Bayesian Inference and Computation

Tutorial Problems 1

These exercises provide some practice in performing basic Bayesian analyses.

1) Water consumption

Water consumption

Water use

More than 5 million ML of water is used in Victoria each year, 90% from surface

water and 10% from groundwater. The majority of Victoria’s water resource is

used for irrigation (78%), while urban uses (both metropolitan and regional)

account for 17% of Victoria’s water consumption.

Efficiency of water use2

Making the best use of our water resource

CSIRO has provided a perspective on water use which is related to the

contribution it makes to Gross National Expenditure (GNE). The prices we pay

for agricultural products tend not to take into account the full environmental

costs of production. When water inputs are considered there is variation in

the amount of water used to produce various commodities. For example,

rice production uses more than 8,000 litres of water for every dollar of GNE,

compared to cereal production which uses around 600 litres to produce the

same value of product.2

Trends in water consumption

While time-series water consumption data are not available for regional Victoria,

records for Melbourne show that per capita water consumption grew steadily

between the 1940s and 1960s, with strong increases in the 1970s. Since the

1980s water consumption has been influenced by drought, and associated water

restrictions, as well as by conservation and efficiency incentives and market

reforms designed to reduce water consumption over the longer term.3

Average daily per capita water use4

Melbourne 1940-2004*

* NOTE: Figure for 2003-04 is forecasted estimation

Consumptive uses of water in Victoria1

2003-04

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Sources 1DSE 2005 State Water Report 2CSIRO and University of Sydney 2005 Balancing Act. A Triple Bottom Line Analysis of the Australian Economy 3Victorian Government 2004 Securing Our Water Future

Together 4Melbourne Water 2005 A Dry History

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4 0

1 9

4 2

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1 9

4 6

1 9

4 8

1 9

5 0

1 9

5 2

1 9

5 4

1 9

5 6

1 9

5 8

1 9

6 0

1 9

6 2

1 9

6 4

1 9

6 6

1 9

6 8

1 9

7 0

1 9

7 2

1 9

7 4

1 9

7 6

1 9

7 8

1 9

8 0

1 9

8 2

1 9

8 4

1 9

8 6

1 9

8 8

1 9

9 0

1 9

9 2

1 9

9 4

1 9

9 6

1 9

9 8

2 0

0 0

2 0

0 2

2 0

0 4

Financial Year Ending

L i

t r

e s

p

e r

p

e r

s o

n

p e

r

d a

y

550

500

450

400

350

300

250

200

150

100

50

0

Drought Years

Water Restrictions

Mid 1990s:

COAG Water

Reforms - Pricing

measures introduced

86

NOTE: These data are for

Australia overall. Cotton is

not grown in Victoria and rice

represents less than 1% of

the State’s grain production

(as at 2001 Agricultural

Census)

I the Melbo rne average daily per capita water use analyis, we modelled the discrete observa-

tions x1, . . . , xn as independent draws from a Poisson(θ) distribution. Assuming a Gamma(α, β)

p ior, which has a density function of

pi(θ) =

βα

Γ(α)

θα−1 exp(−βθ), for α, β, γ > 0,

we computed the posterior as a Gamma (α +

∑n

i=1 xi, β + n) distribution.

(a) Show that the posterior mean of θ is given by

α+

∑

i xi

β+n

.

(b) Show that the posterior variance of θ is given by

α+

∑

i xi

(β+n)2

.

(e) Show that the predictive distribution for a future observation, y, is

NegBin

(

y | α +∑i xi, 1β+n+1), where the probability mass function of a Negative Bino-

mial random variable with parameters a > 0, and 0 ≤ p ≤ 1, is given by

pi(y | a, p) =

(

y + a− 1

y

)

(1− p)apy.

1

2) Rock Strata

(a) Rock strata A and B are difficult to distinguish in the field. Through careful laboratory

studies it has been determined that the only characteristic which might be useful in

aiding discrimination is the presence or absence of a particular brachiopod fossil. In rock

exposures of the size usually encountered, the probabilities of fossil presence are found to

be as in the table below. It is also known that rock type A occurs about four times as

often as type B in this area of study.

Stratum Fossil present Fossil absent

A 0.9 0.1

B 0.2 0.8

If a sample is taken, and the fossil found to be present, calculate the posterior distribution

of rock types.

(b) If the geologist always classifies as A when the fossil is found to be present, and classifies

as B when it is absent, what is the probability she will be correct in a future classification?

3) Coloured Balls

(a) Repeat the Coloured Balls example from Lecture 1, using a different choice of prior

distribution. In what way does this change of prior affect the posterior probability of no

black balls left in the bag?

2

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