D.2 Bayesian Credibility - Outline
D.2.1 Discrete Bayesian Credibility
Terminology
Calculator Approach
Exercises
D.2 Bayesian Credibility
D.2.1 Discrete Bayesian Credibility
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Example
An insurance company classifies its customers as high risk, medium
risk, and low risk.
The annual number of claims N per customer has a Poisson
distribution, with mean 6 for high risk individuals, 2 for medium risk,
and 1 for low risk.
30% of customers are high risk, while 20% are low risk.
A randomly selected insured filed 2 claims last year.
What is the probability that he is high risk?
Before (prior to) looking at the data, P[High] = 0.30
This is called the prior probability
We want P[High | N1 = 2]
The probability after (post) looking at data is called the posterior
probability
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Example (Continued)
Class
Low
Med
High
Prior
0.20
0.50
0.30
P[N1 = 2 | Class]
0.5e−1
2e−2
18e−6
Prior·P[N = 2 | Class]
0.0368
0.1353
0.0134
Posterior
0.198
0.730
0.072
Using Bayes’ Theorem
P[High, N1 = 2]
P[N1 = 2]
P[High] · P[N1 = 2 | High]
= P
P[Class] · P[N1 = 2 | Class]
0.3 · 18e−6
=
0.2 · 0.5e−1 + 0.5 · 2e−2 + 0.3 · 18e−6
= 0.072
P[High | N1 = 2] =
D.2 Bayesian Credibility
D.2.1 Discrete Bayesian Credibility
Example
Consider the same situation as before:
Class Prior E[N | Class] Posterior
Low
0.20
1
0.198
Med
0.50
2
0.730
High 0.30
6
0.072
a) Before looking at any data, what is the expected number of
claims filed by a randomly chosen insured?
X
E[N ] =
E[N | Class] · Prior prob of class
= 1 · 0.2 + 2 · 0.5 + 6 · 0.3 = 3.0
b) Given that an insured had 2 claims last year, what is the
expected number that will be filed this year?
X
E[N2 | N1 = 2] =
E[N2 | Class] · P[Class | N1 = 2]
X
=
E[N2 | Class] · Posterior prob of class
= 1 · 0.198 + 2 · 0.730 + 6 · 0.072 = 2.09
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Textbook Terminology
Before/prior to looking at data:
I
P[Class] = Prior distribution
I
P[N = k | Class] = conditional distribution
I
P[N = k] = unconditional distribution
After/post looking at data:
I
P[Class | N = k] = posterior distribution
I
Conditional distribution P[N = k | Class] is unchanged
I
P[N2 = n | N1 = k] = predictive distribution
I
E[N2 | N1 ] = predictive mean = Bayesian credibility estimate
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Exam Wordings
Suppose we want P[Class | N1 = k].
I
Exam questions will ask for “posterior probability that ... ”
If we want P[N2 = j | N1 = k].
I
Exam questions should ask for “predictive probability that
there will be j claims in Year 2”
I
They occasionally ask for “posterior probability that there will
be j claims in Year 2”
If we want E[N2 | N1 = k].
I
Should ask for Bayesian credibility premium or Bayesian
estimate for year 2
I
Or for predictive mean in year 2
I
May ask for posterior mean of number of claims in year 2.
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Calculator Approach
The MultiView can do the tedious part for us. In our example:
Class Prior E[N | Class] P[N = 2 | Class] Prior · Conditional
L1
L2
old L3
new L3
0.0368
Low
0.20
1
0.1839
Med
0.50
2
0.2707
0.1353
0.0134
High 0.30
6
0.0446
P
a) Find E[N1 ] = Prior · E[N | Class]
I Set Data=L2, FRQ=L1, find x
P
b) Find P[N1 = 2] = Prior · P[N = 2 | Class]
I In L3, sto = e−L2 · L22 /2!
I Set Data=L3, FRQ=L1, find x
c) E[N2 | N1 = 2]
I Ideally, we would let L4 = L1 · L3
I Change L3 formula to L1 · e−L2 · L22 /2!
I Set Data=L2, FRQ=L3, find x
I The MultiView will renormalize our frequency for us!
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Exercise 1
The number of monthly claims N are independent binomial variables
with m = 3 and q = 0.2 for low risk individuals, 0.4 for medium risk,
and 0.7 for high risk. 50% of individuals are low risk, 30% are medium
risk, and 20% are high risk.
A randomly selected insured filed 4 claims in the last 2 months. What
is the expected number of claims he will file over the next 3 months?
