-
Notifications
You must be signed in to change notification settings - Fork 3
Expand file tree
/
Copy pathgamma_poisson.html
More file actions
546 lines (516 loc) · 25.9 KB
/
gamma_poisson.html
File metadata and controls
546 lines (516 loc) · 25.9 KB
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
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<title>Count Data and Ordinal Data with the Gamma-Poisson Distribution</title>
<link rel="stylesheet" href="_static/basic.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
<link rel="stylesheet" href="_static/bootswatch-3.3.4/lumen/bootstrap.min.css" type="text/css" />
<link rel="stylesheet" href="_static/bootstrap-sphinx.css" type="text/css" />
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
VERSION: '0.1.0',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true
};
</script>
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script type="text/javascript" src="_static/js/jquery-1.11.0.min.js"></script>
<script type="text/javascript" src="_static/js/jquery-fix.js"></script>
<script type="text/javascript" src="_static/bootstrap-3.3.4/js/bootstrap.min.js"></script>
<script type="text/javascript" src="_static/bootstrap-sphinx.js"></script>
<link rel="top" title="None" href="index.html" />
<link rel="up" title="Tutorials" href="docs.html" />
<link rel="next" title="Inferring Gaussians with the Dirichlet Process Mixture Model" href="gauss2d.html" />
<link rel="prev" title="Univariate Data with the Normal Inverse Chi-Square Distribution" href="nic.html" />
<meta charset='utf-8'>
<meta http-equiv='X-UA-Compatible' content='IE=edge,chrome=1'>
<meta name='viewport' content='width=device-width, initial-scale=1.0, maximum-scale=1'>
<meta name="apple-mobile-web-app-capable" content="yes">
</head>
<body role="document">
<div id="navbar" class="navbar navbar-inverse navbar-default navbar-fixed-top">
<div class="container">
<div class="navbar-header">
<!-- .btn-navbar is used as the toggle for collapsed navbar content -->
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".nav-collapse">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="index.html">
datamicroscopes</a>
<span class="navbar-text navbar-version pull-left"><b>0.1</b></span>
</div>
<div class="collapse navbar-collapse nav-collapse">
<ul class="nav navbar-nav">
<li><a href="https://github.com/datamicroscopes">GitHub</a></li>
<li><a href="https://qadium.com/">Qadium</a></li>
<li class="dropdown globaltoc-container">
<a role="button"
id="dLabelGlobalToc"
data-toggle="dropdown"
data-target="#"
href="index.html">Site <b class="caret"></b></a>
<ul class="dropdown-menu globaltoc"
role="menu"
aria-labelledby="dLabelGlobalToc"><ul>
<li class="toctree-l1"><a class="reference internal" href="intro.html">Discovering structure in your data: an overview of clustering</a></li>
<li class="toctree-l1"><a class="reference internal" href="ncluster.html">Finding the number of clusters with the Dirichlet Process</a></li>
<li class="toctree-l1"><a class="reference internal" href="enron_blog.html">Network Modeling with the Infinite Relational Model</a></li>
<li class="toctree-l1"><a class="reference internal" href="topic.html">Bayesian Nonparametric Topic Modeling with the Daily Kos</a></li>
</ul>
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="datatypes.html">Datatypes and Bayesian Nonparametric Models</a></li>
<li class="toctree-l1"><a class="reference internal" href="bb.html">Binary Data with the Beta Bernouli Distribution</a></li>
<li class="toctree-l1"><a class="reference internal" href="dd.html">Categorical Data and the Dirichlet Discrete Distribution</a></li>
<li class="toctree-l1"><a class="reference internal" href="niw.html">Real Valued Data and the Normal Inverse-Wishart Distribution</a></li>
<li class="toctree-l1"><a class="reference internal" href="nic.html">Univariate Data with the Normal Inverse Chi-Square Distribution</a></li>
<li class="toctree-l1 current"><a class="current reference internal" href="">Count Data and Ordinal Data with the Gamma-Poisson Distribution</a></li>
</ul>
<ul>
<li class="toctree-l1"><a class="reference internal" href="gauss2d.html">Inferring Gaussians with the Dirichlet Process Mixture Model</a></li>
<li class="toctree-l1"><a class="reference internal" href="mnist_predictions.html">Digit recognition with the MNIST dataset</a></li>
<li class="toctree-l1"><a class="reference internal" href="enron_email.html">Clustering the Enron e-mail corpus using the Infinite Relational Model</a></li>
