My First Stan Model

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# “My First Stan Model”
## Do Bay Leaves Do Any Good?
### Abby Kaplan
### January 10, 2019

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## "My First Experiment"

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## "My First ~~Experiment~~ Stan Model"

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![](bay_leaves.jpg)

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![](experiment_setup_small.png)

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![](rug_lightning_talk_files/figure-html/plot_raw_data-1.png)

---

## The model

`$$\mbox{prefer.bay.leaf} = \mbox{Bernoulli}(\mbox{logit}^{-1}(\beta + \gamma_i\sigma_{\gamma} + \delta_j\sigma_{\delta}))$$`

* `$\beta$`: overall intercept

* `$\gamma_i$`: intercept for batch `$i$`
* `$\sigma_{\gamma}$`: standard deviation of `$\gamma_i$`

* `$\delta_j$`: intercept for subject `$j$`
* `$\sigma_{\delta}$`: standard deviation of `$\delta_j$`

`$$\beta \sim \mbox{N}(0, 2)$$`

`$$\gamma_i, \delta_j \sim \mbox{N}(0, 1)$$`

`$$\sigma_{\gamma}, \sigma_{\delta} \in [0,\inf] \sim \mbox{Cauchy}(0, 1)$$`

---

## What Stan gives you

.center[
<table class="table" style="font-size: 12px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> sample </th>
   <th style="text-align:right;"> beta </th>
   <th style="text-align:right;"> gamma.1. </th>
   <th style="text-align:right;"> gamma.2. </th>
   <th style="text-align:right;"> gamma.3. </th>
   <th style="text-align:right;"> gamma.4. </th>
   <th style="text-align:right;"> sigma_gamma </th>
   <th style="text-align:right;"> delta.1. </th>
   <th style="text-align:right;"> delta.2. </th>
   <th style="text-align:left;"> ... </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:right;"> 0.0381603 </td>
   <td style="text-align:right;"> -0.5866505 </td>
   <td style="text-align:right;"> -1.7717493 </td>
   <td style="text-align:right;"> 0.1951042 </td>
   <td style="text-align:right;"> 0.1842853 </td>
   <td style="text-align:right;"> 0.1286942 </td>
   <td style="text-align:right;"> -0.9239049 </td>
   <td style="text-align:right;"> 0.6042314 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:right;"> 0.2626396 </td>
   <td style="text-align:right;"> 0.3976282 </td>
   <td style="text-align:right;"> -0.0257513 </td>
   <td style="text-align:right;"> -1.0893670 </td>
   <td style="text-align:right;"> -0.7284009 </td>
   <td style="text-align:right;"> 0.4963586 </td>
   <td style="text-align:right;"> 0.0957525 </td>
   <td style="text-align:right;"> -1.2964409 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:right;"> 0.1140987 </td>
   <td style="text-align:right;"> 0.5158181 </td>
   <td style="text-align:right;"> -0.8393754 </td>
   <td style="text-align:right;"> 1.5553642 </td>
   <td style="text-align:right;"> 0.4570358 </td>
   <td style="text-align:right;"> 0.5793863 </td>
   <td style="text-align:right;"> 0.1001774 </td>
   <td style="text-align:right;"> 0.7797288 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 4 </td>
   <td style="text-align:right;"> 0.8345731 </td>
   <td style="text-align:right;"> -0.3933963 </td>
   <td style="text-align:right;"> -0.2986723 </td>
   <td style="text-align:right;"> -1.1892198 </td>
   <td style="text-align:right;"> -0.8657181 </td>
   <td style="text-align:right;"> 0.9091757 </td>
   <td style="text-align:right;"> -2.0567760 </td>
   <td style="text-align:right;"> -0.3299187 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 5 </td>
   <td style="text-align:right;"> 0.7405298 </td>
   <td style="text-align:right;"> 0.1480183 </td>
   <td style="text-align:right;"> -0.8649021 </td>
   <td style="text-align:right;"> -1.3578496 </td>
   <td style="text-align:right;"> -0.9145567 </td>
   <td style="text-align:right;"> 0.6320696 </td>
   <td style="text-align:right;"> -1.2670545 </td>
   <td style="text-align:right;"> -1.3615715 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 6 </td>
   <td style="text-align:right;"> 0.6651553 </td>
   <td style="text-align:right;"> -0.1847768 </td>
   <td style="text-align:right;"> -0.7090246 </td>
   <td style="text-align:right;"> -1.0886230 </td>
   <td style="text-align:right;"> -0.7304895 </td>
   <td style="text-align:right;"> 0.4295265 </td>
