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Emanuel Rodriguez 2022-09-11 01:27:09 -07:00
parent 838f9e05f3
commit 0d08907a15
8 changed files with 399 additions and 114 deletions
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@ -113,10 +113,7 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
<ul>
<li><a href="#likelihood" id="toc-likelihood" class="nav-link active" data-scroll-target="#likelihood">Likelihood</a></li>
<li><a href="#simualation" id="toc-simualation" class="nav-link" data-scroll-target="#simualation">Simualation</a></li>
<li><a href="#binomial-model-and-the-chess-example" id="toc-binomial-model-and-the-chess-example" class="nav-link" data-scroll-target="#binomial-model-and-the-chess-example">Binomial Model and the chess example</a>
<ul class="collapse">
<li><a href="#the-binomial-model" id="toc-the-binomial-model" class="nav-link" data-scroll-target="#the-binomial-model">The Binomial Model</a></li>
</ul></li>
<li><a href="#binomial-model-and-the-chess-example" id="toc-binomial-model-and-the-chess-example" class="nav-link" data-scroll-target="#binomial-model-and-the-chess-example">Binomial Model and the chess example</a></li>
</ul>
</nav>
</div>
@ -244,12 +241,12 @@ Probability and Likelihood
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a> gt<span class="sc">::</span><span class="fu">cols_width</span>(<span class="fu">everything</span>() <span class="sc">~</span> <span class="fu">px</span>(<span class="dv">100</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div id="ibvcfeegcr" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<div id="gqllsnwjsv" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>html {
font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif;
}
#ibvcfeegcr .gt_table {
#gqllsnwjsv .gt_table {
display: table;
border-collapse: collapse;
margin-left: auto;
@ -274,7 +271,7 @@ Probability and Likelihood
border-left-color: #D3D3D3;
}
#ibvcfeegcr .gt_heading {
#gqllsnwjsv .gt_heading {
background-color: #FFFFFF;
text-align: center;
border-bottom-color: #FFFFFF;
@ -286,7 +283,7 @@ Probability and Likelihood
border-right-color: #D3D3D3;
}
#ibvcfeegcr .gt_title {
#gqllsnwjsv .gt_title {
color: #333333;
font-size: 125%;
font-weight: initial;
@ -298,7 +295,7 @@ Probability and Likelihood
border-bottom-width: 0;
}
#ibvcfeegcr .gt_subtitle {
#gqllsnwjsv .gt_subtitle {
color: #333333;
font-size: 85%;
font-weight: initial;
@ -310,13 +307,13 @@ Probability and Likelihood
border-top-width: 0;
}
#ibvcfeegcr .gt_bottom_border {
#gqllsnwjsv .gt_bottom_border {
border-bottom-style: solid;
border-bottom-width: 2px;
border-bottom-color: #D3D3D3;
}
#ibvcfeegcr .gt_col_headings {
#gqllsnwjsv .gt_col_headings {
border-top-style: solid;
border-top-width: 2px;
border-top-color: #D3D3D3;
@ -331,7 +328,7 @@ Probability and Likelihood
border-right-color: #D3D3D3;
}
#ibvcfeegcr .gt_col_heading {
#gqllsnwjsv .gt_col_heading {
color: #333333;
background-color: #FFFFFF;
font-size: 100%;
@ -351,7 +348,7 @@ Probability and Likelihood
overflow-x: hidden;
}
#ibvcfeegcr .gt_column_spanner_outer {
#gqllsnwjsv .gt_column_spanner_outer {
color: #333333;
background-color: #FFFFFF;
font-size: 100%;
@ -363,15 +360,15 @@ Probability and Likelihood
padding-right: 4px;
}
#ibvcfeegcr .gt_column_spanner_outer:first-child {
#gqllsnwjsv .gt_column_spanner_outer:first-child {
padding-left: 0;
}
#ibvcfeegcr .gt_column_spanner_outer:last-child {
#gqllsnwjsv .gt_column_spanner_outer:last-child {
padding-right: 0;
}
#ibvcfeegcr .gt_column_spanner {
#gqllsnwjsv .gt_column_spanner {
border-bottom-style: solid;
border-bottom-width: 2px;
border-bottom-color: #D3D3D3;
@ -383,7 +380,7 @@ Probability and Likelihood
width: 100%;
}
#ibvcfeegcr .gt_group_heading {
#gqllsnwjsv .gt_group_heading {
padding-top: 8px;
padding-bottom: 8px;
padding-left: 5px;
@ -408,7 +405,7 @@ Probability and Likelihood
vertical-align: middle;
}
#ibvcfeegcr .gt_empty_group_heading {
#gqllsnwjsv .gt_empty_group_heading {
padding: 0.5px;
color: #333333;
background-color: #FFFFFF;
@ -423,15 +420,15 @@ Probability and Likelihood
vertical-align: middle;
}
#ibvcfeegcr .gt_from_md > :first-child {
#gqllsnwjsv .gt_from_md > :first-child {
margin-top: 0;
}
#ibvcfeegcr .gt_from_md > :last-child {
#gqllsnwjsv .gt_from_md > :last-child {
margin-bottom: 0;
}
#ibvcfeegcr .gt_row {
#gqllsnwjsv .gt_row {
padding-top: 8px;
padding-bottom: 8px;
padding-left: 5px;
@ -450,7 +447,7 @@ Probability and Likelihood
overflow-x: hidden;
}
#ibvcfeegcr .gt_stub {
#gqllsnwjsv .gt_stub {
color: #333333;
background-color: #FFFFFF;
