... KL divergence is asymmetric and it’s important to understand the differences between forward and reverse KL. So basically: $$[6,11] = \frac{p}{5}; [0,5] = \frac{1-p}{6}$$. Thus, you can calculate the Euclidian distance $\int_x(p(x)-q(x))^2dx$, Cauchy-Schwarz distance, etc. Now… that’s not so much the intelligence part. since \(\text{log }a - \text{log b} = \text{log }\frac{a}{b}\). Determining its relation to well-known distance measures reveals a new way to depict how commonly used distance measures relate to each other. While this data is great, we have a bit of a problem. To do this, we must build a quadrature to estimate the integral from the KDE. KL-Divergence only satisfies the second condition. This distribution is Q. Using the KL-divergence, we can start measuring the changes in frequencies due to close-outs and quality perimeter defenders to help understand when teams are not taking the three they usually take. While this is example is only optimizing a single parameter, we can easily imagine extending this approach to high dimensional models with many parameters. Applications include characterizing the relative (Shannon) entropy in information systems, randomness in continuous time-series, and information gain when comparing statistical models of inference. Since the divergence is not symmetric, we must specify the baseline distribution. What we want to do is reduce this data to a simple model with just one or two parameters. This is where Kullback-Leibler Divergence comes in. In the simple case, a relative entropy of 0 indicates that the two distributions in question are identical. 2 I. Some believe (Huszar, 2015) that one reason behind GANs’ big success is switching the loss function from asymmetric KL divergence in traditional maximum-likelihood approach to symmetric JS divergence. We could rewrite our formula in terms of expectation: $$D_{KL}(p||q) = E[\text{log } p(x) - \text{log } q(x)]$$. Yes… there’s been negligible intelligence obtained thus far. In mathematical statistics, the Kullback–Leibler divergence, D KL {\displaystyle D_{\text{KL}}} (also called relative entropy), is a measure of how one probability distribution is different from a second, reference probability distribution. It may be tempting to think of KL Divergence as a distance metric, however we cannot use KL Divergence to measure the distance between two distributions. Tile Pattern KL-Divergence for Analysing and Evolving Game Levels. We can think of the KL divergence as distance metric (although it isn’t symmetric) that quantifies the difference between two probability distributions. The most important metric in information theory is called Entropy, typically denoted as \(H\). If DeAndre Jordan is swapped with Enes Kanter, we will see a ridiculously different result. What happens if we produce another player with almost an identical table? The reason for this is that KL Divergence is not symmetric. Commonly, we find that much of the analysis about player tendency and capability stops here. Yes, that is a post from four years ago as a knee-jerk response to poorly displayed ESPN shot charts at the time. It’s a little misleading only due to the fact that they combine both frequency and efficiency. Computing the KL Divergence by Smoothing. Field Goal Distribution for PJ Tucker through February 5th, 2019. Pingback: An Example in Kullback-Leibler Divergence, Pingback: Weekly Sports Analytics News Roundup - February 12th, 2019 - StatSheetStuffer. Abstract: Asymmetric information distances are used to define asymmetric norms and quasimetrics on the statistical manifold and its dual space of random variables. With KL divergence we can calculate exactly how much information is lost when we approximate one distribution with another. Variational Bayesian method, including Variational Autoencoders, use KL divergence to generate optimal approximating distributions, allowing for much more efficient inference for very difficult integrals. A common approach to this is called a "Variational Autoencoder" which learns the best way to approximate the information in a data set. The idea is just to realize that pdfs are like any other functions in a L2-space. If we toss the coin once, and it lands heads, we aren’t very surprised and hence the information “transmitted” by such an event is … Optimal compression scheme is to record heads as 0 and tails as 1. After collecting many samples we have come to this empirical probability distribution of the number of teeth in each worm: The empirical probability distribution of the data collected. Our uniform approximation wipes out any nuance in our data. To gain insight of good or bad, we must then build the analytical model that identifies good and bad. KL Divergence has its origins in information theory. The shots in the lane? Due to this, we call it a divergence instead of a measurement. Notation . That would give us a binomal distribution that looks like this: Our binomial approximation has more subtlety, but doesn't perfectly model our data either. On the left we have P.J. Let’s compare this to Rudy Gobert of Utah. More specifically, it quantifies the "amount of information" (in units such as shannons, commonly called bits) obtained about one random variable through observing the other random