency. For these two distributions, it shouldn't come as a surprise that the information gain is precisely 1/4 bits: $$ I[p;q] = \sum_i p_i \log_2 \frac{p_i}{q_i} = 1/4 \textrm{ bits}. $$

For an optimally designed code, the code lengths go as $-\log p(x)$ for any symbol $x$. Our information gain can be interpreted as a difference in expected code lengths under $p$: $$ I[p;q] = \mathbb{E}_p[ -\log q ] - \mathbb{E}_p[-\log p ]. $$ The information gain $I[p;q]$ measures the excess encoding cost for trying to encode messages from $p$ using a code designed for $q$.