title

Entropy (for data science) Clearly Explained!!!

description

Entropy is a fundamental concept in Data Science because it shows up all over the place - from Decision Trees, to similarity metrics, to state of the art dimension reduction algorithms. It's also surprisingly simple, but often poorly explained. Traditionally the equation is presented with the expectation that you memorize it without thoroughly understanding what it means and where it came from. This video takes a very different approach by showing you, step-by-step, where this simple equation comes from, making it easy to remember (and derive), understand and explain to your friends at parties.
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0:00 Awesome song and introduction
1:28 Introduction to surprise
4:34 Equation for surprise
6:09 Calculating surprise for a series of events
9:35 Entropy defined for a coin
10:45 Entropy is the expected value of surprise
11:41 The entropy equation
13:01 Entropy in action!!!
#StatQuest #Entropy

detail

{'title': 'Entropy (for data science) Clearly Explained!!!', 'heatmap': [{'end': 286.525, 'start': 253.879, 'weight': 0.822}, {'end': 526.428, 'start': 486.949, 'weight': 0.96}, {'end': 728.5, 'start': 690.02, 'weight': 0.752}, {'end': 840.16, 'start': 820.274, 'weight': 0.704}, {'end': 947.164, 'start': 921.124, 'weight': 0.766}], 'summary': 'Explains the significance of entropy in data science, its role in classification trees, mutual information, relative entropy, and cross entropy, probability and surprise relationships, entropy calculation for chicken selection based on color, and promotes statquest study guides.', 'chapters': [{'end': 92.856, 'segs': [{'end': 92.856, 'src': 'embed', 'start': 33.877, 'weight': 0, 'content': [{'end': 40.261, 'text': 'For example, entropy can be used to build classification trees, which are used to classify things.', 'start': 33.877, 'duration': 6.384}, {'end': 48.486, 'text': 'Entropy is also the basis of something called mutual information, which quantifies the relationship between two things.', 'start': 41.302, 'duration': 7.184}, {'end': 59.683, 'text': 'And entropy is the basis of relative entropy and cross entropy which show up all over the place,', 'start': 49.795, 'duration': 9.888}, {'end': 64.146, 'text': 'including fancy dimension reduction algorithms like t-SNE and UMAP.', 'start': 59.683, 'duration': 4.463}, {'end': 74.134, 'text': 'What these three things have in common is that they all use entropy, or something derived from it, to quantify similarities and differences.', 'start': 65.346, 'duration': 8.788}, {'end': 79.674, 'text': "So let's learn how entropy quantifies similarities and differences.", 'start': 75.495, 'duration': 4.179}, {'end': 86.152, 'text': 'However, in order to talk about entropy, first we have to understand surprise.', 'start': 80.868, 'duration': 5.284}, {'end': 88.473, 'text': "So let's talk about chickens.", 'start': 86.992, 'duration': 1.481}, {'end': 92.856, 'text': 'Imagine we had two types of chickens, orange and blue.', 'start': 89.614, 'duration': 3.242}], 'summary': 'Entropy quantifies similarities and differences in classification trees and dimension reduction algorithms using mutual information, relative entropy, and cross entropy.', 'duration': 58.979, 'max_score': 33.877, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw33877.jpg'}], 'start': 0.583, 'title': 'Entropy in data science', 'summary': 'Delves into the significance of entropy in data science, its role in classification trees, mutual information, relative entropy, and cross entropy, and its ability to quantify similarities and differences.', 'chapters': [{'end': 92.856, 'start': 0.583, 'title': 'Understanding entropy in data science', 'summary': 'Discusses the importance of entropy in data science, its applications in classification trees, mutual information, relative entropy, and cross entropy, and how it quantifies similarities and differences.', 'duration': 92.273, 'highlights': ['Entropy is used in building classification trees for classifying things and is the basis of mutual information, quantifying the relationship between two things.', 'Relative entropy and cross entropy, derived from entropy, show up in various applications, including dimension reduction algorithms like t-SNE and UMAP.', 'Entropy quantifies similarities and differences, and understanding it requires comprehension of surprise and its application in various scenarios.']