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<h2 class="hd hd-2 unit-title">Information theory - overview</h2>
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<h2 class="hd hd-2 unit-title">The classical data compression problem</h2>
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Your browser does not support this video format. Try using a different browser.
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{
String.prototype.trim = function()
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var rlb = rb + lb;
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get_search_module();
var sterm = $('#site-search-box').val();
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new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
}
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}
$("#site-search-box").keypress(function(event) {
if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
}
return {'course_static_url': course_static_url,
'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
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'get': get,
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var vuid = vblock.data('usage-id').split('@');
var vid;
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vid = vuid[5];
}else{
vid = vuid[2];
}
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var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + ftid).length){
var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb;
html += "contribute transcript fix" + mke("/a") + mke("/span");
console.log("html = ", html);
vblock.after(html)
}
$('#' + ftid).click(function(){
var cst = $('ol.subtitles').find('li.current');
var cindex = Number(cst.data('index'));
var gurl;
if (mfnpre.endsWith('_cq_sol')){
gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/";
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gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/";
}
gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1);
console.log("going to ", gurl);
window.open(gurl, "MITx 8.370x subtitle source");
});
});
}
try{
add_fix_transcript();
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console.log(err);
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setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
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<h2 class="hd hd-2 unit-title">CQ: Noiseless coding theorem</h2>
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Noiseless coding theorem
</h3>
<div class="problem-progress" id="block-v1:MITx+8.371.3x+2T2018+type@problem+block@cq_M2L20_Noiseless_coding_theorem-problem-progress"></div>
<div class="problem">
<div>
<p>
This is a concept question, provided to check your understanding of the previous lecture video clips. </p>
<p>
Which of the following are true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_cq_M2L20_Noiseless_coding_theorem_2_1">
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<text>Strings of length [mathjaxinline]n[/mathjaxinline] sampled from iid samples from the uniform distribution over [mathjaxinline]\{ 1,\ldots ,d\}[/mathjaxinline] may not be compressed below [mathjaxinline]n \log d[/mathjaxinline] bits.</text>
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<text>Entropy is a concave function.</text>
</label>
</div>
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<text>In Shannon's noiseless coding theorem, the probability of error decays only polynomially in the length of the message.</text>
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<div class="field">
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<text>Entropy is always positive.</text>
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<span id="answer_cq_M2L20_Noiseless_coding_theorem_2_1"/>
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String.prototype.startsWith = function (str){
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String.prototype.trim = function()
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var get_search_module_ficus = function(){
var cid = $('div.xblock').data('course-id');
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// search_module_url = "/courses/course-v1:MITx+8.371.3x+2T2018/" + cid + "/courseware/welcome/Search_this_course/";
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var course_root_link = $('span.nav-item-course').find('a').attr('href');
if (course_root_link){
console.log("course_root_link = ", course_root_link);
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console.log("2. search_module_url = ", search_module_url);
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}
});
}
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var sterm = $('#site-search-box').val();
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new_url = search_module_url + "?q=" + sterm;
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+ " id='site-search-box'"
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}
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if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
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'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
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}
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return;
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vid = vuid[5];
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vid = vuid[2];
}
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mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
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var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + ftid).length){
var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb;
html += "contribute transcript fix" + mke("/a") + mke("/span");
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vblock.after(html)
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var cindex = Number(cst.data('index'));
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window.open(gurl, "MITx 8.370x subtitle source");
});
});
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<h2 class="hd hd-2 unit-title">CQ: Typical sequences</h2>
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Typical sequences
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<p>
This is a concept question, provided to check your understanding of the previous lecture video clips. </p>
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Which of the following are true? </p>
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<text>For most distributions, there are exponentially many typical sequences.</text>
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<text>It is possible that not all strings in a typical set have exponentially small probability.</text>
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<text>Asymptotically optimal source coding can be achieved by an encoder which gives incorrect answers on all strings outside the typical set.</text>
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<text>The smaller the image of a decoding map the better probability of success we obtain.</text>
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<h4 class="hd hd-4 video-hls-error is-hidden">
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<div class="closed-captions"></div>
<div class="video-controls is-hidden">
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<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
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String.prototype.startsWith = function (str){
return this.indexOf(str) === 0;
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{
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console.log("3. search_module_url = ", search_module_url);
return;
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var course_root_link = $('span.nav-item-course').find('a').attr('href');
if (course_root_link){
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if (!($('div.course-index').length)){
return get_search_module_ficus();
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$('div.course-index').find('nav').find('a').each(function(){
if ($(this).text().trim().startsWith("Search this course")){
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console.log("search_module_url = ", search_module_url);
}
});
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get_search_module();
var sterm = $('#site-search-box').val();
// new_url = jump_to_url + "/Search_this_module/?q=" + sterm;
new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
}
