Introduction to Computational Bayesian Methods for Actuaries

# Introduction to Computational Bayesian Methods for Actuaries
### Michael Jones
### 10 October 2018

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<span style="font-size:600%">
`$$P(\theta | D) = \frac{P(D|\theta)P(\theta)}{\int d\theta' P(D|\theta')P(\theta')}$$`
</span>

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<span style="font-size:600%">
`$$\color{lightgrey}{P(\theta | D) = \frac{P(D|\theta)\color{black}{P(\theta)}}{\int d\theta' P(D|\theta')P(\theta')}}$$`
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<span style="font-size:600%">
`$$\color{lightgrey}{P(\theta | D) = \frac{\color{black}{P(D|\theta)}{P(\theta)}}{\int d\theta' P(D|\theta')P(\theta')}}$$`
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<span style="font-size:600%">
`$$\color{lightgrey}{\color{black}{P(\theta | D)} = \frac{P(D|\theta){P(\theta)}}{\int d\theta' P(D|\theta')P(\theta')}}$$`
</span>

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<span style="font-size:600%">
`$$\color{lightgrey}{P(\theta | D) = \frac{P(D|\theta){P(\theta)}}{\color{black}{\int d\theta' P(D|\theta')P(\theta')}}}$$`
</span>

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???

It produces results that are testable and given the right models generally are as good (sometimes better) than frequentist analysis.

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Most Bayesian analysis stems from a few foundational ideas compared to frequentist statistics which has a much more varied set of fundamentals.

Posterior information is rich and useful

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???

Humans naturally think in Bayesian terms which are then 'unlearned' when studying frequentist analysis.

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Through the priors, assumptions are formally absorbed into the analysis and cannot be hidden as in classical statistics.

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Multilevel models, choice of priors, e.g. simple to make robust to outliers

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A whole posterior distribution can be difficult to communicate/process

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ok if conjugate pairs, but otherwise, it's difficult to do without MCMC or other computationally intensive programs.

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It's not what people are used to, and the themes are unfamiliar, though it's gaining traction in social sciences and pharmacology (and election prediction)

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- Identify your data
- Define a descriptive model
- Specify a prior
- Compute the Posterior
- Interpret the Posterior
- Check the model is reasonable

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##Parameter

<span style="font-size:400%">
`$$\theta = P(y_i = \text{heads})$$`
</span>

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## Likelihood

### Bernoulli:

<span style="font-size:400%">
$$ p(y|\theta) = \theta^y(1 - \theta)^{(1-y)}$$
</span>

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## Prior
### Beta
<span style="font-size:400%">
`$$\begin{align}
  P(\theta|a,b) &= \text{beta}(a,b)\\\\
  &=\frac{\theta^{(a-1)}(1-\theta)^{(b-1)}}{B(a,b)}
  \end{align}$$`
</span>

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## Posterior
### Another Beta

<span style="font-size:300%">
`$$\begin{align}
  P(\theta|n_{heads},n_{tails},a,b) &= \text{beta}(a+n_{heads},b+n_{tails})\\\\
  &=\frac{\theta^{(a+n_{heads}-1)}(1-\theta)^{(b+n_{tails}-1)}}{B(a+n_{heads},b+n_{tails})}
  \end{align}$$`
</span>

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# Live demo time:

https://mjones.shinyapps.io/coin/

click [here](#47)
]

click [here](#43)
]

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# No idea about the coin
Flat `$\text{beta}(1,1)$` distribution:

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# Collect some data

7 Heads, 5 tails:

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# Posterior

Another Beta: `$Beta(6,8)$`

# Strong Prior

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???

- Conjugate priors
- calculating the normalisation integral

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- If the posterior is narrower, you waste a lot of time/computing power calculating the posterior in places which do not matter
- Not only that, but as your dimensions increase (i.e. number of parameters), the number of points increases exponentially.

