{"id":5115,"date":"2021-11-18T20:19:31","date_gmt":"2021-11-18T19:19:31","guid":{"rendered":"https:\/\/dev.m2hycon.de\/aktuelles\/unkategorisiert\/information-entropy-defined-as-easy-as-pie\/"},"modified":"2024-05-15T17:02:03","modified_gmt":"2024-05-15T15:02:03","slug":"information-entropy-defined-as-easy-as-pie","status":"publish","type":"post","link":"https:\/\/www.m2hycon.de\/en\/news\/practice\/information-entropy-defined-as-easy-as-pie\/","title":{"rendered":"Infor\u00adma\u00adtion entropy defined as easy as pie!"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"5115\" class=\"elementor elementor-5115 elementor-1115\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d5da8b5 elementor-section-full_width elementor-section-height-min-height red noisy elementor-section-height-default elementor-section-items-middle\" data-id=\"d5da8b5\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d22d6c7\" data-id=\"d22d6c7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-adc86b0 elementor-widget elementor-widget-theme-post-featured-image elementor-widget-image\" data-id=\"adc86b0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"theme-post-featured-image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"512\" src=\"https:\/\/www.m2hycon.de\/wp-content\/uploads\/2022\/11\/m2ycon_blog_informationsentropie_1920x960-1024x512.jpg\" class=\"attachment-large size-large wp-image-4845\" alt srcset=\"https:\/\/www.m2hycon.de\/wp-content\/uploads\/2022\/11\/m2ycon_blog_informationsentropie_1920x960-1024x512.jpg 1024w, https:\/\/www.m2hycon.de\/wp-content\/uploads\/2022\/11\/m2ycon_blog_informationsentropie_1920x960-300x150.jpg 300w, https:\/\/www.m2hycon.de\/wp-content\/uploads\/2022\/11\/m2ycon_blog_informationsentropie_1920x960-768x384.jpg 768w, https:\/\/www.m2hycon.de\/wp-content\/uploads\/2022\/11\/m2ycon_blog_informationsentropie_1920x960-1536x768.jpg 1536w, https:\/\/www.m2hycon.de\/wp-content\/uploads\/2022\/11\/m2ycon_blog_informationsentropie_1920x960.jpg 1920w\" sizes=\"(max-width: 1024px) 100vw, 1024px\">\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-be19b45 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"be19b45\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-99bdcb5\" data-id=\"99bdcb5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6688c8f elementor-widget elementor-widget-theme-post-title elementor-page-title elementor-widget-heading\" data-id=\"6688c8f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"theme-post-title.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Infor\u00adma\u00adtion entropy defined as easy as pie!<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3a50170 elementor-widget elementor-widget-post-info\" data-id=\"3a50170\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"post-info.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul class=\"elementor-inline-items elementor-icon-list-items elementor-post-info\">\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item elementor-repeater-item-9fc294b elementor-inline-item\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text elementor-post-info__item elementor-post-info__item--type-custom\">\n\t\t\t\t\t\t\t\t\t\tVon Alexei Kurgansky\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t<li class=\"elementor-icon-list-item elementor-repeater-item-2efc209 elementor-inline-item\" itemprop=\"datePublished\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text elementor-post-info__item elementor-post-info__item--type-date\">\n\t\t\t\t\t\t\t\t\t\t<time>November 18, 2021<\/time>\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t<li class=\"elementor-icon-list-item elementor-repeater-item-84e7708 elementor-inline-item\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text elementor-post-info__item elementor-post-info__item--type-custom\">\n\t\t\t\t\t\t\t\t\t\t<a