font-size: 100%;
@ -463,7 +460,7 @@ Probability and Likelihood
padding-right: 5px;
}
#ibvcfeegcr .gt_stub_row_group {
#gqllsnwjsv .gt_stub_row_group {
color: #333333;
background-color: #FFFFFF;
font-size: 100%;
@ -477,11 +474,11 @@ Probability and Likelihood
vertical-align: top;
}
#ibvcfeegcr .gt_row_group_first td {
#gqllsnwjsv .gt_row_group_first td {
border-top-width: 2px;
}
#ibvcfeegcr .gt_summary_row {
#gqllsnwjsv .gt_summary_row {
color: #333333;
background-color: #FFFFFF;
text-transform: inherit;
@ -491,16 +488,16 @@ Probability and Likelihood
padding-right: 5px;
}
#ibvcfeegcr .gt_first_summary_row {
#gqllsnwjsv .gt_first_summary_row {
border-top-style: solid;
border-top-color: #D3D3D3;
}
#ibvcfeegcr .gt_first_summary_row.thick {
#gqllsnwjsv .gt_first_summary_row.thick {
border-top-width: 2px;
}
#ibvcfeegcr .gt_last_summary_row {
#gqllsnwjsv .gt_last_summary_row {
padding-top: 8px;
padding-bottom: 8px;
padding-left: 5px;
@ -510,7 +507,7 @@ Probability and Likelihood
border-bottom-color: #D3D3D3;
}
#ibvcfeegcr .gt_grand_summary_row {
#gqllsnwjsv .gt_grand_summary_row {
color: #333333;
background-color: #FFFFFF;
text-transform: inherit;
@ -520,7 +517,7 @@ Probability and Likelihood
padding-right: 5px;
}
#ibvcfeegcr .gt_first_grand_summary_row {
#gqllsnwjsv .gt_first_grand_summary_row {
padding-top: 8px;
padding-bottom: 8px;
padding-left: 5px;
@ -530,11 +527,11 @@ Probability and Likelihood
border-top-color: #D3D3D3;
}
#ibvcfeegcr .gt_striped {
#gqllsnwjsv .gt_striped {
background-color: rgba(128, 128, 128, 0.05);
}
#ibvcfeegcr .gt_table_body {
#gqllsnwjsv .gt_table_body {
border-top-style: solid;
border-top-width: 2px;
border-top-color: #D3D3D3;
@ -543,7 +540,7 @@ Probability and Likelihood
border-bottom-color: #D3D3D3;
}
#ibvcfeegcr .gt_footnotes {
#gqllsnwjsv .gt_footnotes {
color: #333333;
background-color: #FFFFFF;
border-bottom-style: none;
@ -557,7 +554,7 @@ Probability and Likelihood
border-right-color: #D3D3D3;
}
#ibvcfeegcr .gt_footnote {
#gqllsnwjsv .gt_footnote {
margin: 0px;
font-size: 90%;
padding-left: 4px;
@ -566,7 +563,7 @@ Probability and Likelihood
padding-right: 5px;
}
#ibvcfeegcr .gt_sourcenotes {
#gqllsnwjsv .gt_sourcenotes {
color: #333333;
background-color: #FFFFFF;
border-bottom-style: none;
@ -580,7 +577,7 @@ Probability and Likelihood
border-right-color: #D3D3D3;
}
#ibvcfeegcr .gt_sourcenote {
#gqllsnwjsv .gt_sourcenote {
font-size: 90%;
padding-top: 4px;
padding-bottom: 4px;
@ -588,64 +585,64 @@ Probability and Likelihood
padding-right: 5px;
}
#ibvcfeegcr .gt_left {
#gqllsnwjsv .gt_left {
text-align: left;
}
#ibvcfeegcr .gt_center {
#gqllsnwjsv .gt_center {
text-align: center;
}
#ibvcfeegcr .gt_right {
#gqllsnwjsv .gt_right {
text-align: right;
font-variant-numeric: tabular-nums;
}
#ibvcfeegcr .gt_font_normal {
#gqllsnwjsv .gt_font_normal {
font-weight: normal;
}
#ibvcfeegcr .gt_font_bold {
#gqllsnwjsv .gt_font_bold {
font-weight: bold;
}
#ibvcfeegcr .gt_font_italic {
#gqllsnwjsv .gt_font_italic {
font-style: italic;
}
#ibvcfeegcr .gt_super {
#gqllsnwjsv .gt_super {
font-size: 65%;
}
#ibvcfeegcr .gt_footnote_marks {
#gqllsnwjsv .gt_footnote_marks {
font-style: italic;
font-weight: normal;
font-size: 75%;
vertical-align: 0.4em;
}
#ibvcfeegcr .gt_asterisk {
#gqllsnwjsv .gt_asterisk {
font-size: 100%;
vertical-align: 0;
}
#ibvcfeegcr .gt_indent_1 {
#gqllsnwjsv .gt_indent_1 {
text-indent: 5px;
}
#ibvcfeegcr .gt_indent_2 {
#gqllsnwjsv .gt_indent_2 {
text-indent: 10px;
}
#ibvcfeegcr .gt_indent_3 {
#gqllsnwjsv .gt_indent_3 {
text-indent: 15px;
}
#ibvcfeegcr .gt_indent_4 {
#gqllsnwjsv .gt_indent_4 {
text-indent: 20px;
}
#ibvcfeegcr .gt_indent_5 {
#gqllsnwjsv .gt_indent_5 {
text-indent: 25px;
}
</style>
@ -854,12 +851,12 @@ Baye’s Rule
<span id="cb10-8"><a href="#cb10-8" aria-hidden="true" tabindex="-1"></a> gt<span class="sc">::</span><span class="fu">cols_width</span>(<span class="fu">everything</span>() <span class="sc">~</span> <span class="fu">px</span>(<span class="dv">100</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div id="dpxebxbyvj" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<div id="roxupfdiiw" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>html {
font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif;
}
#dpxebxbyvj .gt_table {
#roxupfdiiw .gt_table {
display: table;
border-collapse: collapse;
margin-left: auto;
@ -884,7 +881,7 @@ Baye’s Rule
border-left-color: #D3D3D3;
}
#dpxebxbyvj .gt_heading {
#roxupfdiiw .gt_heading {
background-color: #FFFFFF;
text-align: center;
border-bottom-color: #FFFFFF;
