Transcript Decision Trees Posterior probabilities

Incorporating New Information to
Decision Trees (posterior
probabilities)
MGS3100 - Chapter 6
Part 3
How We Will Use Bayes' Theorem
Prior information can be based on the
results of previous experiments, or expert
opinion, and can be expressed as
probabilities. If it is desirable to improve on
this state of knowledge, an experiment can
be conducted. Bayes' Theorem is the
mechanism used to update the state of
knowledge with the results of the
experiment to provide a posterior
distribution.
Bayes’ Theorem
Used to revise probabilities
based upon new data
Prior
probabilities
New data
Posterior
probabilities
How Bayes' Theorem Works
Let the experiment be A and the prediction be B. Let’s
assume that both have occurred. The probability of both A
and B together is P(A∩B), or simply P(AB). The law of
conditional probability says that this probability can be found
as the product of the conditional probability of one, given the
other, times the probability of the other. That is:
P(A|B) * P(B) = P(AB) = P(B|A) * P(A)
Simple algebra shows that:
P(B|A) = P(A|B) * P(B)
P(A)
This is Bayes' Theorem.
Sequential Decisions
• Would you hire a market research group
or a consultant (or a psychic) to get more
info about states of nature?
• How would additional info cause you to
revise your probabilities of states of nature
occuring?
• Draw a new tree depicting the complete
problem.
Problem: Marketing Cellular Phones
The design and product-testing phase has just been
completed for Sonorola’s new line of cellular phones.
Three alternatives are being considered for a
marketing/production strategy for this product:
1. Aggressive (A)
2. Basic (B)
3. Cautious (C)
Management decides to categorize the level of
demand as either strong (S) or weak (W).
Here, we reproduce the last slide of the Sonorola problem
from lecture slides part 2.
Of the three
expected
values, choose
12.85, the
branch
associated
with the Basic
strategy.
This decision is indicated in the TreePlan by
the number 2 in the decision node.
Marketing Department
• Reports on the state of the market
– Encouraging
– Discouraging
First, find out the reliability of the source of
information (in this case, the marketing
research group).
Find the Conditional Probability based on
the prior track record:
For two events A and B, the conditional probability
[P(A|B)], is the probability of event A given that
event B will occur.
For example, P(E|S) is the conditional probability
that marketing gives an encouraging report given
that the market is in fact going to be strong.
If marketing were perfectly reliable, P(E|S) = 1.
However, marketing has the following “track
record” in predicting the market:
P(E|S) = 0.6
P(D|S) = 1 - P(E|S) = 0.4
P(D|W) = 0.7
P(E|W) = 1 - P(D|W) = 0.3
Here is the same information displayed in tabular form:
Reliabilities
Strong
Encouraging
Discouraging
Weak
0.6
0.4
0.3
0.7
Calculating the Posterior Probabilities:
Suppose that marketing has come back with an
encouraging report.
Knowing this, what is the probability that the market is
in fact strong [P(S|E)]?
Note that probabilities such as P(S) and P(W) are initial
estimates called a prior probabilities.
Conditional probabilities such as P(S|E) are called
posterior probabilities.
The domestic tractor division has already estimated the
prior probabilities as P(S) = 0.45 and P(W) = 0.55.
Now, use Bayes’ Theorem (see appendix for a formal
description) to determine the posterior probabilities.
P(E|S)P(E|W)
P(D|S)
P(D|W)
=B3*B$8
=SUM(B12:B13)
=B12/$D12
=SUM(B12:C12)
Appendix
Bayes Theorem
• Bayes' theorem is a result in probability theory, which
gives the conditional probability distribution of a random
variable A given B in terms of the conditional probability
distribution of variable B given A and the marginal
probability distribution of A alone.
• In the context of Bayesian probability theory and
statistical inference, the marginal probability distribution
of A alone is usually called the prior probability
distribution or simply the prior. The conditional
distribution of A given the "data" B is called the posterior
probability distribution or just the posterior.