   <td style="text-align:right;"> -1.1094154 </td>
   <td style="text-align:right;"> -1.3284828 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 7 </td>
   <td style="text-align:right;"> -0.9446587 </td>
   <td style="text-align:right;"> 0.6964131 </td>
   <td style="text-align:right;"> 0.3959616 </td>
   <td style="text-align:right;"> 0.6783410 </td>
   <td style="text-align:right;"> 0.7992607 </td>
   <td style="text-align:right;"> 0.9157164 </td>
   <td style="text-align:right;"> -0.9395835 </td>
   <td style="text-align:right;"> -0.9652781 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 8 </td>
   <td style="text-align:right;"> 0.5688823 </td>
   <td style="text-align:right;"> -0.7402765 </td>
   <td style="text-align:right;"> -1.1801935 </td>
   <td style="text-align:right;"> -0.0955702 </td>
   <td style="text-align:right;"> -1.0452409 </td>
   <td style="text-align:right;"> 0.2957341 </td>
   <td style="text-align:right;"> 1.3606070 </td>
   <td style="text-align:right;"> -0.4098481 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 9 </td>
   <td style="text-align:right;"> 1.1633767 </td>
   <td style="text-align:right;"> 0.5247243 </td>
   <td style="text-align:right;"> -1.4510800 </td>
   <td style="text-align:right;"> 0.1383595 </td>
   <td style="text-align:right;"> -0.8334814 </td>
   <td style="text-align:right;"> 0.9092710 </td>
   <td style="text-align:right;"> 0.0772841 </td>
   <td style="text-align:right;"> -1.4462531 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 10 </td>
   <td style="text-align:right;"> 0.3588886 </td>
   <td style="text-align:right;"> -1.3746020 </td>
   <td style="text-align:right;"> -0.0798747 </td>
   <td style="text-align:right;"> 0.6042118 </td>
   <td style="text-align:right;"> -0.6485768 </td>
   <td style="text-align:right;"> 0.1092848 </td>
   <td style="text-align:right;"> 1.1794727 </td>
   <td style="text-align:right;"> -0.5398320 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 11 </td>
   <td style="text-align:right;"> 0.1585778 </td>
   <td style="text-align:right;"> 0.5717858 </td>
   <td style="text-align:right;"> -1.1179175 </td>
   <td style="text-align:right;"> 0.4109562 </td>
   <td style="text-align:right;"> 0.2118042 </td>
   <td style="text-align:right;"> 0.7897882 </td>
   <td style="text-align:right;"> 0.0583793 </td>
   <td style="text-align:right;"> -0.5520460 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 12 </td>
   <td style="text-align:right;"> 0.3136630 </td>
   <td style="text-align:right;"> 0.7874952 </td>
   <td style="text-align:right;"> -0.6955751 </td>
   <td style="text-align:right;"> 0.1958716 </td>
   <td style="text-align:right;"> 0.3913952 </td>
   <td style="text-align:right;"> 0.4845534 </td>
   <td style="text-align:right;"> -0.2037558 </td>
   <td style="text-align:right;"> -1.5467309 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 13 </td>
   <td style="text-align:right;"> 0.0404615 </td>
   <td style="text-align:right;"> 0.9990253 </td>
   <td style="text-align:right;"> -1.2931172 </td>
   <td style="text-align:right;"> 1.1747395 </td>
   <td style="text-align:right;"> 0.4294918 </td>
   <td style="text-align:right;"> 0.5587113 </td>
   <td style="text-align:right;"> -0.4359608 </td>
   <td style="text-align:right;"> 0.0125138 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 14 </td>
   <td style="text-align:right;"> 0.1788236 </td>
   <td style="text-align:right;"> -0.3799278 </td>
   <td style="text-align:right;"> 1.0026268 </td>
   <td style="text-align:right;"> -0.6759536 </td>
   <td style="text-align:right;"> -1.2857618 </td>
   <td style="text-align:right;"> 0.2832233 </td>
   <td style="text-align:right;"> 1.8629497 </td>
   <td style="text-align:right;"> -1.6790301 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 15 </td>
   <td style="text-align:right;"> 0.3600137 </td>
   <td style="text-align:right;"> 0.3479470 </td>
   <td style="text-align:right;"> -1.3522211 </td>
   <td style="text-align:right;"> 0.8713164 </td>
   <td style="text-align:right;"> -0.3963604 </td>
   <td style="text-align:right;"> 1.0634831 </td>
   <td style="text-align:right;"> -1.2731816 </td>
   <td style="text-align:right;"> 0.4896202 </td>
   <td style="text-align:left;"> ... </td>
  </tr>
</tbody>
</table>
]

