-
Notifications
You must be signed in to change notification settings - Fork 0
/
Statistical Thinking 2023_v2.html
725 lines (697 loc) · 59.3 KB
/
Statistical Thinking 2023_v2.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"><head>
<meta charset="utf-8">
<meta name="generator" content="quarto-1.2.269">
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
<meta name="author" content="Luis Correia">
<title>Statistical Rethinking 2023</title>
<style>
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
width: 0.8em;
margin: 0 0.8em 0.2em -1.6em;
vertical-align: middle;
}
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
color: #aaaaaa;
}
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
div.sourceCode
{ }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span.al { color: #ff0000; font-weight: bold; } /* Alert */
code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */
code span.at { color: #7d9029; } /* Attribute */
code span.bn { color: #40a070; } /* BaseN */
code span.bu { color: #008000; } /* BuiltIn */
code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */
code span.ch { color: #4070a0; } /* Char */
code span.cn { color: #880000; } /* Constant */
code span.co { color: #60a0b0; font-style: italic; } /* Comment */
code span.cv { color: #60a0b0; font-weight: bold; font-style: italic; } /* CommentVar */
code span.do { color: #ba2121; font-style: italic; } /* Documentation */
code span.dt { color: #902000; } /* DataType */
code span.dv { color: #40a070; } /* DecVal */
code span.er { color: #ff0000; font-weight: bold; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #40a070; } /* Float */
code span.fu { color: #06287e; } /* Function */
code span.im { color: #008000; font-weight: bold; } /* Import */
code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */
code span.kw { color: #007020; font-weight: bold; } /* Keyword */
code span.op { color: #666666; } /* Operator */
code span.ot { color: #007020; } /* Other */
code span.pp { color: #bc7a00; } /* Preprocessor */
code span.sc { color: #4070a0; } /* SpecialChar */
code span.ss { color: #bb6688; } /* SpecialString */
code span.st { color: #4070a0; } /* String */
code span.va { color: #19177c; } /* Variable */
code span.vs { color: #4070a0; } /* VerbatimString */
code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */
</style>
<script src="Statistical Thinking 2023_v2_files/libs/clipboard/clipboard.min.js"></script>
<script src="Statistical Thinking 2023_v2_files/libs/quarto-html/quarto.js"></script>
<script src="Statistical Thinking 2023_v2_files/libs/quarto-html/popper.min.js"></script>
<script src="Statistical Thinking 2023_v2_files/libs/quarto-html/tippy.umd.min.js"></script>
<script src="Statistical Thinking 2023_v2_files/libs/quarto-html/anchor.min.js"></script>
<link href="Statistical Thinking 2023_v2_files/libs/quarto-html/tippy.css" rel="stylesheet">
<link href="Statistical Thinking 2023_v2_files/libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
<script src="Statistical Thinking 2023_v2_files/libs/bootstrap/bootstrap.min.js"></script>
<link href="Statistical Thinking 2023_v2_files/libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
<link href="Statistical Thinking 2023_v2_files/libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
<script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js" type="text/javascript"></script>
</head>
<body class="fullcontent">
<div id="quarto-content" class="page-columns page-rows-contents page-layout-article">
<main class="content" id="quarto-document-content">
<header id="title-block-header" class="quarto-title-block default">
<div class="quarto-title">
<h1 class="title">Statistical Rethinking 2023</h1>
</div>
<div class="quarto-title-meta">
<div>
<div class="quarto-title-meta-heading">Author</div>
<div class="quarto-title-meta-contents">
<p>Luis Correia </p>
</div>
</div>
</div>
</header>
<section id="lecture-1" class="level2">
<h2 class="anchored" data-anchor-id="lecture-1">Lecture 1</h2>
<p>The content of this lecture should be reviewed and posted.</p>
<div style="page-break-after: always;"></div>
</section>
<section id="lecture-2" class="level2">
<h2 class="anchored" data-anchor-id="lecture-2">Lecture 2</h2>
<p><To be reviewed with lecture contents></p>
<p>A new model is born when we write:</p>
<p><span class="math display">\[
W\sim Binomial(N,p)
\]</span></p>
<p>with</p>
<p><span class="math display">\[
p\sim Unif(0,1)
\]</span></p>
<p>Simulation of Bayesian Experiment</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="co"># definegrid</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>p_grid <span class="ot"><-</span><span class="fu">seq</span>(<span class="at">from=</span><span class="dv">0</span>,<span class="at">to=</span><span class="dv">1</span>,<span class="at">length.out=</span><span class="dv">20</span>)</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="co"># defineprior</span></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a><span class="co">#prior <-rep(1,20)</span></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>prior <span class="ot"><-</span><span class="fu">ifelse</span>(p_grid<span class="sc"><</span><span class="fl">0.5</span>,<span class="dv">0</span>,<span class="dv">1</span>)</span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a><span class="co"># computelikelihoodateachvalueingrid</span></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a>likelihood <span class="ot"><-</span><span class="fu">dbinom</span>(<span class="dv">6</span>,<span class="at">size=</span><span class="dv">9</span>,<span class="at">prob=</span>p_grid)</span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a><span class="co"># computeproductoflikelihoodandprior</span></span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a>unstd.posterior <span class="ot"><-</span>likelihood<span class="sc">*</span>prior</span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a><span class="co"># standardizetheposterior,soitsumsto1</span></span>