@ -896,7 +893,7 @@ Baye’s Rule
border-right-color: #D3D3D3;
}
#dpxebxbyvj .gt_title {
#roxupfdiiw .gt_title {
color: #333333;
font-size: 125%;
font-weight: initial;
@ -908,7 +905,7 @@ Baye’s Rule
border-bottom-width: 0;
}
#dpxebxbyvj .gt_subtitle {
#roxupfdiiw .gt_subtitle {
color: #333333;
font-size: 85%;
font-weight: initial;
@ -920,13 +917,13 @@ Baye’s Rule
border-top-width: 0;
}
#dpxebxbyvj .gt_bottom_border {
#roxupfdiiw .gt_bottom_border {
border-bottom-style: solid;
border-bottom-width: 2px;
border-bottom-color: #D3D3D3;
}
#dpxebxbyvj .gt_col_headings {
#roxupfdiiw .gt_col_headings {
border-top-style: solid;
border-top-width: 2px;
border-top-color: #D3D3D3;
@ -941,7 +938,7 @@ Baye’s Rule
border-right-color: #D3D3D3;
}
#dpxebxbyvj .gt_col_heading {
#roxupfdiiw .gt_col_heading {
color: #333333;
background-color: #FFFFFF;
font-size: 100%;
@ -961,7 +958,7 @@ Baye’s Rule
overflow-x: hidden;
}
#dpxebxbyvj .gt_column_spanner_outer {
#roxupfdiiw .gt_column_spanner_outer {
color: #333333;
background-color: #FFFFFF;
font-size: 100%;
@ -973,15 +970,15 @@ Baye’s Rule
padding-right: 4px;
}
#dpxebxbyvj .gt_column_spanner_outer:first-child {
#roxupfdiiw .gt_column_spanner_outer:first-child {
padding-left: 0;
}
#dpxebxbyvj .gt_column_spanner_outer:last-child {
#roxupfdiiw .gt_column_spanner_outer:last-child {
padding-right: 0;
}
#dpxebxbyvj .gt_column_spanner {
#roxupfdiiw .gt_column_spanner {
border-bottom-style: solid;
border-bottom-width: 2px;
border-bottom-color: #D3D3D3;
@ -993,7 +990,7 @@ Baye’s Rule
width: 100%;
}
#dpxebxbyvj .gt_group_heading {
#roxupfdiiw .gt_group_heading {
padding-top: 8px;
padding-bottom: 8px;
padding-left: 5px;
@ -1018,7 +1015,7 @@ Baye’s Rule
vertical-align: middle;
}
#dpxebxbyvj .gt_empty_group_heading {
#roxupfdiiw .gt_empty_group_heading {
padding: 0.5px;
color: #333333;
background-color: #FFFFFF;
@ -1033,15 +1030,15 @@ Baye’s Rule
vertical-align: middle;
}
#dpxebxbyvj .gt_from_md > :first-child {
#roxupfdiiw .gt_from_md > :first-child {
margin-top: 0;
}
#dpxebxbyvj .gt_from_md > :last-child {
#roxupfdiiw .gt_from_md > :last-child {
margin-bottom: 0;
}
#dpxebxbyvj .gt_row {
#roxupfdiiw .gt_row {
padding-top: 8px;
padding-bottom: 8px;
padding-left: 5px;
@ -1060,7 +1057,7 @@ Baye’s Rule
overflow-x: hidden;
}
#dpxebxbyvj .gt_stub {
#roxupfdiiw .gt_stub {
color: #333333;
background-color: #FFFFFF;
font-size: 100%;
@ -1073,7 +1070,7 @@ Baye’s Rule
padding-right: 5px;
}
#dpxebxbyvj .gt_stub_row_group {
#roxupfdiiw .gt_stub_row_group {
color: #333333;
background-color: #FFFFFF;
font-size: 100%;
@ -1087,11 +1084,11 @@ Baye’s Rule
vertical-align: top;
}
#dpxebxbyvj .gt_row_group_first td {
#roxupfdiiw .gt_row_group_first td {
border-top-width: 2px;
}
#dpxebxbyvj .gt_summary_row {
#roxupfdiiw .gt_summary_row {
color: #333333;
background-color: #FFFFFF;
text-transform: inherit;
@ -1101,16 +1098,16 @@ Baye’s Rule
padding-right: 5px;
}
#dpxebxbyvj .gt_first_summary_row {
#roxupfdiiw .gt_first_summary_row {
border-top-style: solid;
border-top-color: #D3D3D3;
}
#dpxebxbyvj .gt_first_summary_row.thick {
#roxupfdiiw .gt_first_summary_row.thick {
border-top-width: 2px;
}
#dpxebxbyvj .gt_last_summary_row {
#roxupfdiiw .gt_last_summary_row {
padding-top: 8px;
padding-bottom: 8px;
padding-left: 5px;
@ -1120,7 +1117,7 @@ Baye’s Rule
border-bottom-color: #D3D3D3;
}
#dpxebxbyvj .gt_grand_summary_row {
#roxupfdiiw .gt_grand_summary_row {
color: #333333;
background-color: #FFFFFF;
text-transform: inherit;
@ -1130,7 +1127,7 @@ Baye’s Rule
padding-right: 5px;
}
#dpxebxbyvj .gt_first_grand_summary_row {
#roxupfdiiw .gt_first_grand_summary_row {
padding-top: 8px;
padding-bottom: 8px;
padding-left: 5px;
@ -1140,11 +1137,11 @@ Baye’s Rule
border-top-color: #D3D3D3;
}
#dpxebxbyvj .gt_striped {
#roxupfdiiw .gt_striped {
background-color: rgba(128, 128, 128, 0.05);
}
#dpxebxbyvj .gt_table_body {
#roxupfdiiw .gt_table_body {
border-top-style: solid;
border-top-width: 2px;
border-top-color: #D3D3D3;
@ -1153,7 +1150,7 @@ Baye’s Rule
border-bottom-color: #D3D3D3;
}
#dpxebxbyvj .gt_footnotes {
#roxupfdiiw .gt_footnotes {
color: #333333;
background-color: #FFFFFF;
border-bottom-style: none;
@ -1167,7 +1164,7 @@ Baye’s Rule
border-right-color: #D3D3D3;
}
#dpxebxbyvj .gt_footnote {
#roxupfdiiw .gt_footnote {
margin: 0px;
font-size: 90%;
padding-left: 4px;
@ -1176,7 +1173,7 @@ Baye’s Rule
padding-right: 5px;
}
#dpxebxbyvj .gt_sourcenotes {
#roxupfdiiw .gt_sourcenotes {
color: #333333;
background-color: #FFFFFF;
border-bottom-style: none;
@ -1190,7 +1187,7 @@ Baye’s Rule
