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Introduction to Computational Bayesian Methods for Actuaries

Michael Jones

10 October 2018

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Actuaries
should be
scientific

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Actuaries
should be
Bayesian

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What?

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Why?

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How?

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Bayesian
Recap

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P(θ|D)=P(D|θ)P(θ)∫dθ′P(D|θ′)P(θ′)

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P(θ|D)=P(D|θ)P(θ)∫dθ′P(D|θ′)P(θ′)

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P(θ|D)=P(D|θ)P(θ)∫dθ′P(D|θ′)P(θ′)

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P(θ|D)=P(D|θ)P(θ)∫dθ′P(D|θ′)P(θ′)

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P(θ|D)=P(D|θ)P(θ)∫dθ′P(D|θ′)P(θ′)

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Why be
Bayesian

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It works

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

It's coherent

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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

It's natural

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

It's self
documenting

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

You get the
full Posterior

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It's modular

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

Why not be
Bayesian

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What
prior?

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It's
awkward

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A whole posterior distribution can be difficult to communicate/process

It's
hard

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

It's
weird

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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)

Frequentism
vs
Bayesianism

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Probability

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Data

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Parameters

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Ranges

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0.100 0.125 0.150 0.175 0.200 100 draws from a Binomial with mean 15 Frequentist Confidence Intervals
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95% HDI: 0.05 to 0.57 0 200 400 600 800 0.00 0.25 0.50 0.75 y count Bayesian Highest Density Intervals
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p-values &
significance
testing

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How to be
Bayesian

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The Bayesian
Workflow

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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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Analytically:
Estimating the
Bias of a coin

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Parameter

θ=P(yi=heads)

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Likelihood

Bernoulli:

p(y|θ)=θy(1−θ)(1−y)

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Prior

Beta

P(θ|a,b)=beta(a,b)=θ(a−1)(1−θ)(b−1)B(a,b)

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20, 0.1 20, 1 20, 2 20, 3 20, 4 20, 20 4, 0.1 4, 1 4, 2 4, 3 4, 4 4, 20 3, 0.1 3, 1 3, 2 3, 3 3, 4 3, 20 2, 0.1 2, 1 2, 2 2, 3 2, 4 2, 20 1, 0.1 1, 1 1, 2 1, 3 1, 4 1, 20 0.1, 0.1 0.1, 1 0.1, 2 0.1, 3 0.1, 4 0.1, 20 θ y
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Posterior

Another Beta

P(θ|nheads,ntails,a,b)=beta(a+nheads,b+ntails)=θ(a+nheads−1)(1−θ)(b+ntails−1)B(a+nheads,b+ntails)

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Live demo time:

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

If it works

click here

If it doesn't work

click here

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No idea about the coin

Flat beta(1,1) distribution:

0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 θ Prior
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Collect some data

7 Heads, 5 tails:

0e+00 1e-04 2e-04 3e-04 0.00 0.25 0.50 0.75 1.00 theta likelihood
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Posterior

Another Beta: Beta(6,8)

0e+00 2e-05 4e-05 6e-05 0.00 0.25 0.50 0.75 1.00 theta posterior
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Strong Prior

prior likelihood posterior 0.00 0.25 0.50 0.75 1.00 0 2 4 6 8 0e+00 1e-04 2e-04 3e-04 0 2 4 theta y
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Problems

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  • Conjugate priors
  • calculating the normalisation integral

What prior
to use?

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Normalising

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Computationally:
First Steps

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0 1 2 0.00 0.25 0.50 0.75 1.00 x y
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0 1 2 0.00 0.25 0.50 0.75 1.00 x y
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0 1 2 0.00 0.25 0.50 0.75 1.00 x y
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0 1 2 0.00 0.25 0.50 0.75 1.00 x y
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0 10 20 0.00 0.25 0.50 0.75 1.00 x y
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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.