}], 'duration': 92.273, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw583.jpg', 'highlights': ['Entropy is used in building classification trees for classifying things and is the basis of mutual information, quantifying the relationship between two things.', 'Relative entropy and cross entropy, derived from entropy, show up in various applications, including dimension reduction algorithms like t-SNE and UMAP.', 'Entropy quantifies similarities and differences, and understanding it requires comprehension of surprise and its application in various scenarios.']}, {'end': 764.055, 'segs': [{'end': 120.235, 'src': 'embed', 'start': 93.557, 'weight': 0, 'content': [{'end': 98.34, 'text': 'And instead of just letting them randomly roam all over the screen,', 'start': 93.557, 'duration': 4.783}, {'end': 105.385, 'text': 'our friend Statsquatch chased them around until they were organized into three separate areas A, B and C.', 'start': 98.34, 'duration': 7.045}, {'end': 116.775, 'text': 'Now, if Statsquatch just randomly picked up a chicken in Area A, then because there are six orange chickens and only one blue chicken,', 'start': 106.573, 'duration': 10.202}, {'end': 120.235, 'text': 'there is a higher probability that they will pick up an orange chicken.', 'start': 116.775, 'duration': 3.46}], 'summary': 'Statsquatch organized chickens into 3 areas, a, b, and c, with 6 orange and 1 blue chicken in area a.', 'duration': 26.678, 'max_score': 93.557, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw93557.jpg'}, {'end': 225.291, 'src': 'embed', 'start': 198.73, 'weight': 1, 'content': [{'end': 204.913, 'text': 'Because we know there is a type of inverse relationship between probability and surprise.', 'start': 198.73, 'duration': 6.183}, {'end': 209.454, 'text': "it's tempting to just use the inverse of probability to calculate surprise.", 'start': 204.913, 'duration': 4.541}, {'end': 217.317, 'text': 'Because when we plot the inverse, we see that the closer the probability is to zero, the larger the y-axis value.', 'start': 210.214, 'duration': 7.103}, {'end': 225.291, 'text': 'However, there is at least one problem with just using the inverse of the probability to calculate surprise.', 'start': 218.749, 'duration': 6.542}], 'summary': 'Inverse relationship between probability and surprise, but using inverse alone has limitations.', 'duration': 26.561, 'max_score': 198.73, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw198730.jpg'}, {'end': 286.525, 'src': 'heatmap', 'start': 253.879, 'weight': 0.822, 'content': [{'end': 262, 'text': 'So, when the probability of getting heads is 1, then we want the surprise for getting heads to be 0.', 'start': 253.879, 'duration': 8.121}, {'end': 270.652, 'text': 'However, when we take the inverse of the probability of getting heads, we get 1 instead of what we want, 0.', 'start': 262, 'duration': 8.652}, {'end': 276.417, 'text': "And this is one reason why we can't just use the inverse of the probability to calculate surprise.", 'start': 270.652, 'duration': 5.765}, {'end': 286.525, 'text': 'So, instead of just using the inverse of the probability to calculate surprise, we use the log of the inverse of the probability.', 'start': 277.878, 'duration': 8.647}], 'summary': 'Using log of inverse probability to calculate surprise avoids issues with using inverse directly.', 'duration': 32.646, 'max_score': 253.879, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw253879.jpg'}, {'end': 385.551, 'src': 'embed', 'start': 351.792, 'weight': 3, 'content': [{'end': 355.273, 'text': 'So surprise is the log of the inverse of the probability.', 'start': 351.792, 'duration': 3.481}, {'end': 362.335, 'text': 'Bam Note when calculating surprise for two outputs.', 'start': 356.513, 'duration': 5.822}, {'end': 365.616, 'text': 'in this case the two outputs are heads and tails.', 'start': 362.335, 'duration': 3.281}, {'end': 369.377, 'text': 'then it is customary to use the log base two for the calculations.', 'start': 365.616, 'duration': 3.761}, {'end': 376.419, 'text': "Now that we know what surprise is, let's imagine