<li class="toctree-l1"><a class="reference internal" href="hdp.html">Learning Topics in The Daily Kos with the Hierarchical Dirichlet Process</a></li>
</ul>
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="docs.html">Tutorials</a><ul>
<li class="toctree-l2"><a class="reference internal" href="intro.html">Discovering structure in your data: an overview of clustering</a></li>
<li class="toctree-l2"><a class="reference internal" href="ncluster.html">Finding the number of clusters with the Dirichlet Process</a></li>
<li class="toctree-l2"><a class="reference internal" href="enron_blog.html">Network Modeling with the Infinite Relational Model</a></li>
<li class="toctree-l2"><a class="reference internal" href="topic.html">Bayesian Nonparametric Topic Modeling with the Daily Kos</a></li>
</ul>
</li>
<li class="toctree-l1 current"><a class="reference internal" href="docs.html#datatypes-and-likelihood-models-in-datamicroscopes">Datatypes and likelihood models in datamicroscopes</a><ul class="current">
<li class="toctree-l2"><a class="reference internal" href="datatypes.html">Datatypes and Bayesian Nonparametric Models</a></li>
<li class="toctree-l2"><a class="reference internal" href="bb.html">Binary Data with the Beta Bernouli Distribution</a></li>
<li class="toctree-l2"><a class="reference internal" href="dd.html">Categorical Data and the Dirichlet Discrete Distribution</a></li>
<li class="toctree-l2"><a class="reference internal" href="niw.html">Real Valued Data and the Normal Inverse-Wishart Distribution</a></li>
<li class="toctree-l2"><a class="reference internal" href="nic.html">Univariate Data with the Normal Inverse Chi-Square Distribution</a></li>
<li class="toctree-l2 current"><a class="current reference internal" href="">Count Data and Ordinal Data with the Gamma-Poisson Distribution</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="docs.html#examples">Examples</a><ul>
<li class="toctree-l2"><a class="reference internal" href="gauss2d.html">Inferring Gaussians with the Dirichlet Process Mixture Model</a></li>
<li class="toctree-l2"><a class="reference internal" href="mnist_predictions.html">Digit recognition with the MNIST dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="enron_email.html">Clustering the Enron e-mail corpus using the Infinite Relational Model</a></li>
<li class="toctree-l2"><a class="reference internal" href="hdp.html">Learning Topics in The Daily Kos with the Hierarchical Dirichlet Process</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="api.html">API Reference</a><ul>
<li class="toctree-l2"><a class="reference internal" href="microscopes.common.dataview.html">dataviews</a></li>
<li class="toctree-l2"><a class="reference internal" href="microscopes.common.util.html">util</a></li>
<li class="toctree-l2"><a class="reference internal" href="microscopes.common.random.html">microscopes.common.random</a></li>
<li class="toctree-l2"><a class="reference internal" href="microscopes.common.query.html">query</a></li>
<li class="toctree-l2"><a class="reference internal" href="microscopes.common.validator.html">microscopes.common.validator</a></li>
<li class="toctree-l2"><a class="reference internal" href="microscopes.kernels.parallel.html">parallel</a></li>
<li class="toctree-l2"><a class="reference internal" href="microscopes.mixture.html">mixturemodel</a></li>
<li class="toctree-l2"><a class="reference internal" href="microscopes.irm.html">irm</a></li>
<li class="toctree-l2"><a class="reference internal" href="microscopes.kernels.html">kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="api.html#indices-and-tables">Indices and tables</a></li>
</ul>
</li>
</ul>
</ul>
</li>
<li class="dropdown">
<a role="button"
id="dLabelLocalToc"
data-toggle="dropdown"
data-target="#"
href="#">Contents <b class="caret"></b></a>
<ul class="dropdown-menu localtoc"
role="menu"
aria-labelledby="dLabelLocalToc"><ul>
<li><a class="reference internal" href="#">Count Data and Ordinal Data with the Gamma-Poisson Distribution</a></li>
</ul>
</ul>
</li>
<li class="hidden-sm">
<div id="sourcelink">
<a href="_sources/gamma_poisson.txt"
rel="nofollow">Source</a>
</div></li>
</ul>
<form class="navbar-form navbar-right" action="search.html" method="get">
<div class="form-group">
<input type="text" name="q" class="form-control" placeholder="Search" />
</div>