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class:middle

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# Metropolis Algorithm
![](metropolis_pics/1_current_place.svg)

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# Metropolis Algorithm
![](metropolis_pics/2_choices.svg)

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# Metropolis Algorithm
![](metropolis_pics/3_chosen.svg)

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# Metropolis Algorithm
![](metropolis_pics/4_definitely_move.svg)

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# Metropolis Algorithm
![](metropolis_pics/5_maybe_move.svg)
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# Metropolis Algorithm Example

- 50 Islands

- Populations in ratio 1:2:3:...:50

- Want to visit each in accordance with its population

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# Two Coins
![](two_coins.svg)

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class: center, middle

```
## Markov Chain Monte Carlo (MCMC) output:
## Start = 501 
## End = 531 
## Thinning interval = 1 
##        theta[1]  theta[2]
##  [1,] 0.5331603 0.7960773
##  [2,] 0.7224562 0.4499415
##  [3,] 0.7102999 0.2461300
##  [4,] 0.5370162 0.5314258
##  [5,] 0.5014244 0.2403774
##  [6,] 0.8557981 0.5679354
##  [7,] 0.4511761 0.5866845
##  [8,] 0.5826508 0.4369579
##  [9,] 0.6994008 0.3722830
## [10,] 0.3815623 0.3713527
## [11,] 0.8098616 0.3163093
## [12,] 0.5554198 0.4975122
## [13,] 0.4028894 0.3134066
## [14,] 0.3785868 0.3637059
## [15,] 0.4323854 0.5251539
## [16,] 0.7882131 0.4448705
## [17,] 0.6833106 0.4087616
## [18,] 0.6819669 0.4665725
## [19,] 0.7579478 0.3968828
## [20,] 0.5021557 0.4240982
## [21,] 0.7010325 0.4487093
## [22,] 0.5308746 0.4451829
## [23,] 0.6973850 0.4182190
## [24,] 0.5973255 0.5136492
## [25,] 0.8815459 0.3668754
## [26,] 0.7933463 0.4132620
## [27,] 0.6295374 0.1937704
## [28,] 0.7943369 0.5278090
## [29,] 0.5191301 0.3335732
## [30,] 0.5250645 0.4350766
## [31,] 0.4926411 0.3410629
```