href=\"https:\/\/www.m2hycon.de\/en\/tag\/claude-shannon-en\/\" rel=\"tag\">Claude Shannon<\/a>, <a href=\"https:\/\/www.m2hycon.de\/en\/tag\/cross-entropy\/\" rel=\"tag\">Cross entropy<\/a>, <a href=\"https:\/\/www.m2hycon.de\/en\/tag\/entropy\/\" rel=\"tag\">Entropy<\/a>, <a href=\"https:\/\/www.m2hycon.de\/en\/tag\/information-entropy\/\" rel=\"tag\">Infor\u00adma\u00adtion entropy<\/a>, <a href=\"https:\/\/www.m2hycon.de\/en\/tag\/kullback-leibler-divergence\/\" rel=\"tag\">Kullback-Leibler diver\u00adgence<\/a>\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t<\/ul>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-47a9adc elementor-section-content-middle elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"47a9adc\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-inner-column elementor-element elementor-element-57806cf\" data-id=\"57806cf\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-171bae2 elementor-widget__width-auto elementor-widget elementor-widget-text-editor\" data-id=\"171bae2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<h4>Share post:<\/h4>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3352ed9 elementor-share-buttons--view-icon elementor-share-buttons--skin-minimal elementor-share-buttons--color-custom elementor-widget__width-auto elementor-share-buttons--shape-circle elementor-grid-0 elementor-widget elementor-widget-share-buttons\" data-id=\"3352ed9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"share-buttons.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-grid\" role=\"list\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-grid-item\" role=\"listitem\">\n\t\t\t\t\t\t<div class=\"elementor-share-btn elementor-share-btn_facebook\" role=\"button\" tabindex=\"0\" aria-label=\"Share on facebook\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-share-btn__icon\">\n\t\t\t\t\t\t\t\t<i class=\"fab fa-facebook\" aria-hidden=\"true\"><\/i>\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-grid-item\" role=\"listitem\">\n\t\t\t\t\t\t<div class=\"elementor-share-btn elementor-share-btn_twitter\" role=\"button\" tabindex=\"0\" aria-label=\"Share on twitter\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-share-btn__icon\">\n\t\t\t\t\t\t\t\t<i class=\"fab fa-twitter\" aria-hidden=\"true\"><\/i>\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-grid-item\" role=\"listitem\">\n\t\t\t\t\t\t<div class=\"elementor-share-btn elementor-share-btn_linkedin\" role=\"button\" tabindex=\"0\" aria-label=\"Share on linkedin\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-share-btn__icon\">\n\t\t\t\t\t\t\t\t<i class=\"fab fa-linkedin\" aria-hidden=\"true\"><\/i>\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-grid-item\" role=\"listitem\">\n\t\t\t\t\t\t<div class=\"elementor-share-btn elementor-share-btn_xing\" role=\"button\" tabindex=\"0\" aria-label=\"Share on xing\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-share-btn__icon\">\n\t\t\t\t\t\t\t\t<i class=\"fab fa-xing\" aria-hidden=\"true\"><\/i>\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5c0fa21 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5c0fa21\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ac5c031\" data-id=\"ac5c031\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8dd54b3 maths elementor-widget elementor-widget-text-editor\" data-id=\"8dd54b3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\tLet\u2019s assume that in summer we want to send our friends a telegram from Hamburg to St. Peters\u00adburg every day about the weather condi\u00adtions: sunny, slightly cloudy, overcast, rainy. Weather is a random event for us, however we expect sunny weather for about $1\/2$ of the days in summer, partly cloudy for $1\/4$ of the days, overcast for $1\/8$ of the days and rainy for $1\/8$ of the days. Only <em>zeros<\/em> ($0$) and <em>ones<\/em> (1) can be sent by telegraph. One symbol costs $1$ Euro. If the messages are encoded with symbol pairs $00$, $01$, $10$ and $11$, two bits are sent every day at a total cost of $2$ Euro. The telegrams then cost $184$ Euro for $92$ summer days in total. If you code the messages \u201csunny\u201d, \u201cslightly cloudy\u201d, \u201covercast\u201d, \u201crainy\u201d with $0$, $10$, $110$, $111$, on average every day\n<p class=\"maths\">$\n(1\/2)\\cdot 1 + (1\/4)\\cdot 2 + (1\/8)\\cdot 3 + (1\/8)\\cdot 3 = 7\/4\n$<\/p>\nBit sent, and for $92$ summer days about $7\/4 \\cdot 1 \\cdot 92 = 161$ Euro are spent. You can see the diffe\u00adrence, some codes are more expen\u00adsive and some are cheaper, and it is obvious that there is the cheapest code.\n<blockquote>If the coding is so intel\u00adli\u00adgent that it causes the lowest cost of telegrams with event reports over the summer, then the average number of bits sent per day is referred to as the infor\u00adma\u00adtion entropy of this event.<\/blockquote>\nThe genera\u00adliza\u00adtion of the defini\u00adtion to arbitrary random events is now simple.\n<blockquote>How do you calcu\u00adlate infor\u00adma\u00adtion entropy?<\/blockquote>\nThis question is related to the question of what coding is. Assume that there is a terri\u00adtory of messages, which is repre\u00adsented by the first rectangle in the figure. Let us assume that the area of the entire terri\u00adtory is equal to $1$. If the terri\u00adtory is divided into two parts, each part is given the codenames (coordi\u00adnates) $0$ and $1$ as shown in the second rectangle. The area sizes of $0$ and $1$ are equal to $1\/2$. If each message terri\u00adtory area $0$ and $1$ is also divided into two parts, then these small areas of the terri\u00adtory are given the codes $00$, $01$, $10$, $11$ as shown by the third rectangle. The area sizes of these parts are equal to $1\/4$. This shows what the codes are for the areas with area sizes $1\/8$, $1\/16$ etc. It can be seen that the area size of a message terri\u00adtory area is equal to $\\frac{1}{2^n}$, where $n$ is the length of its code. In other words: if $q$ is the area of the region, its code length is $\\log_2 \\frac{1}{q}=-\\log_2 q$. The larger the area, the shorter the code.\n\nThe 4. and 5. rectan\u00adgles show examples of the place\u00adment of messages over the message terri\u00adtory.\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-db93984 elementor-widget elementor-widget-image\" data-id=\"db93984\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" src=\"https:\/\/www.m2hycon.de\/wp-content\/uploads\/elementor\/thumbs\/m2hycon_blog_informationsentropie_in_text-q1v4xudtoeqbm6dbw34g78lq8g1c97rnutyhv1etf6.jpg\" title=\"Infor\u00adma\u00adti\u00adons\u00aden\u00adtropie kinder\u00adleicht definiert\" alt=\"Informationsentropie kinderleicht definiert\" loading=\"lazy\">\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c8094a4 maths elementor-widget elementor-widget-text-editor\" data-id=\"c8094a4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\tLet $N$ messages be given, so that the proba\u00adbi\u00adlity of the $i$th message is equal to $p_i$ and the $i$th message occupies a message terri\u00adtory area with the area size $q_i$ and a code of length $-\\log_2 q_i$. The average number of bits sent daily with this coding is\n\n$\n-p_1\\cdot \\log_2 q_1 -\\ldots -p_N\\cdot \\log_2 q_N.\n$\n<blockquote>For which values of $q_i$ is this sum minimal?<\/blockquote>\nWe remind you of the fact\n\n$$\np_1 +\u2026+p_N = 1\n$$\n\nand a secon\u00addary condi\u00adtion\n\n$$\nq_1 +\u2026+q_N = 1.\n$$\n\nLet\u2019s solve the problem with Lagrange multi\u00adpliers. We do not forget that $p_i$ are constants and $q_i$ are varia\u00adbles.