border-right-color: #D3D3D3;
}
#dpxebxbyvj .gt_sourcenote {
#roxupfdiiw .gt_sourcenote {
font-size: 90%;
padding-top: 4px;
padding-bottom: 4px;
@ -1198,64 +1195,64 @@ Baye’s Rule
padding-right: 5px;
}
#dpxebxbyvj .gt_left {
#roxupfdiiw .gt_left {
text-align: left;
}
#dpxebxbyvj .gt_center {
#roxupfdiiw .gt_center {
text-align: center;
}
#dpxebxbyvj .gt_right {
#roxupfdiiw .gt_right {
text-align: right;
font-variant-numeric: tabular-nums;
}
#dpxebxbyvj .gt_font_normal {
#roxupfdiiw .gt_font_normal {
font-weight: normal;
}
#dpxebxbyvj .gt_font_bold {
#roxupfdiiw .gt_font_bold {
font-weight: bold;
}
#dpxebxbyvj .gt_font_italic {
#roxupfdiiw .gt_font_italic {
font-style: italic;
}
#dpxebxbyvj .gt_super {
#roxupfdiiw .gt_super {
font-size: 65%;
}
#dpxebxbyvj .gt_footnote_marks {
#roxupfdiiw .gt_footnote_marks {
font-style: italic;
font-weight: normal;
font-size: 75%;
vertical-align: 0.4em;
}
#dpxebxbyvj .gt_asterisk {
#roxupfdiiw .gt_asterisk {
font-size: 100%;
vertical-align: 0;
}
#dpxebxbyvj .gt_indent_1 {
#roxupfdiiw .gt_indent_1 {
text-indent: 5px;
}
#dpxebxbyvj .gt_indent_2 {
#roxupfdiiw .gt_indent_2 {
text-indent: 10px;
}
#dpxebxbyvj .gt_indent_3 {
#roxupfdiiw .gt_indent_3 {
text-indent: 15px;
}
#dpxebxbyvj .gt_indent_4 {
#roxupfdiiw .gt_indent_4 {
text-indent: 20px;
}
#dpxebxbyvj .gt_indent_5 {
#roxupfdiiw .gt_indent_5 {
text-indent: 25px;
}
</style>
@ -1275,11 +1272,11 @@ Baye’s Rule
</thead>
<tbody class="gt_table_body">
<tr><td class="gt_row gt_left">fake</td>
<td class="gt_row gt_right">4031</td>
<td class="gt_row gt_right">0.4031</td></tr>
<td class="gt_row gt_right">3967</td>
<td class="gt_row gt_right">0.3967</td></tr>
<tr><td class="gt_row gt_left">real</td>
<td class="gt_row gt_right">5969</td>
<td class="gt_row gt_right">0.5969</td></tr>
<td class="gt_row gt_right">6033</td>
<td class="gt_row gt_right">0.6033</td></tr>
</tbody>
@ -1316,8 +1313,8 @@ Baye’s Rule
# Groups: usage [2]
usage fake real
&lt;chr&gt; &lt;int&gt; &lt;int&gt;
1 no 2955 5845
2 yes 1076 124</code></pre>
1 no 2891 5910
2 yes 1076 123</code></pre>
</div>
</div>
<div class="cell">
@ -1345,7 +1342,7 @@ Baye’s Rule
type total prop
&lt;chr&gt; &lt;int&gt; &lt;dbl&gt;
1 fake 1076 0.897
2 real 124 0.103</code></pre>
2 real 123 0.103</code></pre>
</div>
</div>
</section>
@ -1373,7 +1370,7 @@ Baye’s Rule
</tr>
</tbody>
</table>
<div class="callout-caution callout callout-style-default no-icon callout-captioned">
<div class="callout-tip callout callout-style-default no-icon callout-captioned">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon no-icon"></i>
@ -1404,7 +1401,7 @@ Discrete Probability Model
</ol>
</div>
</div>
<div class="callout-caution callout callout-style-default no-icon callout-captioned">
<div class="callout-tip callout callout-style-default no-icon callout-captioned">
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@ -1417,8 +1414,7 @@ in emanuel’s words
<p>what does this mean? well its very straightforward a pmf is a function that takes in a some value y and outputs the probability that the random variable <span class="math inline">\(Y\)</span> equals <span class="math inline">\(y\)</span>.</p>
</div>
</div>
<section id="the-binomial-model" class="level3">
<h3 class="anchored" data-anchor-id="the-binomial-model">The Binomial Model</h3>
<p>next we would like add a the dependancy of <span class="math inline">\(Y\)</span> on <span class="math inline">\(\pi\)</span>, we do so by introducing the conditional pmf.</p>
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@ -1438,7 +1434,7 @@ Conditional probability model of data <span class="math inline">\(Y\)</span>
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@ -1451,7 +1447,145 @@ in emanuel’s words
<p>this is essentially the same probability model had defined above, except now we are condition probabilities by some parameter <span class="math inline">\(\pi\)</span></p>
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</section>
<p>in the example of the chess player we must make some assumptions:</p>
<ol type="1">
<li><p>the chances of winning any match in the game stay constant. So if at match number 1 human has a .65% of winning, then that is the same for match 2-6.</p></li>
<li><p>Winning or loosing a game does not affect the chances of winning or loosing the next game, i.e matches are independent of one another.</p></li>
</ol>
<p>These two assumptions lead us to the <strong>Binomial Model</strong>.</p>
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The Binomial Model