<span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a>posterior <span class="ot"><-</span>unstd.posterior<span class="sc">/</span><span class="fu">sum</span>(unstd.posterior)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>( p_grid,posterior,<span class="at">type=</span><span class="st">"b"</span>,</span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="at">xlab=</span><span class="st">"probability ofwater"</span>,<span class="at">ylab=</span><span class="st">"posteriorprobability"</span>)</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="fu">mtext</span>( <span class="st">"20points"</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="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-2-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<section id="homework-1" class="level3">
<h3 class="anchored" data-anchor-id="homework-1">Homework 1</h3>
<p>Suppose the globe tossing data had turned into a 4-water and 11-land. Construct the posterior distribution.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="co"># 2.7 analyticalcalculation</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>W <span class="ot"><-</span><span class="dv">6</span></span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a>L <span class="ot"><-</span><span class="dv">3</span></span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a><span class="fu">curve</span>( <span class="fu">dbeta</span>(x,W<span class="sc">+</span><span class="dv">1</span>,L<span class="sc">+</span><span class="dv">1</span>),<span class="at">from=</span><span class="dv">0</span>,<span class="at">to=</span><span class="dv">1</span>)</span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a><span class="co"># quadraticapproximation</span></span>
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a><span class="fu">curve</span>( <span class="fu">dnorm</span>(x,<span class="fl">0.67</span>,<span class="fl">0.16</span>),<span class="at">lty=</span><span class="dv">2</span>,<span class="at">add=</span><span class="cn">TRUE</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="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-3-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>Simulation of Bayesian Experiment</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="co"># definegrid</span></span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>p_grid <span class="ot"><-</span><span class="fu">seq</span>(<span class="at">from=</span><span class="dv">0</span>,<span class="at">to=</span><span class="dv">1</span>,<span class="at">length.out=</span><span class="dv">20</span>)</span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a><span class="co"># defineprior</span></span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a><span class="co">#prior <-rep(1,20)</span></span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a>prior <span class="ot"><-</span><span class="fu">ifelse</span>(p_grid<span class="sc"><</span><span class="fl">0.5</span>,<span class="dv">0</span>,<span class="dv">1</span>)</span>
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a><span class="co"># computelikelihoodateachvalueingrid</span></span>
<span id="cb4-9"><a href="#cb4-9" aria-hidden="true" tabindex="-1"></a>likelihood <span class="ot"><-</span><span class="fu">dbinom</span>(<span class="dv">6</span>,<span class="at">size=</span><span class="dv">9</span>,<span class="at">prob=</span>p_grid)</span>
<span id="cb4-10"><a href="#cb4-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-11"><a href="#cb4-11" aria-hidden="true" tabindex="-1"></a><span class="co"># computeproductoflikelihoodandprior</span></span>
<span id="cb4-12"><a href="#cb4-12" aria-hidden="true" tabindex="-1"></a>unstd.posterior <span class="ot"><-</span>likelihood<span class="sc">*</span>prior</span>
<span id="cb4-13"><a href="#cb4-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-14"><a href="#cb4-14" aria-hidden="true" tabindex="-1"></a><span class="co"># standardizetheposterior,soitsumsto1</span></span>
<span id="cb4-15"><a href="#cb4-15" aria-hidden="true" tabindex="-1"></a>posterior <span class="ot"><-</span>unstd.posterior<span class="sc">/</span><span class="fu">sum</span>(unstd.posterior)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>( p_grid,posterior,<span class="at">type=</span><span class="st">"b"</span>,</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a><span class="at">xlab=</span><span class="st">"probability ofwater"</span>,<span class="at">ylab=</span><span class="st">"posteriorprobability"</span>)</span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a><span class="fu">mtext</span>( <span class="st">"20points"</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="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-5-1.png" class="img-fluid" width="672"></p>
</div>
</div>
</section>
<section id="homework-1-1" class="level3">
<h3 class="anchored" data-anchor-id="homework-1-1">Homework 1</h3>