\n\nThe Lagrange function is\n\n$$\n\\lambda(q_1,\\ldots,q_N,\\lambda) =-p_1\\cdot \\log_2 q_1 -\u2026-p_N\\cdot \\log_2 q_N + \\lambda\\cdot (q_1+\u2026+q_N).\n$$\n\nLet\u2019s find the deriva\u00adtives of $\\Lambda$ to $q_i$ and set them equal to $0$:\n\n$$\n-\\frac{p_i}{q_i\\cdot \\ln 2} + \\lambda = 0.\n$$\n\nReminder,\n\n$$\n\\frac{d}{dx}\\log_2x = \\frac{d}{dx} \\frac{\\ln x}{\\ln 2} = \\frac{1}{x \\ln 2}.\n$$\n\nFrom this follows\n\n$$\np_i = \\lambda\\cdot q_i\\cdot\\ln 2.\n$$\n\nIf you sum these equations for all $i$, you get\n\n$$\n\\lambda\\cdot\\ln 2 = 1.\n$$\n\nIf you insert $\\lambda\\cdot \\ln 2 = 1$ into $p_i = \\lambda\\cdot q_i\\cdot\\ln 2$, you get $p_i = q_i$. Optimal coding is there\u00adfore achieved when $q_i = p_i$, which means that the formula for calcu\u00adla\u00adting the entropy $H$ is as defined:\n\n$$\nH = -p_1\\cdot \\log_2 p_1 -\u2026-p_N\\cdot \\log_2 p_N.\n$$\n<blockquote>Cross entropy<\/blockquote>\nNote that $q_i$ behave like proba\u00adbi\u00adlity varia\u00adbles: they are greater than $0$ and their sum is equal to $1$. The size\n\n$$\nH(p,q)=-p_1\\cdot \\log_2 q_1 -\u2026-p_N\\cdot \\log_2 q_N\n$$\n\nis called the cross entropy of the two proba\u00adbi\u00adlity distri\u00adbu\u00adtions $(p_i)$ and $(q_i)$, but $q_i$ does not actually have a proba\u00adbi\u00adlity role. The value $H(p,q)$ provides the average bit costs for the trans\u00admis\u00adsion of messages that are coded accor\u00adding to ranges $q_i$ but are trans\u00admitted with proba\u00adbi\u00adli\u00adties $p_i$.\n<blockquote>Kullback-Leibler diver\u00adgence<\/blockquote>\nThe diffe\u00adrence between\n\n$$\n-p_1\\cdot \\log_2 q_1 -\u2026-p_N\\cdot \\log_2 q_N\n$$\n\nand\n\n$$\n-p_1\\cdot \\log_2 p_1 -\u2026-p_N\\cdot \\log_2 p_N\n$$\n\nis the value of average additional bit costs for message trans\u00admis\u00adsion due to subop\u00adtimal coding. This quantity is referred to as the Kullback-Leibler diver\u00adgence between the distri\u00adbu\u00adtions $(p_i)$ and $(q_i)$.\n<blockquote>One remark<\/blockquote>\nFor the sake of simpli\u00adcity, let us now assume that there are only $2$ weather condi\u00adtions in summer: good weather $G$ and bad weather $S$.\nLet us also assume the proba\u00adbi\u00adli\u00adties $P(G)=3\/4$, $P(S)=1\/4$.\nAccor\u00addingly, the infor\u00adma\u00adtion entropy of the weather is equal to\n\n$$\n-\\frac{3}{4}\\cdot \\log_2 \\frac{3}{4} -\\frac{1}{4}\\cdot \\log_2 \\frac{1}{4} \\approx 0.8113.\n$$\n\nFor the optimal coding of the event $S$ you need $\\log_2 \\frac{1}{4}=2$ bits. Every\u00adthing is in the green here.\nHowever, for the $G$ you need about $\\log_2 \\frac{3}{4}\\approx 0.415$ bits.\nThis is less than $1$ bit and does not match reality,\nthat by defini\u00adtion only whole symbols $0$ and $1$ are written in the telegrams.\nThere is no other way and we have to encode both events with $1$ bit: $0$ and $1$,\njust as in the case $P(G)=P(S)=1\/2$.\nSo if we want to send a telegram every day, we can\u2019t save money and are forced to,\nTelegraph $92$ bit for $92$ days. It means the entropy value equals $1$ and seems to contra\u00addict the theory.\n\nHowever, you can actually save money by sending a single telegram with a weather report for all days, not every day but only at the end of the summer.\nYou can see it in an example with $2$ days.\n\nWith $P(G)=P(S)=1\/2$ the proba\u00adbi\u00adli\u00adties for all possible 2-day variants are the same\n\n$$\nP(GG)=P(GS)=P(SG)=P(SS)=1\/4.\n$$\n\nThe infor\u00adma\u00adtion entropy here is $2$ for $2$-day weather events and $1$ for $1$-day weather events.\nThe weather entropy is there\u00adfore $1$.\n\nIn the case $P(G)=3\/4$, $P(S)=1\/4$ it is $P(GG)=9\/16$, $P(GS)=P(SG)=3\/16$, $P(SS)=1\/16$.\nThe most optimal coding for the event $GG$ requires appro\u00adxi\u00adm\u00adately\n$\\log_2 \\frac{9}{16} \\approx 0.83$ Bit.