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<p>Let the random variable <span class="math inline">\(Y\)</span> represent the number of successes in <span class="math inline">\(n\)</span> trials. Assume that each trial is independent, and the probability of sucess in a given trial is <span class="math inline">\(\pi\)</span>. Then the conditional dependence of <span class="math inline">\(Y\)</span> on <span class="math inline">\(\pi\)</span> can be modeled by the <strong>Binomial Model</strong> with parameters <span class="math inline">\(n\)</span> and <span class="math inline">\(\pi\)</span>. We can write this as,</p>
<p><span class="math display">\[Y|\pi \sim Bin(n, \pi)\]</span></p>
<p>the binomial model is specified by the pmf:</p>
<p><span class="math display">\[f(y|\pi) = {n \choose y} \pi^y(1 - \pi)^{n-y}\]</span></p>
</div>
</div>
<p>knowing this we can represent <span class="math inline">\(Y\)</span> the total number of matches out of 6 that the human can win.</p>
<p><span class="math display">\[Y|\pi \sim Bin(6, \pi)\]</span></p>
<p>and conditional pmf:</p>
<p><span class="math display">\[f(y|\pi) = {6 \choose y}\pi^y(1 - \pi)^{6 - y}\;\; \text{for } y \in \{1, 2, 3, 4, 5, 6\}\]</span></p>
<p>with the pmf we can now determine the probability of the human winning <span class="math inline">\(Y\)</span> matches out of 6 for any given value of <span class="math inline">\(\pi\)</span></p>
<div class="cell">
<div class="sourceCode cell-code" id="cb18"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a>chess_pmf <span class="ot">&lt;-</span> <span class="cf">function</span>(y, p, <span class="at">n =</span> <span class="dv">6</span>) {</span>
<span id="cb18-2"><a href="#cb18-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">choose</span>(n, y) <span class="sc">*</span> (p <span class="sc">^</span> y) <span class="sc">*</span> (<span class="dv">1</span> <span class="sc">-</span> p)<span class="sc">^</span>(n <span class="sc">-</span> y)</span>
<span id="cb18-3"><a href="#cb18-3" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb18-4"><a href="#cb18-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb18-5"><a href="#cb18-5" aria-hidden="true" tabindex="-1"></a><span class="co"># what is probability that human wins 6 games given a pi value of .8 </span></span>
<span id="cb18-6"><a href="#cb18-6" aria-hidden="true" tabindex="-1"></a><span class="fu">chess_pmf</span>(<span class="at">y =</span> <span class="dv">5</span>, <span class="at">p =</span> .<span class="dv">8</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 0.393216</code></pre>
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<p>the formula for the binomial is actually pretty intuitive, first you have the scalar <span class="math inline">\({n \choose y}\)</span> this will determine the total number of ways the player can win <span class="math inline">\(y\)</span> games out of the possible <span class="math inline">\(n\)</span>. This is first multiplied by the probablility of success in the <span class="math inline">\(n\)</span> trials since <span class="math inline">\((p ^ y)\)</span> can be re-written as <span class="math inline">\(p\times p\times \cdots \times p\)</span>, and then multiplied by the probability of <span class="math inline">\(n-y\)</span> failures <span class="math inline">\((1 - p)^{n - y}\)</span></p>
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<div class="cell">
<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a>pies <span class="ot">&lt;-</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="at">by =</span> .<span class="dv">05</span>)</span>
<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a>py <span class="ot">&lt;-</span> <span class="fu">chess_pmf</span>(<span class="at">y =</span> <span class="dv">4</span>, <span class="at">p =</span> pies)</span>
<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb20-4"><a href="#cb20-4" aria-hidden="true" tabindex="-1"></a>d <span class="ot">&lt;-</span> <span class="fu">data.frame</span>(<span class="at">pies =</span> pies, <span class="at">py =</span> py)</span>
<span id="cb20-5"><a href="#cb20-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb20-6"><a href="#cb20-6" aria-hidden="true" tabindex="-1"></a>d <span class="sc">|&gt;</span></span>
<span id="cb20-7"><a href="#cb20-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">ggplot</span>(<span class="fu">aes</span>(pies, py)) <span class="sc">+</span> <span class="fu">geom_col</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="ch2_files/figure-html/unnamed-chunk-14-1.png" class="img-fluid" width="672"></p>