<p>Suppose the globe tossing data had turned into a 4-water and 11-land. Construct the posterior distribution.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="co"># 2.7 analyticalcalculation</span></span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>W <span class="ot"><-</span><span class="dv">6</span></span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a>L <span class="ot"><-</span><span class="dv">3</span></span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a><span class="fu">curve</span>( <span class="fu">dbeta</span>(x,W<span class="sc">+</span><span class="dv">1</span>,L<span class="sc">+</span><span class="dv">1</span>),<span class="at">from=</span><span class="dv">0</span>,<span class="at">to=</span><span class="dv">1</span>)</span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a><span class="co"># quadraticapproximation</span></span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a><span class="fu">curve</span>( <span class="fu">dnorm</span>(x,<span class="fl">0.67</span>,<span class="fl">0.16</span>),<span class="at">lty=</span><span class="dv">2</span>,<span class="at">add=</span><span class="cn">TRUE</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="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-6-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a>p <span class="ot"><-</span> <span class="fu">c</span>(<span class="dv">0</span>, .<span class="dv">25</span>, .<span class="dv">5</span>, .<span class="dv">75</span>, <span class="dv">1</span>)</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a>model <span class="ot"><-</span> <span class="fu">sapply</span>(p, <span class="cf">function</span> (p, W, L) <span class="fu">return</span> (<span class="dv">4</span><span class="sc">*</span>p)<span class="sc">^</span>W<span class="sc">*</span>(<span class="dv">4-4</span><span class="sc">*</span>p)<span class="sc">^</span>L)</span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(model)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0 1 2 3 4</code></pre>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>n_samples <span class="ot"><-</span><span class="dv">1000</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a>p <span class="ot"><-</span><span class="fu">rep</span>(<span class="cn">NA</span>,n_samples)</span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a>p[<span class="dv">1</span>] <span class="ot"><-</span><span class="fl">0.5</span></span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a>W <span class="ot"><-</span><span class="dv">6</span></span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a>L <span class="ot"><-</span><span class="dv">3</span></span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">2</span><span class="sc">:</span>n_samples){</span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a> p_new <span class="ot"><-</span><span class="fu">rnorm</span>(<span class="dv">1</span>,p[i<span class="dv">-1</span>],<span class="fl">0.1</span>)</span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (p_new <span class="sc"><</span> <span class="dv">0</span>) p_new <span class="ot"><-</span> <span class="fu">abs</span>(p_new)</span>
<span id="cb9-9"><a href="#cb9-9" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (p_new <span class="sc">></span> <span class="dv">1</span>) p_new <span class="ot"><-</span> <span class="dv">2</span><span class="sc">-</span>p_new</span>
<span id="cb9-10"><a href="#cb9-10" aria-hidden="true" tabindex="-1"></a> q0 <span class="ot"><-</span><span class="fu">dbinom</span>(W,W<span class="sc">+</span>L,p[i<span class="dv">-1</span>])</span>
<span id="cb9-11"><a href="#cb9-11" aria-hidden="true" tabindex="-1"></a> q1 <span class="ot"><-</span><span class="fu">dbinom</span>(W,W<span class="sc">+</span>L,p_new)</span>
<span id="cb9-12"><a href="#cb9-12" aria-hidden="true" tabindex="-1"></a> p[i] <span class="ot"><-</span><span class="fu">ifelse</span>(<span class="fu">runif</span>(<span class="dv">1</span>)<span class="sc"><</span>q1<span class="sc">/</span>q0,p_new,p[i<span class="dv">-1</span>])</span>
<span id="cb9-13"><a href="#cb9-13" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">density</span>(p),<span class="at">xlim=</span><span class="fu">c</span>(<span class="dv">0</span>,<span class="dv">1</span>))</span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a><span class="fu">curve</span>( <span class="fu">dbeta</span>(x, W<span class="sc">+</span><span class="dv">1</span>, L<span class="sc">+</span><span class="dv">1</span> ),<span class="at">lty=</span><span class="dv">2</span>,<span class="at">add=</span><span class="cn">TRUE</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="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-9-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a>sample <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">'W'</span>,<span class="st">'L'</span>,<span class="st">'W'</span>,<span class="st">'W'</span>,<span class="st">'W'</span>,<span class="st">'L'</span>,<span class="st">'W'</span>,<span class="st">'L'</span>,<span class="st">'W'</span>)</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>W <span class="ot"><-</span> <span class="fu">sum</span>(sample<span class="sc">==</span><span class="st">'W'</span>) <span class="co"># No. of W observed</span></span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a>L <span class="ot"><-</span> <span class="fu">sum</span>(sample<span class="sc">==</span><span class="st">'L'</span>) <span class="co"># No. of L observed</span></span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a>p <span class="ot"><-</span> <span class="fu">c</span>(<span class="dv">0</span>,.<span class="dv">25</span>,.<span class="dv">5</span>,.<span class="dv">75</span>,<span class="dv">1</span>) <span class="co"># Proportions W</span></span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a>ways <span class="ot"><-</span> <span class="fu">sapply</span>(p, <span class="cf">function</span> (q) (q<span class="sc">*</span><span class="dv">4</span>)<span class="sc">^</span>W<span class="sc">*</span>((<span class="dv">1</span><span class="sc">-</span>q)<span class="sc">*</span><span class="dv">4</span>)<span class="sc">^</span>L)</span>
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a>prob <span class="ot"><-</span> ways<span class="sc">/</span><span class="fu">sum</span>(ways)</span>
<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a><span class="fu">cbind</span>(p, ways, prob)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> p ways prob
[1,] 0.00 0 0.00000000
[2,] 0.25 27 0.02129338
[3,] 0.50 512 0.40378549
[4,] 0.75 729 0.57492114
