beginners guide bayesian

8/18/2019 Beginners Guide Bayesian

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A Beginner’s Guide to BayesianModelling

Peter England, PhD

EMB

GIRO 2002



Outline

• An easy one paraeter pro!le

• A harder one paraeter pro!le

• Pro!les "ith ultiple paraeters

• Modelling in #inB$G%

• %to&hasti& 'lais Reser(ing• Paraeter un&ertainty in D)A



Bayesian Modelling* General %trategy

• %pe&i+y distri!ution +or the data

• %pe&i+y prior distri!utions +or the paraeters

• #rite do"n the oint distri!ution

• 'olle&t ters in the paraeters o+ interest• Re&ognise the -&onditional. posterior distri!ution/

1es* Estiate the paraeters, or saple dire&tly

o* %aple using an appropriate s&hee

• )ore&asting* Re&ognise the predi&ti(e distri!ution/ 1es* Estiate the paraeters

o* %iulate an o!ser(ation +ro the data distri!ution,&onditional on the siulated paraeters



A One Paraeter Pro!le

• Data %aple 34,5,6,7,6,5,8,5,9,4:

• Distri!uted as a Poisson rando (aria!le/

• $se a Gaa prior +or the ean o+ the

Poisson

• Predi&ting a ne" o!ser(ation/

• egati(e Binoial predi&ti(e distri!ution



Poisson E;aple < Estiation

( )

( )( )

λθ α α θ

θ α

λ θ θ θ

λ α θ

θ

−−

=

−

Γ

=∝

Γ

∏ e y

e y f y f

IPoi y

n

i i

y

i

i<

< =.,->

,?

.-?

( )θ λ α θ n y ei +−−+∑∝ <

( )n yi ++Γ ∑ λ α θ ,?



Poisson E;aple < Predi&tion

( )

<

<<

<<

?

<<

<

<

<

< ,

.,-Binoial egati(e??

,

<

<

<.<?-.-

.?-,>?

<

λ

λ α

λ λ α α

λ λ

λ

α

α θ

α

+==

+=+=

+

++Γ Γ

+Γ =

∑

p x

p x y

n y

y

y y y f

i

y



One Paraeter Pro!le*

%iple 'ase

• #e &an re&ognise the posterior distri!ution

o+ the paraeter • #e &an re&ognise the predi&ti(e distri!ution

• o siulation re@uired

• -#e &an use siulation i+ "e "ant to.



aria!ility o+ a +ore&ast

• In&ludes estiation (arian&e and pro&ess(arian&e

• Analyti& solution* estiate the t"o &oponents

• Bayesian solution* siulate the paraeters, then

siulate the +ore&ast &onditional on the paraeters

2<

(arian&e.estiation(arian&e-pro&esserror predi&tion +=



Main )eatures o+ Bayesian

Analysis• )o&us is on distri!utions -o+ paraeters or

+ore&asts., not ust point estiates• he ode o+ posterior or predi&ti(e

distri!utions is analogous to Ca;iu

lielihood in &lassi&al statisti&s



One Paraeter Pro!le*

Farder 'ase• $se a log lin !et"een the ean and the

paraeter, that is*

• $se a noral distri!ution +or the prior

• #hat is the posterior distri!ution/

• Fo" do "e siulate +ro it/

Mean eθ

=



Poisson E;aple 2 Estiation

( )

( )

( )

2

2<

2

2

<

densitylog

2<

=.,->

.,-?

.-?

2

2

2

2

−

−−∝

∑∝

=∝

∑

∏

−−

−

−−

=

−

σ

µ θ θ

π σ θ θ

σ µ θ

θ

σ

µ θ

θ

σ

µ θ θ

θ

θ

θ

ne y

eee

e y

ee y f y f

N

e IPoi y

i

ne

y

n

i i

e y

i

i

i



Poisson E;aple 2

• %tep <* $se adapti(e ree&tion sapling

-AR%. +ro log density to saple the

paraeter

• %tep 2* )or predi&tion, saple +ro a

Poisson distri!ution "ith ean , "ith

theta siulated at step <

θ e



A MultiParaeter Pro!le

• )ro %&ollni -AAH, 200<.

• 4 Group "orers &opensation poli&ies

• E;posure easured using payroll as a pro;y

• u!er o+ &lais a(aila!le +or ea&h o+ last

8 years

• Pro!le is to des&ri!e &lai +re@uen&ies in

the +ore&ast year



%&ollni E;aple <

Year Group 1 Group 2 Group 3 Group 1 Group 2 Group 3 Group 1 Group 2 Group 3

1 280 260 9 6 0.032 0.023

2 320 275 145 7 4 8 0.022 0.015 0.055

3 265 240 120 6 2 3 0.023 0.008 0.025

4 340 265 105 13 8 4 0.038 0.030 0.038

5 285 115

Average 0.029 0.019 0.039

Payroll P(i,j) lai!" #(i,j) Pro$a$ili%ie"

.<,26-?

.6,6-?

.,-?

.-?