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</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a>pies <span class="ot">&lt;-</span> <span class="fu">c</span>(.<span class="dv">2</span>, .<span class="dv">5</span>, .<span class="dv">8</span>)</span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a>ys <span class="ot">&lt;-</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">6</span></span>
<span id="cb21-3"><a href="#cb21-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb21-4"><a href="#cb21-4" aria-hidden="true" tabindex="-1"></a>d <span class="ot">&lt;-</span> tidyr<span class="sc">::</span><span class="fu">expand_grid</span>(pies, ys)</span>
<span id="cb21-5"><a href="#cb21-5" aria-hidden="true" tabindex="-1"></a>fys <span class="ot">&lt;-</span> purrr<span class="sc">::</span><span class="fu">map2_dbl</span>(d<span class="sc">$</span>ys, d<span class="sc">$</span>pies, <span class="sc">~</span><span class="fu">chess_pmf</span>(.x, .y), <span class="at">n=</span><span class="dv">6</span>)</span>
<span id="cb21-6"><a href="#cb21-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb21-7"><a href="#cb21-7" aria-hidden="true" tabindex="-1"></a>d<span class="sc">$</span>fys <span class="ot">&lt;-</span> fys</span>
<span id="cb21-8"><a href="#cb21-8" aria-hidden="true" tabindex="-1"></a>d<span class="sc">$</span>display_pi <span class="ot">&lt;-</span> <span class="fu">as.factor</span>(<span class="fu">paste</span>(<span class="st">"pi ="</span>, d<span class="sc">$</span>pies))</span>
<span id="cb21-9"><a href="#cb21-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb21-10"><a href="#cb21-10" aria-hidden="true" tabindex="-1"></a>d <span class="sc">|&gt;</span></span>
<span id="cb21-11"><a href="#cb21-11" aria-hidden="true" tabindex="-1"></a> <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> ys, <span class="at">y =</span> fys)) <span class="sc">+</span> </span>
<span id="cb21-12"><a href="#cb21-12" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_col</span>() <span class="sc">+</span> </span>
<span id="cb21-13"><a href="#cb21-13" aria-hidden="true" tabindex="-1"></a> <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">6</span>) <span class="sc">+</span> </span>
<span id="cb21-14"><a href="#cb21-14" aria-hidden="true" tabindex="-1"></a> <span class="fu">facet_wrap</span>(<span class="fu">vars</span>(display_pi))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p><img src="ch2_files/figure-html/unnamed-chunk-15-1.png" class="img-fluid" width="672"></p>
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<p>The plot shows the three possible values for <span class="math inline">\(\pi\)</span> along with the value of the pmf for each of the possible matches the human can win in a game. The values of <span class="math inline">\(f(y|\pi)\)</span> are pretty intuitive, we would expect the random variable <span class="math inline">\(Y\)</span> to be lower when the value of <span class="math inline">\(\pi\)</span> is lower and higher when the value of <span class="math inline">\(\pi\)</span> is higher.</p>
<p>For the sake of the excercise lets add more values of <span class="math inline">\(\pi\)</span> so that we can see this shift happen in more detail.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a>pies <span class="ot">&lt;-</span> <span class="fu">seq</span>(.<span class="dv">1</span>, .<span class="dv">9</span>, <span class="at">by =</span> .<span class="dv">1</span>)</span>
<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a>ys <span class="ot">&lt;-</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">6</span></span>
<span id="cb22-3"><a href="#cb22-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-4"><a href="#cb22-4" aria-hidden="true" tabindex="-1"></a>d <span class="ot">&lt;-</span> tidyr<span class="sc">::</span><span class="fu">expand_grid</span>(pies, ys)</span>
<span id="cb22-5"><a href="#cb22-5" aria-hidden="true" tabindex="-1"></a>fys <span class="ot">&lt;-</span> purrr<span class="sc">::</span><span class="fu">map2_dbl</span>(d<span class="sc">$</span>ys, d<span class="sc">$</span>pies, <span class="sc">~</span><span class="fu">chess_pmf</span>(.x, .y), <span class="at">n=</span><span class="dv">6</span>)</span>
<span id="cb22-6"><a href="#cb22-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-7"><a href="#cb22-7" aria-hidden="true" tabindex="-1"></a>d<span class="sc">$</span>fys <span class="ot">&lt;-</span> fys</span>