[5,] 1.00 0 0.00000000</code></pre>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a>sim_globe <span class="ot"><-</span> <span class="cf">function</span> (<span class="at">p=</span>.<span class="dv">7</span>, <span class="at">N =</span> <span class="dv">9</span>) {</span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">sample</span>(<span class="fu">c</span>(<span class="st">'W'</span>,<span class="st">'L'</span>), <span class="at">size =</span> N, <span class="at">prob =</span> <span class="fu">c</span>(p, <span class="dv">1</span><span class="sc">-</span>p), <span class="at">replace=</span><span class="cn">TRUE</span>)</span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a><span class="fu">sim_globe</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] "W" "W" "W" "W" "W" "W" "W" "W" "L"</code></pre>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="fu">replicate</span>(<span class="fu">sim_globe</span>(<span class="at">p =</span> .<span class="dv">5</span>, <span class="at">N =</span> <span class="dv">9</span>), <span class="at">n =</span> <span class="dv">10</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] "W" "L" "W" "W" "L" "L" "L" "L" "L" "L"
[2,] "W" "W" "W" "L" "W" "W" "W" "L" "W" "W"
[3,] "L" "W" "W" "L" "W" "L" "W" "L" "W" "W"
[4,] "L" "W" "W" "W" "W" "L" "W" "L" "W" "W"
[5,] "W" "L" "L" "L" "W" "W" "L" "L" "L" "W"
[6,] "L" "L" "W" "L" "L" "L" "L" "L" "W" "L"
[7,] "W" "L" "L" "W" "L" "L" "W" "W" "W" "W"
[8,] "L" "W" "W" "L" "W" "W" "L" "L" "L" "L"
[9,] "W" "W" "L" "W" "W" "L" "L" "W" "L" "L" </code></pre>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(<span class="fu">sim_globe</span>(<span class="at">p =</span> .<span class="dv">5</span>, <span class="at">N =</span> <span class="fl">1e4</span> )<span class="sc">==</span><span class="st">'W'</span>)<span class="sc">/</span><span class="fl">1e4</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.5044</code></pre>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ggplot2)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stderr">
<pre><code>Warning: package 'ggplot2' was built under R version 4.2.2</code></pre>
</div>
<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>compute_posterior <span class="ot"><-</span> <span class="cf">function</span> (the_sample, <span class="at">poss =</span> <span class="fu">c</span>(<span class="dv">0</span>,.<span class="dv">25</span>,.<span class="dv">5</span>,.<span class="dv">75</span>,<span class="dv">1</span>)) {</span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a> W <span class="ot"><-</span> <span class="fu">sum</span>(the_sample<span class="sc">==</span><span class="st">'W'</span>) <span class="co"># No. of 'W'</span></span>
<span id="cb21-3"><a href="#cb21-3" aria-hidden="true" tabindex="-1"></a> L <span class="ot"><-</span> <span class="fu">sum</span>(the_sample<span class="sc">==</span><span class="st">'L'</span>) <span class="co"># No. of 'L'</span></span>
<span id="cb21-4"><a href="#cb21-4" aria-hidden="true" tabindex="-1"></a> ways <span class="ot"><-</span> <span class="fu">sapply</span>(p, <span class="cf">function</span> (q) (q<span class="sc">*</span><span class="dv">4</span>)<span class="sc">^</span>W<span class="sc">*</span>((<span class="dv">1</span><span class="sc">-</span>q)<span class="sc">*</span><span class="dv">4</span>)<span class="sc">^</span>L)</span>
<span id="cb21-5"><a href="#cb21-5" aria-hidden="true" tabindex="-1"></a> post <span class="ot"><-</span> ways<span class="sc">/</span><span class="fu">sum</span>(ways)</span>
<span id="cb21-6"><a href="#cb21-6" aria-hidden="true" tabindex="-1"></a> <span class="co"># bars <- sapply(post, function (q) make_bar(q))</span></span>
<span id="cb21-7"><a href="#cb21-7" aria-hidden="true" tabindex="-1"></a> <span class="co"># data.frame(poss, ways, post=round(post,3), bars)</span></span>
<span id="cb21-8"><a href="#cb21-8" aria-hidden="true" tabindex="-1"></a> df <span class="ot"><-</span> <span class="fu">data.frame</span>(poss, ways, <span class="at">post=</span><span class="fu">round</span>(post,<span class="dv">3</span>))</span>
<span id="cb21-9"><a href="#cb21-9" aria-hidden="true" tabindex="-1"></a> <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x=</span>poss, <span class="at">y=</span>post), <span class="at">data=</span>df)<span class="sc">+</span></span>
<span id="cb21-10"><a href="#cb21-10" aria-hidden="true" tabindex="-1"></a> <span class="fu">geom_bar</span>(<span class="at">stat=</span><span class="st">"identity"</span>)</span>
<span id="cb21-11"><a href="#cb21-11" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<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>the_sample <span class="ot"><-</span> <span class="fu">sim_globe</span>()</span>
<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-3"><a href="#cb22-3" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(the_sample)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] "W" "L" "W" "W" "L" "W" "W" "W" "L"</code></pre>
</div>
<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="fu">compute_posterior</span>(the_sample)</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="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-15-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>Calculating the posteriors using the conjugate prior beta</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb25"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(rethinking)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stderr">
<pre><code>Loading required package: rstan</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>Loading required package: StanHeaders</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>
rstan version 2.26.13 (Stan version 2.26.1)</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>For execution on a local, multicore CPU with excess RAM we recommend calling
options(mc.cores = parallel::detectCores()).
To avoid recompilation of unchanged Stan programs, we recommend calling
rstan_options(auto_write = TRUE)