(Plus various summary statistics, diagnostics, etc.)

---

## Exploring a single parameter

.center[
![](rug_lightning_talk_files/figure-html/plot_beta-1.png)
]

---

## Exploring a single parameter

.center[
![](rug_lightning_talk_files/figure-html/plot_beta_transformed-1.png)
]

---

## Exploring multiple parameters together

.center[
<table class="table" style="font-size: 12px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> sample </th>
   <th style="text-align:right;"> beta </th>
   <th style="text-align:right;"> gamma.1. </th>
   <th style="text-align:right;"> gamma.2. </th>
   <th style="text-align:right;"> gamma.3. </th>
   <th style="text-align:right;"> gamma.4. </th>
   <th style="text-align:right;"> sigma_gamma </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:right;"> 0.0381603 </td>
   <td style="text-align:right;background-color: #90EE90;"> -0.5866505 </td>
   <td style="text-align:right;"> -1.7717493 </td>
   <td style="text-align:right;"> 0.1951042 </td>
   <td style="text-align:right;"> 0.1842853 </td>
   <td style="text-align:right;"> 0.1286942 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:right;"> 0.2626396 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.3976282 </td>
   <td style="text-align:right;"> -0.0257513 </td>
   <td style="text-align:right;"> -1.0893670 </td>
   <td style="text-align:right;"> -0.7284009 </td>
   <td style="text-align:right;"> 0.4963586 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:right;"> 0.1140987 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.5158181 </td>
   <td style="text-align:right;"> -0.8393754 </td>
   <td style="text-align:right;"> 1.5553642 </td>
   <td style="text-align:right;"> 0.4570358 </td>
   <td style="text-align:right;"> 0.5793863 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 4 </td>
   <td style="text-align:right;"> 0.8345731 </td>
   <td style="text-align:right;background-color: #90EE90;"> -0.3933963 </td>
   <td style="text-align:right;"> -0.2986723 </td>
   <td style="text-align:right;"> -1.1892198 </td>
   <td style="text-align:right;"> -0.8657181 </td>
   <td style="text-align:right;"> 0.9091757 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 5 </td>
   <td style="text-align:right;"> 0.7405298 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.1480183 </td>
   <td style="text-align:right;"> -0.8649021 </td>
   <td style="text-align:right;"> -1.3578496 </td>
   <td style="text-align:right;"> -0.9145567 </td>
   <td style="text-align:right;"> 0.6320696 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 6 </td>
   <td style="text-align:right;"> 0.6651553 </td>
   <td style="text-align:right;background-color: #90EE90;"> -0.1847768 </td>
   <td style="text-align:right;"> -0.7090246 </td>
   <td style="text-align:right;"> -1.0886230 </td>
   <td style="text-align:right;"> -0.7304895 </td>
   <td style="text-align:right;"> 0.4295265 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 7 </td>
   <td style="text-align:right;"> -0.9446587 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.6964131 </td>
   <td style="text-align:right;"> 0.3959616 </td>
   <td style="text-align:right;"> 0.6783410 </td>