variable. For instance, old plots would not include distance skewing such as a log-transform, a requirement in effort to show actual three-point effects in scoring. While this is helpful in understanding where players are positioned, this is rarely the question that we would like to answer. Hypothesis Testing: Is NBA Scoring Up This Year? Of course, we’d like to play with the bandwidth to make the charts “prettier.” This is simply an out-of-the-box method using Python. Given the data that we have observed, our probability distribution has an entropy of 3.12 bits. Due to this, we call it a divergence instead of a measurement. Developed by Solomon Kullback and Richard Leibler for public release in 1951, KL-Divergence aims to identify the divergence of a probability distribution given a baseline distribution. Comparing West vs. East: If the NBA Playoffs Were Seeded Like the NCAA. Rather than just having our probability distribution \(p\) we add in our approximating distribution \(q\). That value for the minimum KL divergence should look pretty familiar: it's nearly identical to the value we got from our uniform distribution! We’ll also check out on an example for a classification problem using the loss function as Cross-Entropy. Tucker of the Houston Rockets. One option is to represent the distribution of teeth in worms as just a uniform distribution. If we have to choose one to represent our observations, we're better off sticking with the Uniform approximation. We can double check our work by looking at the way KL Divergence changes as we change our values for this parameter. In this tutorial, we write an example to compute kl divergence in … Recently I saw Will Penny explain this (at the Free Energy Principle workshop, of which hopefully more later). We see the same misleading representation with PJ Tucker and again focus on the fractions. Since the divergence is not symmetric, we must specify the baseline distribution. We see the ghost town of field goal attempts in the mid-range, as well as the string of short-range attempts that litter the key. Now that we can quantify this, we want to quantify how much information is lost when we substitute our observed distribution for a parameterized approximation. Optimal encoding of information is a very interesting topic, but not necessary for understanding KL divergence. Remember though that changes in KL-Divergence does not mean good or bad. It arises from geometric considerations similar to those used to derive the Chernoff distance. If we apply the density function formulation here,we can obtain kde plots for both Lopez and Tucker. You train neural networks by minimizing the loss of the objective function. It simply means change. Combining KL divergence with neural networks allows us to learn very complex approximating distribution for our data. Again, if we think in terms of \(log_2\) we can interpret this as "how many bits of information we expect to lose". We're far from Earth and sending data back home is expensive. On the right, we have Brook Lopez of the Milwaukee Bucks. While the Kullback-Leibler distance is asymmetric in the two dis-tributions, the resistor-average distance is not. In fact, the divergence of Gobert from Tucker is 47.5551! We see that almost all FGA occur in the corners. For the uniform distribution we find: $$D_{kl}(\text{Observed } || \text{ Uniform}) = 0.338$$, $$D_{kl}(\text{Observed } || \text{ Binomial}) = 0.477$$. We immediately are able to surgically identify locations of every field goal attempt by both players. EDICS: SPL.SPTM. In this case, the information would be each observation of teeth counts given our empirical distribution. But we can build a distribution and measure the KL-divergence, which helps borrow strength from nearby field goal locations and allows us to start asking which features lead to changes in KL-Divergence. Therefore, the 0.0929 indicates how much PJ Tucker diverges from Brook Lopez in shooting frequency. Either a known cryptosystem in 1945, or a current player of interest. The zonal plots do not capture that activity. Granted, we cannot simply use defensive three point shooting as a metric and we certainly cannot use simple frequencies of shooting (they’re too few in a game). While Monte Carlo simulations can help solve many intractable integrals needed for Bayesian inference, even these methods can be very computationally expensive. We can make the KL divergence concrete with a worked example. Here is a great tutorial that dives into the details of building variational autoencoders. Example 2.24. It’s obvious that if the two distributions are identical, then the integral is zero. I And unlike the “second step further” plots that we skipped over with scatter (hexagon) plotting, we’re not solely dealing with empirical data points, which by the way, are noisy to being with. ( Log Out /  They have very similar distributions and, while still being significantly different according to the Chi-Square test, it’s mainly due to the failure of the Normal assumption for the small values in the table. That is, when perimeter defenders in PnR situations move to a seemingly unfavorable defensive position in an effort to divert the PnR into a favorable defensive match-up. Here's a chart of how those values change together: It turns out we did choose the correct approach for finding the best Binomial distribution to model our data. the rANS encoding algorithm is a very ingenious adaptation of this simple idea! Compared with the original data, it's clear that both approximations are limited. KL(P,Q)!