<span id="cb22-8"><a href="#cb22-8" aria-hidden="true" tabindex="-1"></a>d<span class="sc">$</span>display_pi <span class="ot">&lt;-</span> <span class="fu">as.factor</span>(<span class="fu">paste</span>(<span class="st">"pi ="</span>, d<span class="sc">$</span>pies))</span>
<span id="cb22-9"><a href="#cb22-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-10"><a href="#cb22-10" aria-hidden="true" tabindex="-1"></a>d <span class="sc">|&gt;</span></span>
<span id="cb22-11"><a href="#cb22-11" aria-hidden="true" tabindex="-1"></a> <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> ys, <span class="at">y =</span> fys)) <span class="sc">+</span> </span>
<span id="cb22-12"><a href="#cb22-12" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_col</span>() <span class="sc">+</span> </span>
<span id="cb22-13"><a href="#cb22-13" aria-hidden="true" tabindex="-1"></a> <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">6</span>) <span class="sc">+</span> </span>
<span id="cb22-14"><a href="#cb22-14" aria-hidden="true" tabindex="-1"></a> <span class="fu">facet_wrap</span>(<span class="fu">vars</span>(display_pi), <span class="at">nrow =</span> <span class="dv">3</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="ch2_files/figure-html/unnamed-chunk-16-1.png" class="img-fluid" width="672"></p>
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<p>as it turns out we learn that the human ended up winning just one game in the 1997 rematch, <span class="math inline">\(Y = 1\)</span>. The next step in our analysis is to determine how compatible this new data is with each value of <span class="math inline">\(\pi\)</span>, the likelihood that is.</p>
<p>This is very easy to do with all the work we have done so far:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a>d <span class="sc">|&gt;</span></span>
<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">filter</span>(ys <span class="sc">==</span> <span class="dv">1</span>) <span class="sc">|&gt;</span></span>
<span id="cb23-3"><a href="#cb23-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">ggplot</span>(<span class="fu">aes</span>(pies, fys)) <span class="sc">+</span> </span>
<span id="cb23-4"><a href="#cb23-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_col</span>() <span class="sc">+</span> </span>
<span id="cb23-5"><a href="#cb23-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="fu">seq</span>(.<span class="dv">1</span>, .<span class="dv">9</span>, <span class="at">by =</span> .<span class="dv">1</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="ch2_files/figure-html/unnamed-chunk-17-1.png" class="img-fluid" width="672"></p>
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<p>It’s very important to note the following</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="co"># this will sum to a value greater than 1!!</span></span>
<span id="cb24-2"><a href="#cb24-2" aria-hidden="true" tabindex="-1"></a>d <span class="sc">|&gt;</span></span>
<span id="cb24-3"><a href="#cb24-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">filter</span>(ys <span class="sc">==</span> <span class="dv">1</span>) <span class="sc">|&gt;</span></span>
<span id="cb24-4"><a href="#cb24-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">pull</span>(fys) <span class="sc">|&gt;</span></span>
<span id="cb24-5"><a href="#cb24-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">sum</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 1.37907</code></pre>
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Important
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<p>this has been mentioned before but its an important message to drive home. Note that the reason why thes values sum to a value greater than 1 is that they are <strong>not</strong> probabilities, they are likelihoods. We are determining how likely each value of <span class="math inline">\(\pi\)</span> is given that we have observed <span class="math inline">\(Y = 1\)</span>.</p>
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@ -339,7 +339,7 @@ in the book that we will learn how to build these later on):
|--------|----|----|----|-------|
|$f(\pi)$|.10 |.25 |.65 | 1 |
:::{.callout-caution}
:::{.callout-tip}
## Note
its important to note here that the sum of the values of $\pi$ **do
@ -364,14 +364,15 @@ and has the following properties
:::
:::{.callout-caution}
:::{.callout-tip}
## in emanuel's words
what does this mean? well its very straightforward a pmf is a function that takes
in a some value y and outputs the probability that the random variable
$Y$ equals $y$.
