Lecture 7: Bayes' Theorem and the Value of Imperfect Information

Question 1

Signal Definition

Answer 1

Information obtained to reduce uncertainty

Question 2

Reducing uncertainty

Answer 2

• Core risk management strategy
• Obtain information in order to reduce uncertainty
• Problem: information is costly

Question 3

Bayes’ Theorem

Answer 3

• Used for revising a probability value (prior p(y)) based on additional information that is later obtained (signal s)
• The information is used to construct the posterior distribution (p(y|s)) which represents an updated state of information

p(y|s) =
p(y, s) / p(s) =
[p(s|y) * p(y)] / P(s)

Question 4

How can we calculate the value of imperfect information?

Answer 4

Value of imperfect information = EV with imperfect information (e.g. prototype) - EV without

Question 5

Risk Profile

Answer 5

Probability (1 - F_i) that a certain level of the objective is exceeded.
(F_i: cumulative distribution function)

Question 6

How can we draw the Risk profile?

Answer 6

1. Determine the probability of each outcome (profit)
2. Calculate 1-p (= 1 - F)
3. Draw the graph (risk profile 1 - F dependent on outcome (profit))

Question 7

What can we do in case of increasing complexity? What is the problem of decision trees?

Answer 7

Problem:
Decision trees get very large especially if there are many possible outcomes and uncertain factors

Solution:
Monte Carlo Simulation

Question 8

Monte Carlo Simulation

Answer 8

• Repeated random samples of model input variables over many simulation runs
• Aims to create a model that is close to the real world
• Used to describe or predict how a system will operate

Question 9

Monte Carlo Simulation (Steps)

Answer 9

1. Formulate a simulation model that represents the uncertainties of the problem
2. Define the probability distribution functions for the random input variables
3. Map the model in a computer environment
4. Using a sample of N numbers, you can develop output measures of interest

Question 10

MCS - Formulate a simulation model (1)

Answer 10

• Formulate a model that represents the uncertainties of the problem

Example:

• Profit = sales volume x profit margin - fixed cost
• Known: fixest cost and price per unit
• Unknown: raw material & production cost, sales volume

Question 11

MCS - Distributions of Random Variables (2)

Answer 11

• Define the probability distribution functions for the random input variables

E.g. normal distribution

Question 12

MCS - Implementation of MC Simulation (3)

Answer 12

Map the model in a computer environment:

• Determine the value of the controllable inputs
• Let the computer make a random draw for each of the probabilistic inputs
• Calculate the value of the outcome using the generated numbers
• Repeat the previous steps to generate N Values for the outcome

Question 13

How can we generate random normal distributed values in Excel?

Answer 13

RAND(): generates a uniformly distributed random number x between 0 and 1

NORM.INV(x, y, z): provides the outcome that has the cumulative normal probability x drawn from a normal distribution with mean y and standard deviation z.

Question 14

MCS - Output Measures (4)

Answer 14

Using the sample of N numbers you can develop output measures such as:

• Average outcome (e.g. avg profit)
• Risk profile (e.g. prob. of loss)

Question 15

How many simulation runs do we need?

Answer 15

• Law of large numbers: Sample averages converge to the expected value of a random variable if N goes to infinity
• Opposing pressures: inaccurate vs time to simulate

Question 16

Minimum number of simulation runs (formula)

Answer 16

N = [(z_1 - alpha/2 * sigma) / ε]^2

• > ensures that p(|θ_n − θ| ≤ ε) = 1 − α
• > We have to determine sigma by doing a pilot simulation with a small number of runs (>50)

Question 17

Interpretation of Confidence Interval

Answer 17

There is a 95% chance that the confidence interval contains the true population mean.

Question 18

How can we construct a confidence interval?

Answer 18

[θ_n − z_0.975 * (σ / N^1/2); θ_n + z_0.975 * (σ / N^1/2)]

θ_n = sample mean (different for each run)
θ = population mean