Prior distributions for the amount process

For the Italian daily rainfall application, the prior specification was constructed using suit- able distributions based on the information acquired from previous studies and personal beliefs. The form of the prior here is a recommendation but the choice of hyperparameters is an illustrative example only. However, it is recommended that, in practical applications, prior information should be obtained from experts on rainfall since they are the ones who have a great deal of knowledge about rainfall. While it was not possible, within the scope of this project, to conduct experiments to test elicitation methods, we provide here elicitation questions about observable quantities which could help the experts to specify judgements for prior distributions. By providing these questions we show that it is possible to construct an elicitation scheme based on judgements about observable quantities. We adopt a similar approach with the other examples later in this thesis.

We begin by considering the prior distribution for the parameters of the amount dis- tribution at a particular time of year. We will consider the seasonal effect later, in Section 4.3.3.3.

• Lognormal distribution

Suppose that we imagine a large number of wet days, Y₁, · · · , YTw for a certain time,

for example, in March. Let Zj = log Yj and we have Tw which is large enough so

that the sample mean ¯Z = _T1

w

PTw

j=1Zj is approximately ϑtand the sample variance

S_z2 = _T 1 w−1 PTw t=1  Zt− ¯Z 2

is approximately σ2 = 1/τ . To assess the uncertainty for ¯Z, we can ask the expert as follows:

Q11 “Please give a value such that you think that it is equally likely that ¯Z is less than or greater than the value.” Let the given value be Qz,2.

Q12 “Suppose you were told that ¯Z is less than Qz,2. Please give a new value such

that you think that it is equally likely that ¯Z is less than or greater than this value.” Let the given value be Q .

Q13 “Suppose you were told that ¯Z is greater Qz,2. Please give a new value such

that you think that it is equally likely that ¯Z is less than or greater than this value.” Let the given value be Qz,3.

Then Q_z,1and Q_z,3are our lower and upper quartiles for ¯Z. Hence, the prior mean for

ˆ

ηtis mA0= 1₂(Qz,1+Qz,3) and the prior variance for ˆηtis vA0= [(Qz,3−Qz,1)/1.349]2.

Suppose that our three quartiles for S_Z2 are Q_S,1, Q_S,2and Q_S,3where we can obtain these values by asking the expert as follows:

Q14 “Please now consider the sample variance, S_Z2. Can you give a value QS,2such

that S_Z2 is equally likely to be less than or greater than QS,2.”

Q15 “Suppose you were told that S_Z2 will be less than Q_S,2. Please give a value Q_S,1 such that it is equally that S2_Z is less than or greater than QS,1.”

Q16 “Suppose you were told that S_Z2 will be greater than Q_S,2. Please give a value QS,3 such that it is equally that SZ2 is less than or greater than QS,3.”

Then our three quartiles for τ are Q−1_S,3, Q−1_S,2 and Q−1_S,1. Let τ follow a Ga(g_τ, hτ)

distribution. Then we solve Q−1_S,1/Q−1_S,3 = ˜A3(gτ)/ ˜A1(gτ) interatively to find the

hyperparameter gτ, where ˜Aq(a) is quartile q for a Ga(a, 1) distribution and hτ =

˜

A2(gτ)/Q−1S,2. Using this method, we have chosen suitable numbers for illustration

as follows: Qz,1 = 0.5, Qz,3 = 1.5, QS,1= 1.25, QS,2= 2, QS,3= 2.5, and hence mA0= 1, vA0= 0.55, gτ = 4.08, hτ = 7.51. • G1 and G2 distributions

Suppose that we imagine a large number of wet days, Y1, · · · , YTw in, for exam-

ple, March where T_w is large enough so that the sample mean ¯Y = _T1

w

PTw

j=1Yj is

approximately µ = α/β and the sample variance S_Y2 = _T 1

w−1 PTw j=1  Yj− ¯Y 2 is ap- proximately Var(Yt|α, β) = α/β2. The sample coefficient of variation CY = SY/ ¯Y

is then approximately α−1/2. Furthermore, the sample mean divided by the sample variance, D_Y = ¯Y /S_Y2, is approximately β. In the case of the G1 distribution, we need to assess prior beliefs about the mean ˆηtand the shape parameter α. In the case

of the G2 distribution, we need prior beliefs for the mean ˆηtand the scale parameter

To assess the values of the parameters of the prior distribution for the mean rainfall amount, we can ask a sequence of questions to the expert as follows:

Q17 “Please think about the sample mean ¯Y . Please give a value such that ¯Y is equally likely to be less than or greater than this value.” Let the given value be L.