:::
### The Binomial Model
next we would like add a the dependancy of $Y$ on $\pi$, we do so by introducing
the conditional pmf.
:::{.callout-note}
## Conditional probability model of data $Y$
@ -388,8 +389,158 @@ and has the following properties,
2. $\sum_{\forall y}f(y|\pi) = 1$
:::
:::{.callout-caution}
:::{.callout-tip}
## in emanuel's words
this is essentially the same probability model had defined above, except
now we are condition probabilities by some parameter $\pi$
:::
:::
in the example of the chess player we must make some assumptions:
1. the chances of winning any match in the game stay constant. So if
at match number 1 human has a .65% of winning, then that is the same
for match 2-6.
2. Winning or loosing a game does not affect the chances of winning
or loosing the next game, i.e matches are independent of one another.
These two assumptions lead us to the **Binomial Model**.
:::{.callout-note}
## The Binomial Model
Let the random variable $Y$ represent the number of successes in $n$ trials.
Assume that each trial is independent, and the probability of sucess in a
given trial is $\pi$. Then the conditional dependence of $Y$ on $\pi$ can
be modeled by the **Binomial Model** with parameters $n$ and $\pi$. We can
write this as,
$$Y|\pi \sim Bin(n, \pi)$$
the binomial model is specified by the pmf:
$$f(y|\pi) = {n \choose y} \pi^y(1 - \pi)^{n-y}$$
:::
knowing this we can represent $Y$ the total number of matches out of 6
that the human can win.
$$Y|\pi \sim Bin(6, \pi)$$
and conditional pmf:
$$f(y|\pi) = {6 \choose y}\pi^y(1 - \pi)^{6 - y}\;\; \text{for } y \in \{1, 2, 3, 4, 5, 6\}$$
with the pmf we can now determine the probability of the human winning $Y$ matches
out of 6 for any given value of $\pi$
```{r}
chess_pmf <- function(y, p, n = 6) {
choose(n, y) * (p ^ y) * (1 - p)^(n - y)
}
# what is probability that human wins 6 games given a pi value of .8
chess_pmf(y = 5, p = .8)
```
:::{.callout-tip}
##
the formula for the binomial is actually pretty intuitive, first you have
the scalar ${n \choose y}$ this will determine the total number of ways
the player can win $y$ games out of the possible $n$. This is first multiplied
by the probablility of success in the $n$ trials since $(p ^ y)$ can be
re-written as $p\times p\times \cdots \times p$, and then multiplied by
the probability of $n-y$ failures $(1 - p)^{n - y}$
:::
```{r}
pies <- seq(0, 1, by = .05)
py <- chess_pmf(y = 4, p = pies)
d <- data.frame(pies = pies, py = py)
d |>
ggplot(aes(pies, py)) + geom_col()
```
```{r}
pies <- c(.2, .5, .8)
ys <- 0:6
d <- tidyr::expand_grid(pies, ys)
fys <- purrr::map2_dbl(d$ys, d$pies, ~chess_pmf(.x, .y), n=6)
d$fys <- fys
d$display_pi <- as.factor(paste("pi =", d$pies))
d |>
ggplot(aes(x = ys, y = fys)) +
geom_col() +
scale_x_continuous(breaks = 0:6) +
facet_wrap(vars(display_pi))
```
The plot shows the three possible values for $\pi$ along
with the value of the pmf for each of the possible
matches the human can win in a game. The values of $f(y|\pi)$
are pretty intuitive, we would expect the random variable $Y$
to be lower when the value of $\pi$ is lower and higher when
the value of $\pi$ is higher.
For the sake of the excercise lets add more values of $\pi$
so that we can see this shift happen in more detail.
```{r}
pies <- seq(.1, .9, by = .1)
ys <- 0:6
d <- tidyr::expand_grid(pies, ys)
fys <- purrr::map2_dbl(d$ys, d$pies, ~chess_pmf(.x, .y), n=6)
d$fys <- fys
d$display_pi <- as.factor(paste("pi =", d$pies))
d |>
ggplot(aes(x = ys, y = fys)) +
geom_col() +
scale_x_continuous(breaks = 0:6) +
facet_wrap(vars(display_pi), nrow = 3)
```
as it turns out we learn that the human ended up winning just
one game in the 1997 rematch, $Y = 1$. The next step in our
analysis is to determine how compatible this new data is with
each value of $\pi$, the likelihood that is.
This is very easy to do with all the work we have done so far:
```{r}
d |>
filter(ys == 1) |>
ggplot(aes(pies, fys)) +
geom_col() +
scale_x_continuous(breaks = seq(.1, .9, by = .1))
```
It's very important to note the following
```{r}
# this will sum to a value greater than 1!!
d |>
filter(ys == 1) |>
pull(fys) |>
sum()
```
:::{.callout-important icon="true"}
this has been mentioned before but its an important message
to drive home. Note that the reason why thes values sum to a
value greater than 1 is that they are **not** probabilities, they
are likelihoods. We are determining how likely each value of
$\pi$ is given that we have observed $Y = 1$.
:::

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