Gamma

Gamma

Gamma

P Poi X

i

iijij

β

α

β α θ

θ



%&ollni E;aple <

Posterior Distri!utions

( )

( )

( )

( )

α α α

α θ α

β β θ θ θ α

θ α α θ θ θ β

β α β α θ θ θ



684

<

4

42<

42<

442<4

224<2

<<42<

.-.,,,,>-

<,284?,,,,>

,<?,,,,>

,<?,,,,>

,<?,,,,>

−

=

Γ ∝

++

+++

+++

+++

∏

∑∑ ∑∑ ∑

∑ ∑

e X f

Gamma X

P X Gamma X

P X Gamma X

P X Gamma X

i

i

i

j j

j j

j j



%&ollni E;aple <

• $se Gi!!s %apling

Iterate through ea&h paraeter in turn

%aple +ro the &onditional posterior

distri!ution, treating the other paraeters as

+i;ed

• %apling is easy +or• $se AR% +or

β θ θ θ ,,, 42<

α



#inB$G%

• #inB$G% is an e;pert syste +or Bayesian analysis

• 1ou spe&i+y he distri!ution o+ the data

he prior distri!utions o+ the paraeters• #inB$G% "ors out the &onditional posterior

distri!utions

• #inB$G% de&ides ho" to saple the paraeters

• #inB$G% uses Gi!!s sapling +or ultiple paraeter pro!les



%to&hasti& 'lais Reser(ing

• 'hanges the +o&us +ro a C!est estiate o+ reser(esto a predi&ti(e distri!ution o+ outstanding lia!ilities

• Most sto&hasti& ethods to date ha(e only&onsidered 2nd oent properties -(arian&e. in

addition to a C!est estiate• Bayesian ethods &an !e used to in(estigate a +ull

predi&ti(e distri!ution, and in&orporate udgeent-through the &hoi&e o+ priors.

• )or ore in+oration, see England and errall -BAH,2002.



he Bornhuetter)erguson Method

• $se+ul "hen the data are unsta!le

• )irst get an initial estiate o+ ultiate

• Estiate &hainladder de(elopent +a&tors

• Apply these to the initial estiate o+

ultiate to get an estiate o+ outstanding

&lais



'on&eptual )rae"or

P r e & i ' % i v e i " % r i $ u % i o

* a r i a $ i l i % y( P r e & i ' % i o + r r o r )

, e " e r v e e " % i ! a % e

( - e a " u r e o . l o ' a % i o )



10000 14000 18000 22000 26000 30000 34000

/o%al ,e"erve"

igure 1. Pre&i'%ive Aggrega%e (i"%ri$u%io) o /o%al ,e"erve"



Estiates o+ outstanding &lais

o estiate ultiate &lais using the &hain ladder te&hni@ue, you"ould ultiply the latest &uulati(e &lais in ea&h ro" !y f , a

produ&t o+ de(elopent +a&tors

Fen&e, an estiate o+ "hat the latest &uulati(e &lais should !e iso!tained !y di(iding the estiate o+ ultiate !y f %u!tra&ting this

+ro the estiate o+ ultiate gi(es an estiate o+ outstanding

&lais*

<Estiated $ltiate <

f

× − ÷



he Bornhuetter)erguson Method

Jet the initial estiate o+ ultiate &lais +or

a&&ident year i !e

he estiate o+ outstanding &lais +or a&&ident

year i is

−

+−+− ninin

i M

λ λ λ

42

<<

( )<<

42

42

−= +−+−+−+−

ninin

ninin

i M λ λ λ

λ λ λ

i M



'oparison "ith 'hainladder

repla&es the latest &uulati(e

&lais +or a&&ident year i, to "hi&h the usual &hainladder

paraeters are applied to o!tain the estiate o+ outstanding

&lais )or the &hainladder te&hni@ue, the estiate o+ outstanding

&lais is

ninin

i M

λ λ λ

42

<

+−+−

( )<42<, −+−+−+− nininini D λ λ λ



Multipli&ati(e Model +or 'hainJadder

( )<

? - .

- .

"ith <

is the e;pe&ted ultiate +or origin yearis the proportion paid in de(elopent year

ij ij

ij ij ij

n

ij i j k

k

i

j

C IPoi

C

E C x y y

x i y j

µ

µ η

=

Ε = =

= =∑



B) as a Bayesian Model

Put a prior distri!ution on the ro" paraeters

he Bornhuetter)erguson ethod assues there

is prior no"ledge a!out these paraeters, andthere+ore uses a Bayesian approa&h he prior

in+oration &ould !e suarised as the

+ollo"ing prior distri!utions +or the ro"

paraeters*

( )iii x β α ,tindependen? Γ



B) as a Bayesian Model

• $sing a per+e&t prior -(ery sall (arian&e.

gi(es results analogous to the B) ethod

• $sing a (ague prior -(ery large (arian&e.

gi(es results analogous to the standard

&hain ladder odel

• In a Bayesian &onte;t, un&ertainty

asso&iated "ith a B) prior &an !e

in&orporated



Paraeter $n&ertainty in D)A

• O+ten, in D)A, +ore&asts are o!tained usingsiulation, assuing the underlying

paraeters are +i;ed -+or e;aple, a

standard appli&ation o+ #ilie’s odel.• In&luding paraeter un&ertainty ay not !e

straight+or"ard in the a!sen&e o+ a Bayesian

+rae"or, "hi&h in&ludes it naturally• Ignoring paraeter un&ertainty "ill

underestiate the true un&ertainty=



%uary

• Bayesian odelling using siulationethods &an !e used to +it &ople; odels

• )o&us is on distri!utions o+ paraeters or+ore&asts

• Mode is analogous to Ca;iu

lielihood• It is a natural "ay to in&lude paraeter

un&ertainty "hen +ore&asting -eg in D)A.



Re+eren&es

%&ollni, DPM -200<. Actuarial Modeling wit MCMC

and !"G# , orth Aeri&an A&tuarial Hournal, 6 -2., pages

7K<28

England, PD and errall, RH -2002. #toca$tic Claim$

%e$er&ing in General In$urance, British A&tuarial Hournal

olue 5 Part II -to appear.

%piegelhalter, DH, hoas, A and Best, G -<777.,'in!"G# (er$ion )*+ "$er Manual, MR' Biostatisti&s

$nit

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