For within-chain threading using `reduce_sum()` or `map_rect()` Stan functions,
change `threads_per_chain` option:
rstan_options(threads_per_chain = 1)</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>Do not specify '-march=native' in 'LOCAL_CPPFLAGS' or a Makevars file</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>Loading required package: cmdstanr</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>This is cmdstanr version 0.5.3</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>- CmdStanR documentation and vignettes: mc-stan.org/cmdstanr</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>- CmdStan path: C:/Users/lacor/OneDrive/Documents/.cmdstan/cmdstan-2.31.0</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>- CmdStan version: 2.31.0</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>Loading required package: parallel</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>rethinking (Version 2.21)</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>
Attaching package: 'rethinking'</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>The following object is masked from 'package:rstan':
stan</code></pre>
</div>
<div class="cell-output cell-output-stderr">
<pre><code>The following object is masked from 'package:stats':
rstudent</code></pre>
</div>
<div class="sourceCode cell-code" id="cb41"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb41-1"><a href="#cb41-1" aria-hidden="true" tabindex="-1"></a>post_samples <span class="ot"><-</span> <span class="fu">rbeta</span>(<span class="fl">1e3</span>, <span class="dv">6</span><span class="sc">+</span><span class="dv">1</span>, <span class="dv">3</span><span class="sc">+</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>
<div class="cell">
<div class="sourceCode cell-code" id="cb42"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a><span class="fu">dens</span>(post_samples, <span class="at">lwd=</span><span class="dv">4</span>, <span class="at">col=</span><span class="dv">2</span>, <span class="at">xlab=</span><span class="st">"proportion water"</span>, <span class="at">adj=</span>.<span class="dv">1</span>)</span>
<span id="cb42-2"><a href="#cb42-2" aria-hidden="true" tabindex="-1"></a><span class="fu">curve</span>(<span class="fu">dbeta</span>(x, <span class="dv">6</span><span class="sc">+</span><span class="dv">1</span>, <span class="dv">3</span><span class="sc">+</span><span class="dv">1</span>), <span class="at">add=</span><span class="cn">TRUE</span>, <span class="at">lty=</span><span class="dv">2</span>, <span class="at">lwd=</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="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-17-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb43"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb43-1"><a href="#cb43-1" aria-hidden="true" tabindex="-1"></a>w <span class="ot"><-</span><span class="dv">6</span>;n<span class="ot"><-</span><span class="dv">9</span>;</span>
<span id="cb43-2"><a href="#cb43-2" aria-hidden="true" tabindex="-1"></a>p_grid <span class="ot"><-</span><span class="fu">seq</span>(<span class="at">from=</span><span class="dv">0</span>,<span class="at">to=</span><span class="dv">1</span>,<span class="at">length.out=</span><span class="dv">100</span>)</span>
<span id="cb43-3"><a href="#cb43-3" aria-hidden="true" tabindex="-1"></a>posterior <span class="ot"><-</span><span class="fu">dbinom</span>(w,n,p_grid)<span class="sc">*</span><span class="fu">dunif</span>(p_grid,<span class="dv">0</span>,<span class="dv">1</span>)</span>
<span id="cb43-4"><a href="#cb43-4" aria-hidden="true" tabindex="-1"></a>posterior <span class="ot"><-</span>posterior<span class="sc">/</span><span class="fu">sum</span>(posterior)</span>
<span id="cb43-5"><a href="#cb43-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(posterior,<span class="at">col=</span><span class="dv">2</span>, <span class="at">type=</span><span class="st">"l"</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="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-18-1.png" class="img-fluid" width="672"></p>
</div>
</div>
</section>
</section>
<section id="lecture-3" class="level2">
<h2 class="anchored" data-anchor-id="lecture-3">Lecture 3</h2>
<section id="workflow" class="level3">
<h3 class="anchored" data-anchor-id="workflow">Workflow</h3>
<blockquote class="blockquote">
<ol type="1">
<li>State a clear Question</li>
<li>State your causal Assumptions</li>
<li>Use the causal sketch to define the Generative Model</li>
<li>Use the Generative Model to build an Estimator</li>
<li>Profit</li>
</ol>
</blockquote>
<p>Using the Howell database, we will construct a model to analyse the causal relationship between Height and Weight.</p>
<p>A new model is born when we write:</p>
<p><span class="math display">\[
W = \beta H + U
\]</span></p>
<div class="cell">
<div class="sourceCode cell-code" id="cb44"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(rethinking)</span>
<span id="cb44-2"><a href="#cb44-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb44-3"><a href="#cb44-3" aria-hidden="true" tabindex="-1"></a>sim_weight <span class="ot"><-</span> <span class="cf">function</span> (H, b, sd) {</span>
<span id="cb44-4"><a href="#cb44-4" aria-hidden="true" tabindex="-1"></a> U <span class="ot"><-</span> <span class="fu">rnorm</span>(<span class="fu">length</span>(H),<span class="dv">0</span>,sd)</span>
<span id="cb44-5"><a href="#cb44-5" aria-hidden="true" tabindex="-1"></a> W <span class="ot"><-</span> b<span class="sc">*</span>H <span class="sc">+</span> U</span>
<span id="cb44-6"><a href="#cb44-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span>(W)</span>
<span id="cb44-7"><a href="#cb44-7" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb45"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb45-1"><a href="#cb45-1" aria-hidden="true" tabindex="-1"></a>H <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">200</span>, <span class="at">min=</span><span class="dv">130</span>, <span class="at">max=</span><span class="dv">170</span>)</span>