\nSimilarly for the events $GS$, $SG$, $SS$ you need about $2.415$, $2.415$ and $4$ bits accor\u00addingly.\n\nThe most optimal coding looks like this $GG\\rightarrow 0$, $GS\\rightarrow 10$, $SG\\rightarrow 110$, $SS\\rightarrow 111$.\nWith this coding, the average report for both days is then only\n\n$$\n\\frac{9}{16}\\cdot 1 + \\frac{3}{16}\\cdot 2 + \\frac{3}{16}\\cdot 3 + \\frac{1}{16}\\cdot 3 = 1.6875 \\quad (\\text{Bit})\n$$\n\nand for one day:\n\n$$\n\\frac{1.6875}{2} = 0.84375 \\quad (\\text{Bit}).\n$$\n\nThe infor\u00adma\u00adtion entropy of the weather event is equal to $0.84375$. Already cheaper.\n\nThe longer the sequence of events is coded at once, the more optimal the coding is. If you go to infinity, you approach the theore\u00adtical infor\u00adma\u00adtion entropy value in praxi.\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2479d6b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2479d6b\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-8052f17\" data-id=\"8052f17\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-01eb1b9 elementor-widget elementor-widget-author-box\" data-id=\"01eb1b9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"author-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-author-box\">\n\t\t\t\t\t\t\t<div class=\"elementor-author-box__avatar\">\n\t\t\t\t\t<img decoding=\"async\" src=\"https:\/\/www.m2hycon.de\/wp-content\/uploads\/2023\/09\/m2hycon_ueberuns_team_Aleksei_Kurgansky_800x800-300x300.jpg\" alt=\"Picture of Alexei Kurgansky\" loading=\"lazy\">\n\t\t\t\t<\/div>\n\t\t\t\n\t\t\t<div class=\"elementor-author-box__text\">\n\t\t\t\t\t\t\t\t\t<div>\n\t\t\t\t\t\t<h4 class=\"elementor-author-box__name\">\n\t\t\t\t\t\t\tAlexei Kurgansky\t\t\t\t\t\t<\/h4>\n\t\t\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-c0197e8 elementor-hidden-tablet elementor-hidden-mobile\" data-id=\"c0197e8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6daf353 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6daf353\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ae400ad\" data-id=\"ae400ad\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2169cf8 elementor-widget elementor-widget-post-navigation\" data-id=\"2169cf8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"post-navigation.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-post-navigation\" role=\"navigation\" aria-label=\"Post Navigation\">\n\t\t\t<div class=\"elementor-post-navigation__prev elementor-post-navigation__link\">\n\t\t\t\t<a href=\"https:\/\/www.m2hycon.de\/en\/news\/practice\/estimation-of-a-customer-specific-offer-acceptance-probability\/\" rel=\"prev\"><span class=\"post-navigation__arrow-wrapper post-navigation__arrow-prev\"><i aria-hidden=\"true\" class=\"fas fa-angle-left\"><\/i><span class=\"elementor-screen-only\">Prev<\/span><\/span><span class=\"elementor-post-navigation__link__prev\"><span class=\"post-navigation__prev--label\">Previous post<\/span><span class=\"post-navigation__prev--title\">Estima\u00adtion of a customer-specific offer accep\u00adtance proba\u00adbi\u00adlity<\/span><\/span><\/a>\t\t\t<\/div>\n\t\t\t\t\t\t<div class=\"elementor-post-navigation__next elementor-post-navigation__link\">\n\t\t\t\t<a href=\"https:\/\/www.m2hycon.de\/en\/news\/expert-knowledge\/what-is-preventive-maintenance\/\" rel=\"next\"><span class=\"elementor-post-navigation__link__next\"><span class=\"post-navigation__next--label\">Next post<\/span><span class=\"post-navigation__next--title\">What is preven\u00adtive mainten\u00adance?<\/span><\/span><span class=\"post-navigation__arrow-wrapper post-navigation__arrow-next\"><i aria-hidden=\"true\" class=\"fas fa-angle-right\"><\/i><span class=\"elementor-screen-only\">Next<\/span><\/span><\/a>\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>What is infor\u00adma\u00adtion entropy? 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