if (!$('#site-search-box').length){
$("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'"
+ " id='site-search-box'"
+ rlb + "img src='" + course_static_url
+ "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb);
}
$("#site-search-box").keypress(function(event) {
if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
}
return {'course_static_url': course_static_url,
'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
}
}
var the_site_search = add_site_search();
var add_fix_transcript = function(){
if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
var vblock = $(vblock_e);
var vuid = vblock.data('usage-id').split('@');
var vid;
if (vuid.length==1){
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vid = vuid[5];
}else{
vid = vuid[2];
}
var mfnpre = vid.split("_video",1)[0];
var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + ftid).length){
var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb;
html += "contribute transcript fix" + mke("/a") + mke("/span");
console.log("html = ", html);
vblock.after(html)
}
$('#' + ftid).click(function(){
var cst = $('ol.subtitles').find('li.current');
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gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/";
}
gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1);
console.log("going to ", gurl);
window.open(gurl, "MITx 8.370x subtitle source");
});
});
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try{
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catch(err){
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<h3 class="hd hd-2">Converse of the noiseless coding theorem</h3>
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class="video closed"
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<div class="video-player-pre"></div>
<div class="video-player">
<div id="M3L20k"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
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<div class="focus_grabber last"></div>
</div>
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String.prototype.startsWith = function (str){
return this.indexOf(str) === 0;
};
}
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{
String.prototype.trim = function()
{
return String(this).replace(/^\s+|\s+$/g, '');
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var rb = String.fromCharCode(62);
var amp = String.fromCharCode(38);
var rlb = rb + lb;
var mke = function(x){ return lb + x + rb; }
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var get_search_module_ficus = function(){
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return;
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var course_root_link = $('span.nav-item-course').find('a').attr('href');
if (course_root_link){
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if ($(this).text().trim().startsWith("Search this course")){
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}
});
}
var go_to_search = function(){
get_search_module();
var sterm = $('#site-search-box').val();
// new_url = jump_to_url + "/Search_this_module/?q=" + sterm;
new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
}
if (!$('#site-search-box').length){
$("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'"
+ " id='site-search-box'"
+ rlb + "img src='" + course_static_url
+ "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb);
}
$("#site-search-box").keypress(function(event) {
if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
}
return {'course_static_url': course_static_url,
'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
}
}
var the_site_search = add_site_search();
var add_fix_transcript = function(){
if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
var vblock = $(vblock_e);
var vuid = vblock.data('usage-id').split('@');
var vid;
if (vuid.length==1){
vuid = vblock.data('usage-id').split(';_')
vid = vuid[5];
}else{
vid = vuid[2];
}
var mfnpre = vid.split("_video",1)[0];
var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + ftid).length){
var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb;
html += "contribute transcript fix" + mke("/a") + mke("/span");
console.log("html = ", html);
vblock.after(html)
}
$('#' + ftid).click(function(){
var cst = $('ol.subtitles').find('li.current');
var cindex = Number(cst.data('index'));
var gurl;
if (mfnpre.endsWith('_cq_sol')){
gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/";
}else{
gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/";
}
gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1);
console.log("going to ", gurl);
window.open(gurl, "MITx 8.370x subtitle source");
});
});
}
try{
add_fix_transcript();
}
catch(err){
console.log(err);
}
try{
var rb = String.fromCharCode(62);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
}
catch(err){
console.log(err);
}
</script></span>
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<h2 class="hd hd-2 unit-title">Shannon's noisy coding theorem - statement</h2>
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Noisy coding theorem
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Which of the following are true? </p>
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<text>There is no hope in recovering a message from a binary symmetric channel if the probability of bit flip is 1.</text>
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<text>We can transmit a message with constant size overhead through a binary symmetric channel using a repetition code.</text>
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<text>The noisy coding theorem allows us to transmit a message that is encoded with constant size overhead.</text>
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<text>The capacity of a channel is a constant that only depends on the channel.</text>
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Information theory - overview
Information theory - overview
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Video transcript
- Start of transcript. Skip to the end.
- The field of information theory asks questions about--
- how efficiently can we store and transmit information,
- protect it against errors, communicate it secretly,
- figure out things in a distributed setting?
- Basically, all the questions about-- what
- is the information content of a message,
- and what is its nature, and what can we do with it?
- It generally avoids questions of--
- how hard is it to compute these things?
- So when we talk about error-correcting codes,
- we worry a lot about making the decoder efficient.
- Can I do this in a fault-tolerant way?
- And so on.
- Information theory, in general, does not
- look at these questions.
- Or maybe it doesn't first look at these questions.
- It first looks at the question of--
- what is the ultimate rate of bits that I
- can get through this channel?
- And then only later says--
- and can I do that efficiently on a computer?
- And as a result, it lets us sometimes
- give very sharp answers to questions of--
- how much can I transmit?
- Or how large of a secret message can I
- transmit, if my only limitation is the communication channel
- that I have?
- And one thing that's nice about this--
- so you get the sharp bounds.
- It also gives you a number of information theoretic
- quantities, things like entropies
- and mutual information and ways of measuring correlation, that
- have been useful not only for the study of transmitting
- information, but for problems in statistical physics,
- things like topological order.
- Sometimes there's application to algorithms,
- to understanding data sets.
- So the ideas of entropy that started off
- in information theory have been used pretty widely.
- And so today, I'm going to describe
- the core ideas of classical information theory that
- came from Claude Shannon's 1948 paper.
- So some of you may be super familiar with it.
- Some of you may not have come across it so much.
- And here he laid out two key problems in information theory.
- One of them is compression, and the other one
- is channel coding.
- So compression he referred to as his--
- he did not give these the best names,
- I would say-- the noiseless coding theorem.
- And channel coding-- meaning, if I want to send a message
- through a noisy channel and have the receiver reconstruct
- the message faithfully, how can I do that--
- he called the noisy coding theorem.
- And he gave the optimal rates at which both of these tasks
- can be achieved.
- And he defined them in terms of various quantities
- involving entropies.
- End of transcript. Skip to the start.
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