<input type="hidden" name="check_keywords" value="yes" />
<input type="hidden" name="area" value="default" />
</form>
</div>
</div>
</div>
<div class="container">
<div class="row">
<div class="col-md-12">
<div class="section" id="count-data-and-ordinal-data-with-the-gamma-poisson-distribution">
<span id="gamma-poisson"></span><h1>Count Data and Ordinal Data with the Gamma-Poisson Distribution<a class="headerlink" href="#count-data-and-ordinal-data-with-the-gamma-poisson-distribution" title="Permalink to this headline">¶</a></h1>
<p>Typically, we model count data, or integer valued data, with the
gamma-Poisson distribution</p>
<p>Recall that the Poisson distribution is a distribution over integer
values parameterized by <span class="math">\(\lambda\)</span>. One interpretation behind
<span class="math">\(\lambda\)</span> is that it parameterizes the rate at which events occur
with a fixed interval, assuming these events occur independently. The
gamma distribution is conjugate to the Poisson distribution, so the
gamma-Poisson distribution allows us to learn both the distribution over
counts and the rate parameter <span class="math">\(\lambda\)</span>.</p>
<p>Let’s set up our environment and consider some examples of count data</p>
<div class="code python highlight-python"><div class="highlight"><pre>import pandas as pd
import seaborn as sns
import numpy as np
import matplotlib.pyplot as plt
sns.set_context('talk')
import csv
import urllib2
import StringIO
%matplotlib inline
</pre></div>
</div>
<p><a class="reference external" href="http://data.princeton.edu/wws509/datasets/#ceb">Children Ever Born</a>
is a dataset of birthrates in Fiji from the <em>World Fertility Survey</em>
with the following columns:</p>
<ul class="simple">
<li><code class="docutils literal"><span class="pre">dur</span></code>: marriage duration</li>
<li><code class="docutils literal"><span class="pre">res</span></code>: residence,</li>
<li><code class="docutils literal"><span class="pre">educ</span></code>: level of education,</li>
<li><code class="docutils literal"><span class="pre">mean</span></code>: mean number of born,</li>
<li><code class="docutils literal"><span class="pre">var</span></code>: variance of children born</li>
<li><code class="docutils literal"><span class="pre">y</span></code>: number of women</li>
</ul>
<p>Ordinal columns <code class="docutils literal"><span class="pre">dur</span></code>, <code class="docutils literal"><span class="pre">res</span></code>, and <code class="docutils literal"><span class="pre">educ</span></code> are shown as text in the
following dataset</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">ceb</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s">'http://data.princeton.edu/wws509/datasets/ceb.dat'</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s">'\s+'</span><span class="p">)</span>
<span class="n">ceb</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
</pre></div>
</div>
<div>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>dur</th>
<th>res</th>
<th>educ</th>
<th>mean</th>
<th>var</th>
<th>n</th>
<th>y</th>
</tr>
</thead>
<tbody>
<tr>
<th>1</th>
<td>0-4</td>
<td>Suva</td>
<td>none</td>
<td>0.50</td>
<td>1.14</td>
<td>8</td>
<td>4.00</td>
</tr>
<tr>
<th>2</th>
<td>0-4</td>
<td>Suva</td>
<td>lower</td>
<td>1.14</td>
<td>0.73</td>
<td>21</td>
<td>23.94</td>
</tr>
<tr>
<th>3</th>
<td>0-4</td>
<td>Suva</td>
<td>upper</td>
<td>0.90</td>
<td>0.67</td>
<td>42</td>
<td>37.80</td>
</tr>
<tr>
<th>4</th>
<td>0-4</td>
<td>Suva</td>
<td>sec+</td>
<td>0.73</td>
<td>0.48</td>
<td>51</td>
<td>37.23</td>
</tr>
<tr>
<th>5</th>
<td>0-4</td>
<td>urban</td>
<td>none</td>
<td>1.17</td>
<td>1.06</td>
<td>12</td>
<td>14.04</td>
</tr>
</tbody>
</table>
</div><p>With the these columns encoded, we can now represent them as integers</p>
<p><code class="docutils literal"><span class="pre">dur</span></code> and <code class="docutils literal"><span class="pre">educ</span></code> are ordinal columns. Additionally, number of women,
<code class="docutils literal"><span class="pre">n</span></code>, is integer valued.</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">ceb_int</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s">'http://data.princeton.edu/wws509/datasets/ceb.raw'</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s">'\s+'</span><span class="p">,</span> <span class="n">names</span> <span class="o">=</span> <span class="p">[</span><span class="s">'index'</span><span class="p">]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">ceb</span><span class="o">.</span><span class="n">columns</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="n">index_col</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">ceb_int</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