<span id="cb45-2"><a href="#cb45-2" aria-hidden="true" tabindex="-1"></a>W <span class="ot"><-</span> <span class="fu">sim_weight</span>(H, <span class="at">b=</span>.<span class="dv">5</span>, <span class="at">sd=</span><span class="dv">5</span>)</span>
<span id="cb45-3"><a href="#cb45-3" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(W <span class="sc">~</span> H, <span class="at">col=</span><span class="dv">2</span>, <span class="at">lwd=</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="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-20-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>Our model can be written as follows</p>
<p><span class="math display">\[
W_i=\beta H_i+U_i
\]</span></p>
<p><span class="math display">\[
H_i\sim Uniform(130, 170)
\]</span></p>
<p><span class="math display">\[
U_i\sim Normal(0,\sigma^2)
\]</span></p>
<p>and our estimator can then be written as</p>
<p><span class="math display">\[
E(W_i|H_i)=\alpha+\beta H_i
\]</span></p>
<p>The posterior distribution is now written as<br>
<span class="math display">\[
Pr(\alpha,\beta,\sigma^2|H_i,W_i)\propto Pr(W_i|H_i,\alpha,\beta,\sigma^2)Pr(\alpha,\beta,\sigma^2)
\]</span></p>
<p>Now our Bayesian Model is defined as follows:</p>
<p><span class="math display">\[
W_i\sim Normal(\mu_i,\sigma^2)
\]</span></p>
<p><span class="math display">\[
\mu_i = \alpha+\beta H_i
\]</span></p>
<div class="cell">
<div class="sourceCode cell-code" id="cb46"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb46-1"><a href="#cb46-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(rethinking)</span>
<span id="cb46-2"><a href="#cb46-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb46-3"><a href="#cb46-3" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(Howell1)</span>
<span id="cb46-4"><a href="#cb46-4" aria-hidden="true" tabindex="-1"></a>d <span class="ot"><-</span>Howell1[Howell1<span class="sc">$</span>age<span class="sc">>=</span><span class="dv">18</span>,]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb47"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb47-1"><a href="#cb47-1" aria-hidden="true" tabindex="-1"></a><span class="co">#Similate a sample of 10 people</span></span>
<span id="cb47-2"><a href="#cb47-2" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">93</span>)</span>
<span id="cb47-3"><a href="#cb47-3" aria-hidden="true" tabindex="-1"></a>H <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">10</span>,<span class="dv">130</span>,<span class="dv">170</span>)</span>
<span id="cb47-4"><a href="#cb47-4" aria-hidden="true" tabindex="-1"></a>W <span class="ot"><-</span> <span class="fu">sim_weight</span>(H,<span class="at">b=</span><span class="fl">0.5</span>, <span class="at">sd=</span><span class="dv">5</span>)</span>
<span id="cb47-5"><a href="#cb47-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb47-6"><a href="#cb47-6" aria-hidden="true" tabindex="-1"></a><span class="co">#Run Model</span></span>
<span id="cb47-7"><a href="#cb47-7" aria-hidden="true" tabindex="-1"></a>m3<span class="fl">.1</span> <span class="ot"><-</span> <span class="fu">quap</span>(</span>
<span id="cb47-8"><a href="#cb47-8" aria-hidden="true" tabindex="-1"></a> <span class="fu">alist</span>(</span>
<span id="cb47-9"><a href="#cb47-9" aria-hidden="true" tabindex="-1"></a> W <span class="sc">~</span> <span class="fu">dnorm</span>(mu,sigma),</span>
<span id="cb47-10"><a href="#cb47-10" aria-hidden="true" tabindex="-1"></a> mu <span class="ot"><-</span> a <span class="sc">+</span> b<span class="sc">*</span>H,</span>
<span id="cb47-11"><a href="#cb47-11" aria-hidden="true" tabindex="-1"></a> a <span class="sc">~</span> <span class="fu">dnorm</span>(<span class="dv">0</span>,<span class="dv">10</span>),</span>
<span id="cb47-12"><a href="#cb47-12" aria-hidden="true" tabindex="-1"></a> b <span class="sc">~</span> <span class="fu">dunif</span>(<span class="dv">0</span>,<span class="dv">1</span>),</span>
<span id="cb47-13"><a href="#cb47-13" aria-hidden="true" tabindex="-1"></a> sigma<span class="sc">~</span><span class="fu">dunif</span>(<span class="dv">0</span>,<span class="dv">10</span>)</span>
<span id="cb47-14"><a href="#cb47-14" aria-hidden="true" tabindex="-1"></a>),<span class="at">data=</span><span class="fu">list</span>(<span class="at">W=</span>W, <span class="at">H=</span>H))</span>
<span id="cb47-15"><a href="#cb47-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb47-16"><a href="#cb47-16" aria-hidden="true" tabindex="-1"></a><span class="co">#Summary</span></span>
<span id="cb47-17"><a href="#cb47-17" aria-hidden="true" tabindex="-1"></a><span class="fu">precis</span>(m3<span class="fl">.1</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> mean sd 5.5% 94.5%
a 5.1936596 9.42967209 -9.8767776 20.2640969
b 0.4858079 0.06515645 0.3816753 0.5899405
sigma 5.6393361 1.29435789 3.5707022 7.7079700</code></pre>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb49"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb49-1"><a href="#cb49-1" aria-hidden="true" tabindex="-1"></a>n <span class="ot"><-</span> <span class="fl">1e3</span></span>
<span id="cb49-2"><a href="#cb49-2" aria-hidden="true" tabindex="-1"></a>a <span class="ot"><-</span> <span class="fu">rnorm</span>(n,<span class="dv">0</span>,<span class="dv">10</span>)</span>
<span id="cb49-3"><a href="#cb49-3" aria-hidden="true" tabindex="-1"></a>b <span class="ot"><-</span> <span class="fu">runif</span>(n,<span class="dv">0</span>,<span class="dv">1</span>)</span>
<span id="cb49-4"><a href="#cb49-4" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="cn">NULL</span>,<span class="at">xlim=</span><span class="fu">c</span>(<span class="dv">130</span>,<span class="dv">170</span>),<span class="at">ylim=</span><span class="fu">c</span>(<span class="dv">50</span>,<span class="dv">90</span>),</span>