   <td style="text-align:right;"> 0.7992607 </td>
   <td style="text-align:right;"> 0.9157164 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 8 </td>
   <td style="text-align:right;"> 0.5688823 </td>
   <td style="text-align:right;background-color: #90EE90;"> -0.7402765 </td>
   <td style="text-align:right;"> -1.1801935 </td>
   <td style="text-align:right;"> -0.0955702 </td>
   <td style="text-align:right;"> -1.0452409 </td>
   <td style="text-align:right;"> 0.2957341 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 9 </td>
   <td style="text-align:right;"> 1.1633767 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.5247243 </td>
   <td style="text-align:right;"> -1.4510800 </td>
   <td style="text-align:right;"> 0.1383595 </td>
   <td style="text-align:right;"> -0.8334814 </td>
   <td style="text-align:right;"> 0.9092710 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 10 </td>
   <td style="text-align:right;"> 0.3588886 </td>
   <td style="text-align:right;background-color: #90EE90;"> -1.3746020 </td>
   <td style="text-align:right;"> -0.0798747 </td>
   <td style="text-align:right;"> 0.6042118 </td>
   <td style="text-align:right;"> -0.6485768 </td>
   <td style="text-align:right;"> 0.1092848 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 11 </td>
   <td style="text-align:right;"> 0.1585778 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.5717858 </td>
   <td style="text-align:right;"> -1.1179175 </td>
   <td style="text-align:right;"> 0.4109562 </td>
   <td style="text-align:right;"> 0.2118042 </td>
   <td style="text-align:right;"> 0.7897882 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 12 </td>
   <td style="text-align:right;"> 0.3136630 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.7874952 </td>
   <td style="text-align:right;"> -0.6955751 </td>
   <td style="text-align:right;"> 0.1958716 </td>
   <td style="text-align:right;"> 0.3913952 </td>
   <td style="text-align:right;"> 0.4845534 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 13 </td>
   <td style="text-align:right;"> 0.0404615 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.9990253 </td>
   <td style="text-align:right;"> -1.2931172 </td>
   <td style="text-align:right;"> 1.1747395 </td>
   <td style="text-align:right;"> 0.4294918 </td>
   <td style="text-align:right;"> 0.5587113 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 14 </td>
   <td style="text-align:right;"> 0.1788236 </td>
   <td style="text-align:right;background-color: #90EE90;"> -0.3799278 </td>
   <td style="text-align:right;"> 1.0026268 </td>
   <td style="text-align:right;"> -0.6759536 </td>
   <td style="text-align:right;"> -1.2857618 </td>
   <td style="text-align:right;"> 0.2832233 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 15 </td>
   <td style="text-align:right;"> 0.3600137 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.3479470 </td>
   <td style="text-align:right;"> -1.3522211 </td>
   <td style="text-align:right;"> 0.8713164 </td>
   <td style="text-align:right;"> -0.3963604 </td>
   <td style="text-align:right;"> 1.0634831 </td>
  </tr>
</tbody>
</table>
]