that our coin gets heads 90% of the time and it gets tails 10% of the time.", 'start': 370.537, 'duration': 5.882}, {'end': 385.551, 'text': "Now let's calculate the surprise for getting heads and tails.", 'start': 380.99, 'duration': 4.561}], 'summary': 'Surprise is calculated using the log of the inverse of probability, with heads having a surprise of 0.15 and tails having a surprise of 3.32.', 'duration': 33.759, 'max_score': 351.792, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw351792.jpg'}, {'end': 526.428, 'src': 'heatmap', 'start': 486.949, 'weight': 0.96, 'content': [{'end': 496.234, 'text': 'Now, if we wanted to estimate the total surprise, after flipping the coin 100 times, we approximate how many times we will get heads.', 'start': 486.949, 'duration': 9.285}, {'end': 502.438, 'text': 'by multiplying the probability, we will get heads 0.9, by 100..', 'start': 496.234, 'duration': 6.204}, {'end': 509.262, 'text': 'And we estimate the total surprise from getting heads by multiplying by 0.15.', 'start': 502.438, 'duration': 6.824}, {'end': 514.063, 'text': 'So this term represents how much surprise we expect from getting heads in 100 coin flips.', 'start': 509.262, 'duration': 4.801}, {'end': 526.428, 'text': 'Likewise, we can approximate how many times we will get tails by multiplying the probability we will get tails, 0.1, by 100.', 'start': 516.345, 'duration': 10.083}], 'summary': 'Estimate total surprise from 90 heads and 10 tails in 100 flips.', 'duration': 39.479, 'max_score': 486.949, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw486949.jpg'}, {'end': 587.342, 'src': 'embed', 'start': 556.214, 'weight': 5, 'content': [{'end': 558.817, 'text': "But aren't we supposed to be talking about entropy?", 'start': 556.214, 'duration': 2.603}, {'end': 560.959, 'text': 'Funny, you should ask!.', 'start': 559.918, 'duration': 1.041}, {'end': 573.053, 'text': 'If we divide everything by the number of coin tosses 100, then we get the average amount of surprise per coin toss 0.47..', 'start': 561.826, 'duration': 11.227}, {'end': 579.617, 'text': 'So, on average, we expect the surprise to be 0.47 every time we flip the coin.', 'start': 573.053, 'duration': 6.564}, {'end': 583.139, 'text': 'And that is the entropy of the coin.', 'start': 580.558, 'duration': 2.581}, {'end': 587.342, 'text': 'The expected surprise every time we flip the coin.', 'start': 584.06, 'duration': 3.282}], 'summary': 'Entropy of the coin is 0.47, representing average surprise per flip.', 'duration': 31.128, 'max_score': 556.214, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw556214.jpg'}, {'end': 728.5, 'src': 'heatmap', 'start': 690.02, 'weight': 0.752, 'content': [{'end': 697.261, 'text': 'Now, personally, once I saw that entropy was just the average surprise that we could expect.', 'start': 690.02, 'duration': 7.241}, {'end': 701.902, 'text': 'entropy went from something that I had to memorize to something I could derive.', 'start': 697.261, 'duration': 4.641}, {'end': 708.324, 'text': 'Because now, we can plug the equation for surprise in for x, the specific value.', 'start': 702.662, 'duration': 5.662}, {'end': 711.064, 'text': 'And we can plug in the probability.', 'start': 709.264, 'duration': 1.8}, {'end': 714.945, 'text': 'And we end up with the equation for entropy.', 'start': 712.044, 'duration': 2.901}, {'end': 728.5, 'text': 'Bam!. Unfortunately, even though this equation is made from two relatively easy to interpret terms the surprise times,', 'start': 716.212, 'duration': 12.288}], 'summary': 'Entropy was redefined as the average surprise, making it derivable from the equation for surprise and probability.', 'duration': 38.48, 'max_score': 690.02, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw690020.jpg'}], 'start': 93.557, 'title': 'Probability and surprise relationships', 'summary': 'Explores the correlation between probability and surprise when picking up a chicken, demonstrates surprise calculation based on the inverse relationship with probability, and discusses the concept of surprise, entropy calculation, and the standard form of the entropy equation.', 'chapters': [{'end': 225.291, 'start': 93.557, 'title': 'Probability and surprise relationship', 'summary': 'Explores the relationship between probability