</pre></div>
</div>
<div>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>dur</th>
<th>res</th>
<th>educ</th>
<th>mean</th>
<th>var</th>
<th>n</th>
</tr>
<tr>
<th>index</th>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<th>1</th>
<td>1</td>
<td>1</td>
<td>1</td>
<td>0.50</td>
<td>1.14</td>
<td>8</td>
</tr>
<tr>
<th>2</th>
<td>1</td>
<td>1</td>
<td>2</td>
<td>1.14</td>
<td>0.73</td>
<td>21</td>
</tr>
<tr>
<th>3</th>
<td>1</td>
<td>1</td>
<td>3</td>
<td>0.90</td>
<td>0.67</td>
<td>42</td>
</tr>
<tr>
<th>4</th>
<td>1</td>
<td>1</td>
<td>4</td>
<td>0.73</td>
<td>0.48</td>
<td>51</td>
</tr>
<tr>
<th>5</th>
<td>1</td>
<td>2</td>
<td>1</td>
<td>1.17</td>
<td>1.06</td>
<td>12</td>
</tr>
</tbody>
</table>
</div><p>We can map these orderings of <code class="docutils literal"><span class="pre">dur</span></code> and <code class="docutils literal"><span class="pre">educ</span></code> to produce a crosstab
heatmap of <code class="docutils literal"><span class="pre">n</span></code>, numbe of women</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
<span class="n">ct</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">crosstab</span><span class="p">(</span><span class="n">ceb_int</span><span class="p">[</span><span class="s">'dur'</span><span class="p">],</span> <span class="n">ceb_int</span><span class="p">[</span><span class="s">'educ'</span><span class="p">],</span> <span class="n">values</span><span class="o">=</span><span class="n">ceb_int</span><span class="p">[</span><span class="s">'n'</span><span class="p">],</span> <span class="n">aggfunc</span><span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">)</span><span class="o">.</span><span class="n">sort_index</span><span class="p">(</span><span class="n">ascending</span> <span class="o">=</span> <span class="bp">False</span><span class="p">)</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">ct</span><span class="p">,</span> <span class="n">annot</span> <span class="o">=</span> <span class="bp">True</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">ceb_int</span><span class="p">[</span><span class="s">'dur'</span><span class="p">]</span><span class="o">.</span><span class="n">drop_duplicates</span><span class="p">()</span><span class="o">.</span><span class="n">values</span> <span class="o">-</span> <span class="o">.</span><span class="mi">5</span><span class="p">,</span> <span class="n">ceb</span><span class="p">[</span><span class="s">'dur'</span><span class="p">]</span><span class="o">.</span><span class="n">drop_duplicates</span><span class="p">()</span><span class="o">.</span><span class="n">values</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(</span><span class="n">ceb_int</span><span class="p">[</span><span class="s">'educ'</span><span class="p">]</span><span class="o">.</span><span class="n">drop_duplicates</span><span class="p">()</span><span class="o">.</span><span class="n">values</span> <span class="o">-</span> <span class="o">.</span><span class="mi">5</span><span class="p">,</span> <span class="n">ceb</span><span class="p">[</span><span class="s">'educ'</span><span class="p">]</span><span class="o">.</span><span class="n">drop_duplicates</span><span class="p">()</span><span class="o">.</span><span class="n">values</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">'duration of marriage (years)'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">'level of education'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'heatmap of marriage duration by level of education'</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><matplotlib.text.Text at 0x11a8f48d0>
</pre></div>
</div>
<img alt="_images/gamma_poisson_7_1.png" src="_images/gamma_poisson_7_1.png" />
<p>Since <code class="docutils literal"><span class="pre">dur</span></code> and <code class="docutils literal"><span class="pre">education</span></code> are ordinal valued, the columns assume a
small number of integer values</p>
<p>Additionally, the <a class="reference external" href="http://stanford.edu/class/psych252/data/index.html">caffeine
dataset</a> below