<span id="cb49-5"><a href="#cb49-5" aria-hidden="true" tabindex="-1"></a> <span class="at">xlab=</span><span class="st">"height(cm)"</span>, <span class="at">ylab=</span><span class="st">"weight(kg)"</span>,)</span>
<span id="cb49-6"><a href="#cb49-6" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (j <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">50</span> ) <span class="fu">abline</span>(<span class="at">a=</span>a[j], <span class="at">b=</span>b[j], <span class="at">lwd=</span><span class="dv">2</span>, <span class="at">col=</span><span class="dv">2</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="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-23-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb50"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb50-1"><a href="#cb50-1" aria-hidden="true" tabindex="-1"></a>dat <span class="ot"><-</span> <span class="fu">list</span>(<span class="at">W=</span>d<span class="sc">$</span>weight, <span class="at">H=</span>d<span class="sc">$</span>height)</span>
<span id="cb50-2"><a href="#cb50-2" aria-hidden="true" tabindex="-1"></a><span class="co">#k <- 10</span></span>
<span id="cb50-3"><a href="#cb50-3" aria-hidden="true" tabindex="-1"></a><span class="co">#dat <- list(W=d$weight[1:k], H=d$height[1:k])</span></span>
<span id="cb50-4"><a href="#cb50-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb50-5"><a href="#cb50-5" aria-hidden="true" tabindex="-1"></a><span class="co">#Run Model</span></span>
<span id="cb50-6"><a href="#cb50-6" aria-hidden="true" tabindex="-1"></a>m3<span class="fl">.2</span> <span class="ot"><-</span> <span class="fu">quap</span>(</span>
<span id="cb50-7"><a href="#cb50-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">alist</span>(</span>
<span id="cb50-8"><a href="#cb50-8" aria-hidden="true" tabindex="-1"></a> W <span class="sc">~</span> <span class="fu">dnorm</span>(mu,sigma),</span>
<span id="cb50-9"><a href="#cb50-9" aria-hidden="true" tabindex="-1"></a> mu <span class="ot"><-</span> a <span class="sc">+</span> b<span class="sc">*</span>H,</span>
<span id="cb50-10"><a href="#cb50-10" aria-hidden="true" tabindex="-1"></a> a <span class="sc">~</span> <span class="fu">dnorm</span>(<span class="dv">0</span>,<span class="dv">10</span>),</span>
<span id="cb50-11"><a href="#cb50-11" aria-hidden="true" tabindex="-1"></a> b <span class="sc">~</span> <span class="fu">dunif</span>(<span class="dv">0</span>,<span class="dv">1</span>),</span>
<span id="cb50-12"><a href="#cb50-12" aria-hidden="true" tabindex="-1"></a> sigma<span class="sc">~</span><span class="fu">dunif</span>(<span class="dv">0</span>,<span class="dv">10</span>)</span>
<span id="cb50-13"><a href="#cb50-13" aria-hidden="true" tabindex="-1"></a>),<span class="at">data=</span>dat)</span>
<span id="cb50-14"><a href="#cb50-14" aria-hidden="true" tabindex="-1"></a><span class="fu">precis</span>(m3<span class="fl">.2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> mean sd 5.5% 94.5%
a -43.3834467 4.17074113 -50.0490966 -36.7177968
b 0.5717844 0.02695064 0.5287121 0.6148568
sigma 4.2526514 0.16173248 3.9941717 4.5111311</code></pre>
</div>
<div class="sourceCode cell-code" id="cb52"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb52-1"><a href="#cb52-1" aria-hidden="true" tabindex="-1"></a>post <span class="ot"><-</span> <span class="fu">extract.samples</span>(m3<span class="fl">.2</span>)</span>
<span id="cb52-2"><a href="#cb52-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(dat<span class="sc">$</span>H,dat<span class="sc">$</span>W,<span class="at">col=</span><span class="dv">2</span>,<span class="at">lwd=</span><span class="dv">3</span>,</span>
<span id="cb52-3"><a href="#cb52-3" aria-hidden="true" tabindex="-1"></a> <span class="at">xlim=</span><span class="fu">c</span>(<span class="fu">min</span>(dat<span class="sc">$</span>H),<span class="fu">max</span>(dat<span class="sc">$</span>H)), <span class="at">ylim=</span><span class="fu">c</span>(<span class="fu">min</span>(<span class="dv">30</span>,dat<span class="sc">$</span>w),<span class="fu">max</span>(dat<span class="sc">$</span>W)),</span>
<span id="cb52-4"><a href="#cb52-4" aria-hidden="true" tabindex="-1"></a> <span class="at">xlab=</span><span class="st">"height(cm)"</span>, <span class="at">ylab=</span><span class="st">"weight(kg)"</span>)</span>
<span id="cb52-5"><a href="#cb52-5" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (j <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">20</span>)</span>
<span id="cb52-6"><a href="#cb52-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">abline</span>(<span class="at">a=</span>post<span class="sc">$</span>a[j],<span class="at">b=</span>post<span class="sc">$</span>b[j], <span class="at">lty=</span><span class="dv">1</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-24-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb53"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb53-1"><a href="#cb53-1" aria-hidden="true" tabindex="-1"></a>post <span class="ot"><-</span> <span class="fu">extract.samples</span>(m3<span class="fl">.2</span>)</span>
<span id="cb53-2"><a href="#cb53-2" aria-hidden="true" tabindex="-1"></a><span class="fu">pairs</span>(post)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-25-1.png" class="img-fluid" width="672"></p>
</div>
<div class="sourceCode cell-code" id="cb54"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb54-1"><a href="#cb54-1" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span>(post)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> a b sigma
1 -39.89113 0.5483387 4.279574
2 -39.70825 0.5487077 3.989240
3 -46.81695 0.5954161 4.482863
4 -43.63421 0.5706343 4.377119
5 -42.21878 0.5643710 4.199400
6 -36.38253 0.5265746 4.076836</code></pre>
</div>