and surprise, demonstrating how the probability of picking up a chicken correlates with the level of surprise, and introduces the idea of calculating surprise based on the inverse relationship with probability.', 'duration': 131.734, 'highlights': ['Statsquatch organized chickens into three areas A, B, and C, with varying numbers of blue and orange chickens, demonstrating how the probability of picking up a chicken correlates with the level of surprise, providing a tangible example of the relationship between probability and surprise.', 'The inverse relationship between probability and surprise is illustrated by the fact that a low probability of picking a blue chicken results in high surprise, while a high probability of picking a blue chicken results in low surprise, offering a clear understanding of the inverse correlation between probability and surprise.', 'The chapter introduces the idea of calculating surprise based on the inverse relationship with probability, highlighting the potential complexities and challenges in using the inverse of probability to calculate surprise, emphasizing the need for a more refined approach to calculating surprise.']}, {'end': 764.055, 'start': 226.311, 'title': 'Understanding surprise and entropy', 'summary': 'Discusses the concept of surprise associated with flipping a coin, explains how surprise is calculated using the log of the inverse of the probability, demonstrates the application of surprise in calculating average surprise and entropy, and concludes with the standard form of the equation for entropy.', 'duration': 537.744, 'highlights': ['The concept of surprise is introduced in the context of flipping a coin, where the probability of getting heads is 1, resulting in a surprise of 0, and the inverse of the probability is used to calculate surprise. When the probability of getting heads is 1, the surprise for getting heads is 0, and the inverse of the probability is used to calculate surprise, leading to the introduction of the log of the inverse of the probability to properly quantify surprise.', 'The application of the log of the inverse of the probability in calculating surprise for two outputs, heads and tails, using the log base two, and demonstrating the calculation of surprise for a sequence of coin tosses. The log of the inverse of the probability is used to calculate surprise for two outputs (heads and tails) using the log base two, and the total surprise for a sequence of coin tosses is shown to be the sum of the surprises for each individual toss.', 'The calculation of average surprise, referred to as entropy, by dividing the total surprise by the number of coin tosses, and the derivation of the equation for entropy, which represents the expected value of surprise per coin toss. The average surprise, or entropy, is calculated by dividing the total surprise by the number of coin tosses, and the equation for entropy is derived to represent the expected value of surprise per coin toss.']}], 'duration': 670.498, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw93557.jpg', 'highlights': ['Statsquatch organized chickens into areas A, B, and C, demonstrating the correlation between probability and surprise.', 'The inverse relationship between probability and surprise is illustrated, providing a clear understanding of the correlation.', 'The chapter introduces the idea of calculating surprise based on the inverse relationship with probability, emphasizing the need for a refined approach.', 'The concept of surprise is introduced in the context of flipping a coin, leading to the introduction of the log of the inverse of the probability to quantify surprise.', 'The application of the log of the inverse of the probability in calculating surprise for two outputs, and the total surprise for a sequence of coin tosses is shown to be the sum of the surprises for each individual toss.', 'The calculation of average surprise, referred to as entropy, and the derivation of the equation for entropy, representing the expected value of surprise per coin toss.']