---

## Exploring multiple parameters together

.center[
<table class="table" style="font-size: 12px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> sample </th>
   <th style="text-align:right;"> beta </th>
   <th style="text-align:right;"> gamma.1. </th>
   <th style="text-align:right;"> gamma.2. </th>
   <th style="text-align:right;"> gamma.3. </th>
   <th style="text-align:right;"> gamma.4. </th>
   <th style="text-align:right;"> sigma_gamma </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.0381603 </td>
   <td style="text-align:right;background-color: #90EE90;"> -0.5866505 </td>
   <td style="text-align:right;"> -1.7717493 </td>
   <td style="text-align:right;"> 0.1951042 </td>
   <td style="text-align:right;"> 0.1842853 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.1286942 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.2626396 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.3976282 </td>
   <td style="text-align:right;"> -0.0257513 </td>
   <td style="text-align:right;"> -1.0893670 </td>
   <td style="text-align:right;"> -0.7284009 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.4963586 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.1140987 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.5158181 </td>
   <td style="text-align:right;"> -0.8393754 </td>
   <td style="text-align:right;"> 1.5553642 </td>
   <td style="text-align:right;"> 0.4570358 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.5793863 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 4 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.8345731 </td>
   <td style="text-align:right;background-color: #90EE90;"> -0.3933963 </td>
   <td style="text-align:right;"> -0.2986723 </td>
   <td style="text-align:right;"> -1.1892198 </td>
   <td style="text-align:right;"> -0.8657181 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.9091757 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 5 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.7405298 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.1480183 </td>
   <td style="text-align:right;"> -0.8649021 </td>
   <td style="text-align:right;"> -1.3578496 </td>
   <td style="text-align:right;"> -0.9145567 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.6320696 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 6 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.6651553 </td>
   <td style="text-align:right;background-color: #90EE90;"> -0.1847768 </td>
   <td style="text-align:right;"> -0.7090246 </td>
   <td style="text-align:right;"> -1.0886230 </td>
   <td style="text-align:right;"> -0.7304895 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.4295265 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 7 </td>
   <td style="text-align:right;background-color: #CCEECC;"> -0.9446587 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.6964131 </td>
   <td style="text-align:right;"> 0.3959616 </td>
   <td style="text-align:right;"> 0.6783410 </td>
   <td style="text-align:right;"> 0.7992607 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.9157164 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 8 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.5688823 </td>
   <td style="text-align:right;background-color: #90EE90;"> -0.7402765 </td>
   <td style="text-align:right;"> -1.1801935 </td>
   <td style="text-align:right;"> -0.0955702 </td>
   <td style="text-align:right;"> -1.0452409 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.2957341 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 9 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 1.1633767 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.5247243 </td>
   <td style="text-align:right;"> -1.4510800 </td>
   <td style="text-align:right;"> 0.1383595 </td>
   <td style="text-align:right;"> -0.8334814 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.9092710 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 10 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.3588886 </td>
   <td style="text-align:right;background-color: #90EE90;"> -1.3746020 </td>
   <td style="text-align:right;"> -0.0798747 </td>
   <td style="text-align:right;"> 0.6042118 </td>
   <td style="text-align:right;"> -0.6485768 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.1092848 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 11 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.1585778 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.5717858 </td>
   <td style="text-align:right;"> -1.1179175 </td>
   <td style="text-align:right;"> 0.4109562 </td>
   <td style="text-align:right;"> 0.2118042 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.7897882 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 12 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.3136630 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.7874952 </td>
   <td style="text-align:right;"> -0.6955751 </td>
   <td style="text-align:right;"> 0.1958716 </td>
   <td style="text-align:right;"> 0.3913952 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.4845534 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 13 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.0404615 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.9990253 </td>
   <td style="text-align:right;"> -1.2931172 </td>
   <td style="text-align:right;"> 1.1747395 </td>
   <td style="text-align:right;"> 0.4294918 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.5587113 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 14 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.1788236 </td>
   <td style="text-align:right;background-color: #90EE90;"> -0.3799278 </td>
   <td style="text-align:right;"> 1.0026268 </td>
   <td style="text-align:right;"> -0.6759536 </td>
   <td style="text-align:right;"> -1.2857618 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.2832233 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 15 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.3600137 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.3479470 </td>
   <td style="text-align:right;"> -1.3522211 </td>
   <td style="text-align:right;"> 0.8713164 </td>
   <td style="text-align:right;"> -0.3963604 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 1.0634831 </td>
  </tr>
</tbody>
</table>
]

probability of preferring bay leaf in batch 1: `$\mbox{logit}^{-1}(\beta + \gamma_1\sigma_{\gamma})$`

---

## Exploring multiple parameters together

.center[
![](rug_lightning_talk_files/figure-html/plot_gamma_correlations-1.png)
]