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---

<table>
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   <th style="text-align:left;"> AY </th>
   <th style="text-align:right;"> premium </th>
   <th style="text-align:right;"> 6 </th>
   <th style="text-align:right;"> 18 </th>
   <th style="text-align:right;"> 30 </th>
   <th style="text-align:right;"> 42 </th>
   <th style="text-align:right;"> 54 </th>
   <th style="text-align:right;"> 66 </th>
   <th style="text-align:right;"> 78 </th>
   <th style="text-align:right;"> 90 </th>
   <th style="text-align:right;"> 102 </th>
   <th style="text-align:right;"> 114 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 1991 </td>
   <td style="text-align:right;"> 10,000 </td>
   <td style="text-align:right;"> 358 </td>
   <td style="text-align:right;"> 1,125 </td>
   <td style="text-align:right;"> 1,735 </td>
   <td style="text-align:right;"> 2,183 </td>
   <td style="text-align:right;"> 2,746 </td>
   <td style="text-align:right;"> 3,320 </td>
   <td style="text-align:right;"> 3,466 </td>
   <td style="text-align:right;"> 3,606 </td>
   <td style="text-align:right;"> 3,834 </td>
   <td style="text-align:right;"> 3,901 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 1992 </td>
   <td style="text-align:right;"> 10,400 </td>
   <td style="text-align:right;"> 352 </td>
   <td style="text-align:right;"> 1,236 </td>
   <td style="text-align:right;"> 2,170 </td>
   <td style="text-align:right;"> 3,353 </td>
   <td style="text-align:right;"> 3,799 </td>
   <td style="text-align:right;"> 4,120 </td>
   <td style="text-align:right;"> 4,648 </td>
   <td style="text-align:right;"> 4,914 </td>
   <td style="text-align:right;"> 5,339 </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 1993 </td>
   <td style="text-align:right;"> 10,800 </td>
   <td style="text-align:right;"> 291 </td>
   <td style="text-align:right;"> 1,292 </td>
   <td style="text-align:right;"> 2,219 </td>
   <td style="text-align:right;"> 3,235 </td>
   <td style="text-align:right;"> 3,986 </td>
   <td style="text-align:right;"> 4,133 </td>
   <td style="text-align:right;"> 4,629 </td>
   <td style="text-align:right;"> 4,909 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 1994 </td>
   <td style="text-align:right;"> 11,200 </td>
   <td style="text-align:right;"> 311 </td>
   <td style="text-align:right;"> 1,419 </td>
   <td style="text-align:right;"> 2,195 </td>
   <td style="text-align:right;"> 3,757 </td>
   <td style="text-align:right;"> 4,030 </td>
   <td style="text-align:right;"> 4,382 </td>
   <td style="text-align:right;"> 4,588 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 1995 </td>
   <td style="text-align:right;"> 11,600 </td>
   <td style="text-align:right;"> 443 </td>
   <td style="text-align:right;"> 1,136 </td>
   <td style="text-align:right;"> 2,128 </td>
   <td style="text-align:right;"> 2,898 </td>
   <td style="text-align:right;"> 3,403 </td>
   <td style="text-align:right;"> 3,873 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 1996 </td>
   <td style="text-align:right;"> 12,000 </td>
   <td style="text-align:right;"> 396 </td>
   <td style="text-align:right;"> 1,333 </td>
   <td style="text-align:right;"> 2,181 </td>
   <td style="text-align:right;"> 2,986 </td>
   <td style="text-align:right;"> 3,692 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 1997 </td>
   <td style="text-align:right;"> 12,400 </td>
   <td style="text-align:right;"> 441 </td>
   <td style="text-align:right;"> 1,288 </td>
   <td style="text-align:right;"> 2,420 </td>
   <td style="text-align:right;"> 3,483 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 1998 </td>
   <td style="text-align:right;"> 12,800 </td>
   <td style="text-align:right;"> 359 </td>
   <td style="text-align:right;"> 1,421 </td>
   <td style="text-align:right;"> 2,864 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 1999 </td>
   <td style="text-align:right;"> 13,200 </td>
   <td style="text-align:right;"> 377 </td>
   <td style="text-align:right;"> 1,363 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
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  <tr>
   <td style="text-align:left;"> 2000 </td>
   <td style="text-align:right;"> 13,600 </td>
   <td style="text-align:right;"> 344 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
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---
# The model
See [here](https://magesblog.com/post/2015-11-10-hierarchical-loss-reserving-with-stan/)
`$$\begin{align}
CL_{AY,dev} &\sim \mathcal{N}(\mu_{AY, dev}, \sigma^2_{dev})\\
\mu_{AY,dev} &= ULT_{AY} \cdot G(dev|\omega, \theta)\\
\sigma_{dev}&=\sigma \sqrt{\mu_{dev}}\\
ULT_{AY} &\sim \mathcal{N}(\mu_{ult}, \sigma^2_{ult})\\
G(dev|\omega,\theta) &=1-\exp\left(-\left(\frac{dev}{\theta}\right)^{\omega}\right)
\end{align}$$`
---

# Stan model output

```
## Inference for Stan model: MultiLevelGrowthCurve.
## 4 chains, each with iter=7000; warmup=2000; thin=2; 
## post-warmup draws per chain=2500, total post-warmup draws=10000.
## 
##              mean se_mean     sd     50%     75%   97.5% n_eff Rhat
## mu_ult    5355.32    5.74 275.18 5344.66 5532.70 5914.99  2300    1
## omega        1.30    0.00   0.03    1.30    1.32    1.36  2933    1
## theta       47.50    0.06   2.51   47.32   49.02   52.91  1712    1
## sigma_ult  595.88    2.44 170.08  567.25  682.87 1014.74  4868    1
## sigma        3.08    0.01   0.33    3.05    3.28    3.82  4228    1
## 
## Samples were drawn using NUTS(diag_e) at Tue Oct  9 18:44:13 2018.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
```

---

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# Shinystan
Available [here](https://mjones.shinyapps.io/staninsurance/)

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# References and Further Reading

- *Doing Bayesian Data Analysis* by John K. Kruschke
- *Computational Actuarial Science* edited by Arthur Charpentier
- *Bayesian Data Analysis (3rd edition)* by Andrew Gelman *et al*
- Markus Gesmann's Blog (https://magesblog.com/)
- Arthur Charpentier's blog (https://freakonometrics.hypotheses.org/)

# Interactive

- http://mjones.shinyapps.io/coin/
- http://mjones.shinyapps.io/staninsurance/