Computationally:
MCMC

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Markov Chain
Monte Carlo

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Markov Chain
Monte Carlo

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Markov Chain
Monte Carlo

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Metropolis

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Metropolis Algorithm

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Metropolis Algorithm

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Metropolis Algorithm

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Metropolis Algorithm

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Metropolis Algorithm

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Metropolis Algorithm Example

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Metropolis Algorithm Example

  • 50 Islands
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Metropolis Algorithm Example

  • 50 Islands

  • Populations in ratio 1:2:3:...:50

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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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beginning end 0 100 200 300 400 500 999500 999600 999700 999800 999900 1000000 0 10 20 30 40 50 Step Number Island Number
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1000 10000 1e+06 100 200 300 1 10 20 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.03 0.06 0.09 0.00 0.01 0.02 0.03 0.04 0.0 0.1 0.2 0.3 0.4 0.00 0.05 0.10 0.15 0.00 0.01 0.02 0.03 0.04 0.00 0.25 0.50 0.75 1.00 0.00 0.05 0.10 0.15 0.00 0.01 0.02 0.03 0.04 Step Number Visiting proportion
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Don't make
your own

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Computationally:
In practice

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Two Coins

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## 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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theta[1] 0.2 0.4 0.6 0.8 1.0 m o d e = 0.657 95 % HDI 0.401 0.872 + theta[1]-theta[2] -0.4 -0.2 0.0 0.2 0.4 0.6 m o d e = 0.239 11.8 % < 0 < 88.2 % 95 % HDI -0.134 0.551 + 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.4 0.6 0.8 theta[1] theta[2] theta[2] 0.2 0.4 0.6 0.8 m o d e = 0.424 95 % HDI 0.19 0.682 +
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Bayesianism
applied to
Insurance

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AY premium 6 18 30 42 54 66 78 90 102 114
1991 10,000 358 1,125 1,735 2,183 2,746 3,320 3,466 3,606 3,834 3,901
1992 10,400 352 1,236 2,170 3,353 3,799 4,120 4,648 4,914 5,339
1993 10,800 291 1,292 2,219 3,235 3,986 4,133 4,629 4,909
1994 11,200 311 1,419 2,195 3,757 4,030 4,382 4,588
1995 11,600 443 1,136 2,128 2,898 3,403 3,873
1996 12,000 396 1,333 2,181 2,986 3,692
1997 12,400 441 1,288 2,420 3,483
1998 12,800 359 1,421 2,864
1999 13,200 377 1,363
2000 13,600 344
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1000 2000 3000 4000 5000 30 60 90 Development Period Amount 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
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The model

See here CLAY,dev∼N(μAY,dev,σ2dev)μAY,dev=ULTAY⋅G(dev|ω,θ)σdev=σ√μdevULTAY∼N(μult,σ2ult)G(dev|ω,θ)=1−exp(−(devθ)ω)

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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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ult.9. ult.10. ult.5. ult.6. ult.7. ult.8. ult.1. ult.2. ult.3. ult.4. 4000 5000 6000 7000 4000 5000 6000 7000 8000 4500 5000 5500 6000 4500 5000 5500 6000 6500 5000 5500 6000 6500 7000 5000 6000 7000 8000 3900 4200 4500 4800 4800 5200 5600 6000 6400 5000 5500 6000 6500 5000 5500 6000 6500 0.0000 0.0005 0.0010 0.0015 0e+00 3e-04 6e-04 9e-04 0.0000 0.0005 0.0010 0.0015 0.0020 0e+00 5e-04 1e-03 0.0000 0.0005 0.0010 0.0015 0.0020 0.0000 0.0005 0.0010 0.0015 0e+00 2e-04 4e-04 6e-04 0.000 0.001 0.002 0.003 0.0000 0.0005 0.0010 0.0015 0.00000 0.00025 0.00050 0.00075 0.00100 value density
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1999 2000 1995 1996 1997 1998 1991 1992 1993 1994 0 50 100 0 50 100 0 50 100 0 50 100 0 2000 4000 6000 0 2000 4000 6000 0 2000 4000 6000 dev cum
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Shinystan

Available here

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Further
Topics

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Multilevel
Models

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Bayesian
Model
Averaging

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Causal
Networks

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References and Further Reading

Interactive

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Actuaries
should be
scientific

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