}, {'end': 993.995, 'segs': [{'end': 852.749, 'src': 'heatmap', 'start': 820.274, 'weight': 0, 'content': [{'end': 825.896, 'text': 'there is a much higher probability that we will pick up an orange chicken than pick up a blue chicken.', 'start': 820.274, 'duration': 5.622}, {'end': 835.479, 'text': 'Thus, the total entropy, 0.59, is much closer to the surprise associated with orange chickens than blue chickens.', 'start': 826.956, 'duration': 8.523}, {'end': 840.16, 'text': 'Likewise, we can calculate the entropy for area B.', 'start': 837.139, 'duration': 3.021}, {'end': 842.541, 'text': 'Only this time.', 'start': 841.3, 'duration': 1.241}, {'end': 852.749, 'text': 'the probability of randomly picking up an orange chicken is 1, divided by 11, and the probability of picking up a blue chicken is 10, divided by 11,', 'start': 842.541, 'duration': 10.208}], 'summary': 'The total entropy of 0.59 is closer to the surprise associated with orange chickens than blue chickens.', 'duration': 32.475, 'max_score': 820.274, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw820274.jpg'}, {'end': 993.995, 'src': 'heatmap', 'start': 921.124, 'weight': 2, 'content': [{'end': 929.55, 'text': 'As a result, we can use entropy to quantify the similarity or difference in the number of orange and blue chickens in each area.', 'start': 921.124, 'duration': 8.426}, {'end': 935.395, 'text': 'Entropy is highest when we have the same number of both types of chickens.', 'start': 931.151, 'duration': 4.244}, {'end': 942.3, 'text': 'And as we increase the difference in the number of orange and blue chickens, we lower the entropy.', 'start': 936.335, 'duration': 5.965}, {'end': 947.164, 'text': 'Triple bam! P.S.', 'start': 943.803, 'duration': 3.361}, {'end': 953.686, 'text': 'The next time you want to surprise someone, just whisper, The log of the inverse of the probability.', 'start': 947.504, 'duration': 6.182}, {'end': 959.188, 'text': "Bam Now it's time for some shameless self-promotion.", 'start': 954.446, 'duration': 4.742}, {'end': 967.09, 'text': 'If you want to review statistics and machine learning offline, check out the StatQuest study guides at StatQuest.org.', 'start': 959.928, 'duration': 7.162}, {'end': 968.931, 'text': "There's something for everyone.", 'start': 967.49, 'duration': 1.441}, {'end': 973.495, 'text': "Hooray! We've made it to the end of another exciting StatQuest.", 'start': 969.631, 'duration': 3.864}, {'end': 977.038, 'text': 'If you like this StatQuest and want to see more, please subscribe.', 'start': 973.915, 'duration': 3.123}, {'end': 983.925, 'text': 'And if you want to support StatQuest, consider contributing to my Patreon campaign, becoming a channel member,', 'start': 977.539, 'duration': 6.386}, {'end': 988.249, 'text': 'buying one or two of my original songs or a t-shirt or a hoodie, or just donate.', 'start': 983.925, 'duration': 4.324}, {'end': 990.171, 'text': 'The links are in the description below.', 'start': 988.589, 'duration': 1.582}, {'end': 993.995, 'text': 'Alright, until next time, quest on!.', 'start': 990.932, 'duration': 3.063}], 'summary': 'Entropy quantifies similarity between orange and blue chickens. support statquest for more content.', 'duration': 72.871, 'max_score': 921.124, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw921124.jpg'}], 'start': 767.674, 'title': 'Entropy and surprise in chicken selection', 'summary': 'Explains entropy calculation for different areas based on the number of orange and blue chickens, highlighting the relationship between surprise and probability. it quantifies similarity or difference in chicken distribution. the chapter also introduces a surprise element using the log of the inverse of the probability and promotes statquest study guides, seeking support through subscriptions, patreon, and merchandise sales.', 'chapters': [{'end': 947.164, 'start': 767.674, 'title': 'Entropy and surprise in chicken selection', 'summary': 'Explains entropy calculation for different areas based on the number of orange and blue chickens, highlighting the relationship between surprise and probability, and how entropy quantifies similarity or difference in chicken distribution.', 'duration': 179.49, 'highlights': ['Entropy calculation for area C results in the highest value of 1, indicating the same moderate surprise for both orange and blue chickens every time we pick up a chicken.', 'Area B has a lower entropy value compared to area A, indicating a higher probability of picking a chicken with a lower surprise due to the distribution of orange and blue chickens.', 'Entropy quantifies the similarity or difference in the number of orange and blue chickens in each area, with the highest entropy observed when there is an equal number of both types of chickens.']