---

## Exploring multiple parameters together

.center[
<table class="table" style="font-size: 12px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> sample </th>
   <th style="text-align:right;"> beta </th>
   <th style="text-align:right;"> gamma.1. </th>
   <th style="text-align:right;"> gamma.2. </th>
   <th style="text-align:right;"> gamma.3. </th>
   <th style="text-align:right;"> gamma.4. </th>
   <th style="text-align:right;"> sigma_gamma </th>
   <th style="text-align:right;"> batch.1.prob </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.0381603 </td>
   <td style="text-align:right;background-color: #CCEECC;"> -0.5866505 </td>
   <td style="text-align:right;"> -1.7717493 </td>
   <td style="text-align:right;"> 0.1951042 </td>
   <td style="text-align:right;"> 0.1842853 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.1286942 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.4906665 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.2626396 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.3976282 </td>
   <td style="text-align:right;"> -0.0257513 </td>
   <td style="text-align:right;"> -1.0893670 </td>
   <td style="text-align:right;"> -0.7284009 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.4963586 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.6130156 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.1140987 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.5158181 </td>
   <td style="text-align:right;"> -0.8393754 </td>
   <td style="text-align:right;"> 1.5553642 </td>
   <td style="text-align:right;"> 0.4570358 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.5793863 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.6017966 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 4 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.8345731 </td>
   <td style="text-align:right;background-color: #CCEECC;"> -0.3933963 </td>
   <td style="text-align:right;"> -0.2986723 </td>
   <td style="text-align:right;"> -1.1892198 </td>
   <td style="text-align:right;"> -0.8657181 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.9091757 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.6170172 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 5 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.7405298 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.1480183 </td>
   <td style="text-align:right;"> -0.8649021 </td>
   <td style="text-align:right;"> -1.3578496 </td>
   <td style="text-align:right;"> -0.9145567 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.6320696 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.6972185 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 6 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.6651553 </td>
   <td style="text-align:right;background-color: #CCEECC;"> -0.1847768 </td>
   <td style="text-align:right;"> -0.7090246 </td>
   <td style="text-align:right;"> -1.0886230 </td>
   <td style="text-align:right;"> -0.7304895 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.4295265 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.6423983 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 7 </td>
   <td style="text-align:right;background-color: #CCEECC;"> -0.9446587 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.6964131 </td>
   <td style="text-align:right;"> 0.3959616 </td>
   <td style="text-align:right;"> 0.6783410 </td>
   <td style="text-align:right;"> 0.7992607 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.9157164 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.4238614 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 8 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.5688823 </td>
   <td style="text-align:right;background-color: #CCEECC;"> -0.7402765 </td>
   <td style="text-align:right;"> -1.1801935 </td>
   <td style="text-align:right;"> -0.0955702 </td>
   <td style="text-align:right;"> -1.0452409 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.2957341 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.5866072 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 9 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 1.1633767 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.5247243 </td>
   <td style="text-align:right;"> -1.4510800 </td>
   <td style="text-align:right;"> 0.1383595 </td>
   <td style="text-align:right;"> -0.8334814 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.9092710 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.8376020 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 10 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.3588886 </td>
   <td style="text-align:right;background-color: #CCEECC;"> -1.3746020 </td>
   <td style="text-align:right;"> -0.0798747 </td>
   <td style="text-align:right;"> 0.6042118 </td>
   <td style="text-align:right;"> -0.6485768 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.1092848 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.5519779 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 11 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.1585778 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.5717858 </td>
   <td style="text-align:right;"> -1.1179175 </td>
   <td style="text-align:right;"> 0.4109562 </td>
   <td style="text-align:right;"> 0.2118042 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.7897882 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.6479790 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 12 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.3136630 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.7874952 </td>
   <td style="text-align:right;"> -0.6955751 </td>
   <td style="text-align:right;"> 0.1958716 </td>
   <td style="text-align:right;"> 0.3913952 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.4845534 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.6671330 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 13 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.0404615 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.9990253 </td>
   <td style="text-align:right;"> -1.2931172 </td>
   <td style="text-align:right;"> 1.1747395 </td>
   <td style="text-align:right;"> 0.4294918 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.5587113 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.6453424 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 14 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.1788236 </td>
   <td style="text-align:right;background-color: #CCEECC;"> -0.3799278 </td>
   <td style="text-align:right;"> 1.0026268 </td>
   <td style="text-align:right;"> -0.6759536 </td>
   <td style="text-align:right;"> -1.2857618 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.2832233 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.5177973 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 15 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.3600137 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 0.3479470 </td>
   <td style="text-align:right;"> -1.3522211 </td>
   <td style="text-align:right;"> 0.8713164 </td>
   <td style="text-align:right;"> -0.3963604 </td>
   <td style="text-align:right;background-color: #CCEECC;"> 1.0634831 </td>
   <td style="text-align:right;background-color: #90EE90;"> 0.6748161 </td>
  </tr>
</tbody>
</table>
]

probability of preferring bay leaf in batch 1: `$\mbox{logit}^{-1}(\beta + \gamma_1\sigma_{\gamma})$`

---

## Exploring multiple parameters together

.center[
![](rug_lightning_talk_files/figure-html/plot_batch_intercepts-1.png)
]

---
class: middle

# Working with samples transforms a problem of integral calculus into a problem of data summary.

Richard McElreath, _Statistical Rethinking: A Bayesian Course with Examples in R and Stan_, p. 68

---
class: middle

GitHub repository:

https://github.com/kaplanas/Bay-Leaf-Experiment

GitPages write-up:

https://kaplanas.github.io/Bay-Leaf-Experiment/bay_leaf_analysis.html

this presentation:

https://kaplanas.github.io/Bay-Leaf-Experiment/rug_lightning_talk.html