= KL(Q,P)): order matters and we should compare our predicted distribution with our target distribution in that order. For example we if used our observed data as way of approximating the Binomial distribution we get a very different result: KL(x, y) denotes the generalized KL-Divergence = P i xi log xi yi −xi+yi (also called I-divergence). Kullback-Leibler (KL) Divergence ... As an example, this can be used in Variational Autoencoders (VAEs), and reinforcement learning policy networks such as Trust Region Policy Optimization (TRPO). For example, let’s look at a typical image classification problem where we classify an image into a semantic class such as car, person etc. Published: July 19, 2020 In the previous post, I mentioned about the basic concept of two-sample Kolmogorov-Smirnov (KS) test and its implementation in Spark (Scala API).. Either way, the answers are similar. We think of Q as prior knowledge. This is a relatively small KL-Divergence, but it could be smaller! The most explosive revelation leveraging KL-Divergence is measuring field goal attempts with respect to BLUE action. Suppose that we're space-scientists visiting a distant, new planet and we've discovered a species of biting worms that we'd like to study. Similarly, we are able to differentiate between a left-corner three versus a right-corner three. Detour on KL-divergence To compress, it is useful to know the probability distribution the data is sampled from For example, let X 1; ;X 100 be samples of an unbiased coin. While recently studying about KL divergence, I came across the following intuitive explanation from Ian Goodfellow via an example. In doing this, for this given year, you’ll immediately start seeing the defensive differences in two former Spurs: Danny Green and Jonathon Simmons. In contrast to variation of information, it is a distribution-wise asymmetric measure and thus does not qualify as a statistical metric of spread – it also does not satisfy the triangle inequality. This distribution is Q. Kullback-Leibler Divergence is a method for measuring the similarity between two distributions. Meaning, that the smallest possible value is zero (distributions are equal) and the maximum value is infinity. Another thing to note is that there are two ways to use KLDivLoss that depend on how we set from_logits (which has a default value of true). Field Goal Distribution for Brook Lopez through February 5th, 2019. We can peel back the integral and see exactly where the spatial locations vary and understand how those locations impact the divergence. This is a relatively straightforward method that can be exploited using the scipy.integrate.dblquad package in Python, or crudely using the midpoint rule. How can we choose which one to use? Let's start our exploration by looking at a problem. Just be sure to assign the shot charts to be numpy arrays. But since we're optimizing for minimizing information loss, it's possible this wasn't really the best way choose the parameter. Are these two players the same? As you can see, our estimate for the binomial distribution (marked by the dot) was the best estimate to minimize KL divergence. Neural networks, in the most general sense, are function approximators. Most datasets use a mapping from a string (“Car”) to a numeric value so that we can handle the dataset in a computer easily. While this “one step further” plot helps us, there’s still a ton of information left on the cutting room floor. Change ), You are commenting using your Google account. Therefore, it is common to assume both distributions exists on the same support. But how do we measure their difference? We also see the ghost town of mid-range attempts. Our discussion started by asking about the similarities between two players. We've found that these worms have 10 teeth, but because of all the chomping away, many of them end up missing teeth. Change ), You are commenting using your Facebook account. In order to understand the new player, we consider the new player as new information introduced to the old player. More importantly, are we able to make proper decisions about the style of play for Kevin Durant? Immediately, we gain an idea of differentiation between the players’s shot location tendencies. In order to understand the question we really want to answer (and we haven’t asked just yet), we will tackle this thought exercise first in an effort to understand Kullback-Leibler Divergence. … the KL divergence is the average number of extra bits needed to encode the data, due to the fact that we used distribution q to encode the data instead of the true distribution p. — Page 58, Machine Learning: A Probabilistic Perspective, 2012. No transforms applied. The reason for this is that KL Divergence is not symmetric. Is the NBA Draft Lottery Fixed? Richard Liebler, who would eventually become the Director of Mathematical Research, and Solomon Kullback, who then focused on COMSEC operations, developed the methodology while analyzing bit strings