measures caffeine intake and performance on a 10 question quiz. The
variables are:</p>
<ul class="simple">
<li><code class="docutils literal"><span class="pre">coffee</span></code>: coffee intake (1 = 0 cups, 2 = 2 cups, 3 = 4 cups)</li>
<li><code class="docutils literal"><span class="pre">perf</span></code>: quiz score</li>
<li><code class="docutils literal"><span class="pre">numprob</span></code>: problems attempted</li>
<li><code class="docutils literal"><span class="pre">accur</span></code>: accuracy</li>
</ul>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">response</span> <span class="o">=</span> <span class="n">urllib2</span><span class="o">.</span><span class="n">urlopen</span><span class="p">(</span><span class="s">'http://stanford.edu/class/psych252/_downloads/caffeine.csv'</span><span class="p">)</span>
<span class="n">html</span> <span class="o">=</span> <span class="n">response</span><span class="o">.</span><span class="n">read</span><span class="p">()</span>
<span class="n">caf</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">StringIO</span><span class="o">.</span><span class="n">StringIO</span><span class="p">(</span><span class="n">html</span><span class="p">[:</span><span class="o">-</span><span class="mi">16</span><span class="p">]))</span>
<span class="n">caf</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
</pre></div>
</div>
<div>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>coffee</th>
<th>perf</th>
<th>accur</th>
<th>numprob</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>1</td>
<td>53</td>
<td>0.449877</td>
<td>7</td>
</tr>
<tr>
<th>1</th>
<td>1</td>
<td>9</td>
<td>0.499534</td>
<td>6</td>
</tr>
<tr>
<th>2</th>
<td>1</td>
<td>31</td>
<td>0.498590</td>
<td>6</td>
</tr>
<tr>
<th>3</th>
<td>1</td>
<td>38</td>
<td>0.454312</td>
<td>7</td>
</tr>
<tr>
<th>4</th>
<td>2</td>
<td>40</td>
<td>0.421212</td>
<td>8</td>
</tr>
</tbody>
</table>
</div><p>Based on the characteristics of each column, <code class="docutils literal"><span class="pre">coffee</span></code> and <code class="docutils literal"><span class="pre">numprob</span></code>
easily fit into the category of count data appropriate to a
gamma-Poisson distribution</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="n">caf</span><span class="o">.</span><span class="n">describe</span><span class="p">()</span>
</pre></div>
</div>
<div>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>coffee</th>
<th>perf</th>
<th>accur</th>
<th>numprob</th>
</tr>
</thead>
<tbody>
<tr>
<th>count</th>
<td>60.000000</td>
<td>60.000000</td>
<td>60.000000</td>
<td>60.000000</td>
</tr>
<tr>
<th>mean</th>
<td>2.000000</td>
<td>42.366667</td>
<td>0.510854</td>
<td>7.950000</td>
</tr>
<tr>
<th>std</th>
<td>0.823387</td>
<td>18.350603</td>
<td>0.107704</td>
<td>1.185005</td>
</tr>
<tr>
<th>min</th>
<td>1.000000</td>
<td>5.000000</td>
<td>0.240238</td>
<td>6.000000</td>
</tr>
<tr>
<th>25%</th>
<td>1.000000</td>
<td>31.000000</td>
<td>0.425859</td>
<td>7.000000</td>
</tr>
<tr>
<th>50%</th>
<td>2.000000</td>
<td>40.000000</td>
<td>0.509806</td>
<td>8.000000</td>
</tr>
<tr>
<th>75%</th>
<td>3.000000</td>
<td>53.500000</td>
<td>0.594445</td>
<td>9.000000</td>
</tr>
<tr>
<th>max</th>
<td>3.000000</td>
<td>89.000000</td>
<td>0.748692</td>
<td>10.000000</td>
</tr>
</tbody>
</table>
</div><p>Note that while integer valued data with high values is sometimes
modeled with a gamma-Poisson ditribution, remember that the
gamma-Poisson distribution has equal mean and variance <span class="math">\(\lambda\)</span>:</p>
<div class="math">
\[E(X) = Var(X) = \lambda\]</div>
<p>If you want to be more flexible with this assumption, you may want to
consider using a normal inverse-chisquare or a normal inverse-Wishart
distribution depending on your data</p>
<p>To import the gamma-poisson likelihood, call:</p>
<div class="code python highlight-python"><div class="highlight"><pre><span class="kn">from</span> <span class="nn">microscopes.models</span> <span class="kn">import</span> <span class="n">gp</span> <span class="k">as</span> <span class="n">gamma_poisson</span>
</pre></div>
</div>
</div>
</div>
</div>
</div>
<!-- your html code here -->
<center> Datamicroscopes is developed by <a href="http://www.qadium.com">Qadium</a>, with funding from the <a href="http://www.darpa.mil">DARPA</a> <a href="http://www.darpa.mil/program/xdata">XDATA</a> program. Copyright Qadium 2015. </center>
</body>
</html>