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Exercise 1
The number of monthly claims N are independent binomial variables
with m = 3 and q = 0.2 for low risk individuals, 0.4 for medium risk,
and 0.7 for high risk. 50% of individuals are low risk, 30% are medium
risk, and 20% are high risk.
A randomly selected insured filed 4 claims in the last 2 months. What
is the expected number of claims he will file over the next 3 months?
Let X be the number of claims in the last 2 months, and Y the
number in the next 3 months.
(X | class) ∼ Binomial(m = 2 · 3 = 6, q)
6 4
P[X = 4 | Class] =
q (1 − q)2 = 15q 4 (1 − q)2
4
(Y | Class) ∼ Binomial(m = 3 · 3 = 9, q)
E[Y | Class] = 9q
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D.2.1 Discrete Bayesian Credibility
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Exercise 1 (Continued)
Class
Low
Med
High
Prior
0.5
0.3
0.2
P[data | Class]
15 · 0.24 · 0.82
15 · 0.44 · 0.62
15 · 0.74 · 0.32
Prior·P[data | Class]
0.0077
0.0415
0.0648
Posterior
0.067
0.364
0.569
P[Data, Low]
P[Data]
0.5 · 15 · 0.24 · 0.82
= P
Prior · P[data | Class]
= 0.067
P[Low | Data] =
E[Y | data] = E[E[Y | Class] | data]
= E[9q | data]
= 9 · 0.2 · 0.067 + 9 · 0.4 · 0.364 + 9 · 0.7 · 0.569
= 5.01
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D.2.1 Discrete Bayesian Credibility
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Exercise 1 Calculator Approach
L1
q
0.2
0.4
0.7
L2
Prior
0.5
0.3
0.2
L3
Prior·P[data | Class]
0.0077
0.0415
0.0648
I
Enter q in L1
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Enter prior in L2
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Let L3 = L2 · (6 nCr 4)L14 (1 − L1)2
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In stat mode, set Data = L1, FRQ = L3
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E[q | Data] = x = 0.557
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E[Y | Data] = 9E[q | Data] = 5.01
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Exercise 2
The number of monthly claims N are independent binomial variables
with m = 3 and q = 0.2 for low risk individuals, 0.4 for medium risk,
and 0.7 for high risk. 50% of individuals are low risk, 30% are medium
risk, and 20% are high risk.
A randomly selected insured filed 4 claims in the last 2 months. What
is the variance of the number of claims he will file over the next 3
months?
D.2 Bayesian Credibility
D.2.1 Discrete Bayesian Credibility
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Exercise 2
The number of monthly claims N are independent binomial variables
with m = 3 and q = 0.2 for low risk individuals, 0.4 for medium risk,
and 0.7 for high risk. 50% of individuals are low risk, 30% are medium
risk, and 20% are high risk.
A randomly selected insured filed 4 claims in the last 2 months. What
is the variance of the number of claims he will file over the next 3
months?
Let X = # claims in last 2 months and Y = # in next 3 months.
(Y | Class) ∼ Binomial(m = 9, q)
Var[Y | X = 4] = E[Var[Y | Class] | X = 4] + Var[E[Y | Class] | X = 4]
= E[9q(1 − q) | X = 4] + Var[9q | X = 4]
= 9E[q − q 2 | X = 4] + 92 Var[q | X = 4]
D.2 Bayesian Credibility
D.2.1 Discrete Bayesian Credibility
Exercise 2 (Continued)
L1
L2
L3
q
Prior Prior·P[data | Class] Posterior
0.2
0.5
0.0077
0.067
0.4
0.3
0.0415
0.364
0.7
0.2
0.0648
0.569
Using the calculator (Data=L1, FRQ=L3)
I E[q | X = 4] = x = 0.5572
I Var[q | X = 4] = σ 2 = 0.0292
X
I E[q 2 | X = 4] = σ 2 + (x)2 = 0.3396
X P
I Or: E[q 2 | X = 4] =
x2 ÷ n = 0.3396
Var[Y | X = 4] = 9E[q − q 2 | X = 4] + 92 Var[q | X = 4]
= 9 · 0.5572 − 9 · 0.3396 + 81 · 0.0292 = 4.3
Or: E[q | X = 4] = 0.067 · 0.2 + 0.364 · 0.4 + 0.569 · 0.7
E[q 2 | X = 4] = 0.067 · 0.22 + 0.364 · 0.42 + 0.569 · 0.72
Var[q | X = 4] = E[q 2 | X = 4] − (E[q | X = 4])2
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