in relation to known coding algorithms. We know that if we have \(n\) trials and a probabily is \(p\) then the expectation is just \(E[x] = n \cdot p\). Sup-pose there are two sample distributions P and Q as follows: P: (a: 3/5,b: 2 1/5,c: 1/5) and Q: (a: 5/9,b: 3/9,d: 1/9). It’s a step in the right direction as we can now differentiate between a corner three and a top-of-the-key three. In MXNet Gluon, we can use KLDivLoss to compare categorical distributions. For example, eyetracking studies (e.g., (Itti and Baldi 2005)) showed that surprise, as measured by KL divergence, was a better predictor of visual attention than information, measured by entropy. This tutorial discusses a simple way to use the KL-Divergence as a distance metric to compute the similarity between documents. Let’s take a look at these two players: Can you guess the two players? For example, when using a Gaussian distribution to model a bimodal distribution in blue, the reverse KL-divergence solutions will be either the red curve in the diagram (b) or (c). In this case \(n = 10\), and the expectation is just the mean of our data, which we'll say is 5.7, so our best estimate of p is 0.57. ∙ Queen Mary University of London ∙ 0 ∙ share . If you want to learn more about Bayesian Statistics and probability: Order your copy of Bayesian Statistics the Fun Way No Starch Press! If you are familiar with neural networks, you may have guessed where we were headed after the last section. KL-Divergence only satisfies the second condition. It’s widely known that the KL divergence is not symmetric, i.e. Similarly, if an offense uses a PnR action that leads to a rim-running event, where are the field goal attempts likely going to be generated. In other posts we've seen how powerful Monte Carlo simulations can be to solve a range of probability problems. Let's go back to our data and see what the results look like. Essentially, what we're looking at with the KL divergence is the expectation of the log difference between the probability of data in the original distribution with the approximating distribution. For instance, Brook Lopez is a -45 to 45 degree shooter. And, more importantly, how we might want to rotate on defense. NBA Stats Zone Distribution for PJ Tucker. Since its public release, KL-Divergence has been used extensively across many fields; and still is considered one of the most important entropy measuring tools in cryptography and information theory. We obtain infinity when P is defined in a region where Q can never exist. They are both three-ball-dominant shooters with a tendency to attack the rim. Example: For some subjects, both Rand Sunobserved. It is clear that the distributions are no longer the same. Generative Adversarial Network (GAN) GAN … Suppose we wanted to create an ad hoc distribution to model our data. To learn more about Variational Inference check out the Edward library for python. I am going to “borrow” very liberally from his talk. Kullback-Leibler Divergence is just a slight modification of our formula for entropy. drop-out in longitudinal studies. Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. in surveys. ( Log Out /  Therefore, the new player is a posterior distribution. One important thing to note is that the KL Divergence is an asymmetric measure (i.e. The definition of Entropy for a probability distribution is: $$H = -\sum_{i=1}^{N} p(x_i) \cdot \text{log }p(x_i)$$. Blog: Kullback-Leibler Divergence Explained). Out-of-the-box kde estimate, using Python, for PJ Tucker. Here we see the high volume along the top-of-the-key zones. Quasimetric topology, generated by the Kullback-Leibler (KL) divergence, is considered as the main example, and some of its topological properties are investigated. No transforms applied. What entropy doesn't tell us is the optimal encoding scheme to help us achieve this compression. The key point here is that we can use KL Divergence as an objective function to find the optimal value for any approximating distribution we can come up with. We talk about at what distance a player takes their shots and then typically jump to effective field goal percentage and translate that to rudimentary calculations of expected point value per field goal attempt. In this case all we need to do is estimate that probability parameter of the Binomial distribution. Note: Because \(log\) is undefined for 0, the only time we can allow probabilities that are zero is when \(p(x_i) = 0\) implies \(q(x_i) = 0\). As Milwaukee has modeled their offense much like the Houston Rockets, it’s no surprise these two shooters appear to have the same distribution of field goal attempts. 04/24/2019 ∙ by Simon M. Lucas, et al. And we see a nearly “inverted” plot as majority of PJ Tucker’s three-point attempts are located in the corners. We think of Q as prior knowledge. We can represent this using set notation as {0.99, 0.01}. NBA Stats Zone Distribution for Brook Lopez. ( Log Out /  This seems counter-intuitive since the expectation is taken with respect to P. But there’s a simple explanation for this. This indicates that the same action with different personnel yields different results. Let's look at an example: (The example here is borrowed from the following link. 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