Q18 “Please provide a value K such that the events ¯Y < L/K, L/K < ¯Y < L, L < ¯Y < KL and KL < ¯Y are all equally likely.”

Then L is our median for ¯Y and the lower and upper quartiles for ¯Y are respectively L/K and KL. Hence, our prior median and lower and upper quartiles for ˆηt are

log L, log L − log K and log L + log K, respectively. Therefore, the prior mean for ˆηt

is m_A0= log L and the prior variance for ˆηt is vA0= (log K/0.6745)2.

For the G1 distribution, we need to assess beliefs about CY. We can use a direct

approach to ask the expert as follows:

Q19 “Please give a value QC,2such that CY is equally likely to be less than or greater

than Q_C,2.” The given value, Q_C,2 is the median for C_Y.

Q20 “Suppose you were told that CY will be less than QC,2. Please give a number

QC,1such that it is equally that CY is less than or greater than QC,1.” The given

value, Q_C,1 is the lower quartile for C_Y.

Q21 “Suppose you were told that CY is greater QC,2. Please give a number QC,3

such that it is equally that CY is less than or greater than QC,3.” The given

value, Q_C,3 is the upper quartile for C_Y.

Then our prior quartiles for α are Q−2_C,3, Q−2_C,2 and Q−2_C,1. We suppose that α fol- lows a gamma Ga(gα, hα) distribution and then gα is chosen iteratively solving

(QC,3/QC,1)2 = ˜A3(gα)/ ˜A1(gα) where ˜Aq(a) is quartile q for a Ga(a, 1) distribu-

tions, and h_α= ˜A2(gα)Q2C,2.

For the G2 distribution, we need to assess beliefs about DY. We can use a direct

approach to ask the expert as follows:

Q20 “Please give a value QD,2 such that DY is equally likely to be less than or

greater than Q_D,2.” The given value, Q_D,2 is the median for D_Y.

Q21 “Suppose you were told that DY will be less than QD,2. Please give a number

QD,1 such that it is equally that DY is less than or greater than QD,1.” The

given value, Q_D,1 is the lower quartile for D_Y.

Q19 “Suppose you were told that DY is greater QD,2. Please give a number QD,3

such that it is equally that DY is less than or greater than QD,3.” The given

Then our prior quartiles for β are QD,1, QD,2and QD,3. We suppose that β follows a

gamma Ga(g_β, hβ) distribution and then gβis chosen iteratively solving QD,3/QD,1=

˜

A3(gβ)/ ˜A1(gβ) where ˜Aq(a) is quartile q for a Ga(a, 1) distributions, and hβ =

˜

A2(gβ)QD,2.

Suppose that we choose a day of the year when we believe that the seasonal effects in (4.19) will cancel out. That is, it is a time of the year when mean rainfall amounts will be expected to be close to the annual average. Then our distribution for ˆηt on

this day is, in effect, our distribution for η₀. Using the elicitation procedure above we obtain, for illustration, η0 ∼ N(1.95, 1.06). That is mA0= 1.95 and vA0 = 1.06.

We will return to the question of beliefs about the seasonal effects, and therefore the Fourier coefficients in (4.19) in Section 4.3.3.3.

For the G1 distribution, we obtain α ∼ Ga(4.31, 6.75). That is g_α = 4.31 and

hα = 6.75.

For the G2 distribution, we obtain β ∼ Ga(3.87, 44.33). That is g_β = 3.87 and

hβ = 44.33.