<div class="sourceCode cell-code" id="cb56"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(d<span class="sc">$</span>height,d<span class="sc">$</span>weight,<span class="at">col=</span><span class="dv">2</span>,<span class="at">lwd=</span><span class="dv">3</span>,</span>
<span id="cb56-2"><a href="#cb56-2" aria-hidden="true" tabindex="-1"></a> <span class="at">xlab=</span><span class="st">"height(cm)"</span>, <span class="at">ylab=</span><span class="st">"weight(kg)"</span>)</span>
<span id="cb56-3"><a href="#cb56-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (j <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">20</span>)</span>
<span id="cb56-4"><a href="#cb56-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">abline</span>(<span class="at">a=</span>post<span class="sc">$</span>a[j],<span class="at">b=</span>post<span class="sc">$</span>b[j], <span class="at">lty=</span><span class="dv">1</span>)</span>
<span id="cb56-5"><a href="#cb56-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb56-6"><a href="#cb56-6" aria-hidden="true" tabindex="-1"></a>height_seq <span class="ot"><-</span> <span class="fu">seq</span>(<span class="dv">130</span>,<span class="dv">190</span>,<span class="at">len=</span><span class="dv">20</span>)</span>
<span id="cb56-7"><a href="#cb56-7" aria-hidden="true" tabindex="-1"></a>W_postpred <span class="ot"><-</span> <span class="fu">sim</span>(m3<span class="fl">.2</span>,<span class="at">data=</span><span class="fu">list</span>(<span class="at">H=</span>height_seq))</span>
<span id="cb56-8"><a href="#cb56-8" aria-hidden="true" tabindex="-1"></a>W_PI <span class="ot"><-</span> <span class="fu">apply</span>(W_postpred,<span class="dv">2</span>,PI)</span>
<span id="cb56-9"><a href="#cb56-9" aria-hidden="true" tabindex="-1"></a><span class="fu">lines</span>(height_seq,W_PI[<span class="dv">1</span>,],<span class="at">lty=</span><span class="dv">2</span>,<span class="at">lwd=</span><span class="dv">2</span>)</span>
<span id="cb56-10"><a href="#cb56-10" aria-hidden="true" tabindex="-1"></a><span class="fu">lines</span>(height_seq,W_PI[<span class="dv">2</span>,],<span class="at">lty=</span><span class="dv">2</span>,<span class="at">lwd=</span><span class="dv">2</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="Statistical-Thinking-2023_v2_files/figure-html/unnamed-chunk-25-2.png" class="img-fluid" width="672"></p>
</div>
</div>
</section>
</section>
</main>
<!-- /main column -->
<script id="quarto-html-after-body" type="application/javascript">
window.document.addEventListener("DOMContentLoaded", function (event) {
const toggleBodyColorMode = (bsSheetEl) => {
const mode = bsSheetEl.getAttribute("data-mode");
const bodyEl = window.document.querySelector("body");
if (mode === "dark") {
bodyEl.classList.add("quarto-dark");
bodyEl.classList.remove("quarto-light");
} else {
bodyEl.classList.add("quarto-light");
bodyEl.classList.remove("quarto-dark");
}
}
const toggleBodyColorPrimary = () => {
const bsSheetEl = window.document.querySelector("link#quarto-bootstrap");
if (bsSheetEl) {
toggleBodyColorMode(bsSheetEl);
}
}
toggleBodyColorPrimary();
const icon = "";
const anchorJS = new window.AnchorJS();
anchorJS.options = {
placement: 'right',
icon: icon
};
anchorJS.add('.anchored');
const clipboard = new window.ClipboardJS('.code-copy-button', {
target: function(trigger) {
return trigger.previousElementSibling;
}
});
clipboard.on('success', function(e) {
// button target
const button = e.trigger;
// don't keep focus
button.blur();
// flash "checked"
button.classList.add('code-copy-button-checked');
var currentTitle = button.getAttribute("title");
button.setAttribute("title", "Copied!");
let tooltip;
if (window.bootstrap) {
button.setAttribute("data-bs-toggle", "tooltip");
button.setAttribute("data-bs-placement", "left");
button.setAttribute("data-bs-title", "Copied!");
tooltip = new bootstrap.Tooltip(button,
{ trigger: "manual",
customClass: "code-copy-button-tooltip",
offset: [0, -8]});
tooltip.show();
}
setTimeout(function() {
if (tooltip) {
tooltip.hide();
button.removeAttribute("data-bs-title");
button.removeAttribute("data-bs-toggle");
button.removeAttribute("data-bs-placement");
}
button.setAttribute("title", currentTitle);
button.classList.remove('code-copy-button-checked');
}, 1000);
// clear code selection
e.clearSelection();
});
function tippyHover(el, contentFn) {
const config = {
allowHTML: true,
content: contentFn,
maxWidth: 500,
delay: 100,
arrow: false,
appendTo: function(el) {
return el.parentElement;
},
interactive: true,
interactiveBorder: 10,
theme: 'quarto',
placement: 'bottom-start'
};
window.tippy(el, config);
}
const noterefs = window.document.querySelectorAll('a[role="doc-noteref"]');
for (var i=0; i<noterefs.length; i++) {
const ref = noterefs[i];
tippyHover(ref, function() {
// use id or data attribute instead here
let href = ref.getAttribute('data-footnote-href') || ref.getAttribute('href');
try { href = new URL(href).hash; } catch {}
const id = href.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
return note.innerHTML;
});
}
const findCites = (el) => {
const parentEl = el.parentElement;
if (parentEl) {
const cites = parentEl.dataset.cites;
if (cites) {
return {
el,
cites: cites.split(' ')
};
} else {
return findCites(el.parentElement)
}
} else {
return undefined;
}
};
var bibliorefs = window.document.querySelectorAll('a[role="doc-biblioref"]');
for (var i=0; i<bibliorefs.length; i++) {
const ref = bibliorefs[i];
const citeInfo = findCites(ref);
if (citeInfo) {
tippyHover(citeInfo.el, function() {
var popup = window.document.createElement('div');
citeInfo.cites.forEach(function(cite) {
var citeDiv = window.document.createElement('div');
citeDiv.classList.add('hanging-indent');
citeDiv.classList.add('csl-entry');
var biblioDiv = window.document.getElementById('ref-' + cite);
if (biblioDiv) {
citeDiv.innerHTML = biblioDiv.innerHTML;
}
popup.appendChild(citeDiv);
});
return popup.innerHTML;
});
}
}
});
</script>
</div> <!-- /content -->
</body></html>