}, {'end': 993.995, 'start': 947.504, 'title': 'Statquest: surprise with probability', 'summary': 'Introduces a surprise element using the log of the inverse of the probability, while also promoting statquest study guides and seeking support through subscriptions, patreon, and merchandise sales.', 'duration': 46.491, 'highlights': ['The chapter introduces a surprise element using the log of the inverse of the probability.', 'The transcript promotes StatQuest study guides at StatQuest.org for offline review of statistics and machine learning.', 'The transcript encourages support through subscriptions, Patreon contributions, merchandise sales, and donations.']}], 'duration': 226.321, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/YtebGVx-Fxw/pics/YtebGVx-Fxw767674.jpg', 'highlights': ['Entropy calculation for area C results in the highest value of 1, indicating the same moderate surprise for both orange and blue chickens every time we pick up a chicken.', 'Area B has a lower entropy value compared to area A, indicating a higher probability of picking a chicken with a lower surprise due to the distribution of orange and blue chickens.', 'Entropy quantifies the similarity or difference in the number of orange and blue chickens in each area, with the highest entropy observed when there is an equal number of both types of chickens.', 'The chapter introduces a surprise element using the log of the inverse of the probability.', 'The transcript promotes StatQuest study guides at StatQuest.org for offline review of statistics and machine learning.', 'The transcript encourages support through subscriptions, Patreon contributions, merchandise sales, and donations.']}], 'highlights': ['Entropy is used in building classification trees for classifying things and is the basis of mutual information, quantifying the relationship between two things.', 'Relative entropy and cross entropy, derived from entropy, show up in various applications, including dimension reduction algorithms like t-SNE and UMAP.', 'Entropy quantifies similarities and differences, and understanding it requires comprehension of surprise and its application in various scenarios.', 'Statsquatch organized chickens into areas A, B, and C, demonstrating the correlation between probability and surprise.', 'The inverse relationship between probability and surprise is illustrated, providing a clear understanding of the correlation.', 'The chapter introduces the idea of calculating surprise based on the inverse relationship with probability, emphasizing the need for a refined approach.', 'The concept of surprise is introduced in the context of flipping a coin, leading to the introduction of the log of the inverse of the probability to quantify surprise.', 'The application of the log of the inverse of the probability in calculating surprise for two outputs, and the total surprise for a sequence of coin tosses is shown to be the sum of the surprises for each individual toss.', 'The calculation of average surprise, referred to as entropy, and the derivation of the equation for entropy, representing the expected value of surprise per coin toss.', 'Entropy calculation for area C results in the highest value of 1, indicating the same moderate surprise for both orange and blue chickens every time we pick up a chicken.', 'Area B has a lower entropy value compared to area A, indicating a higher probability of picking a chicken with a lower surprise due to the distribution of orange and blue chickens.', 'Entropy quantifies the similarity or difference in the number of orange and blue chickens in each area, with the highest entropy observed when there is an equal number of both types of chickens.', 'The chapter introduces a surprise element using the log of the inverse of the probability.', 'The transcript promotes StatQuest study guides at StatQuest.org for offline review of statistics and machine learning.', 'The transcript encourages support through subscriptions, Patreon contributions, merchandise sales, and donations.']}