LIBRARY
OF THE
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
CONSTRUCTION OF OPTIMAL
SEQUENTIAL DECISION TREES
by
CZrfhu^r
William A. Martin
April 1971
MASSACHUSETTS
^UTE OF TECHNOLOGY
50 MEMORIAL DRIVE
[BRIDGE. MASSACHUSETTS
(
I
MASS. INST. TECH.
JUL
12
1971
DEWEr LIBRARY
CONSTRUCTION OF OPTIMAL
SEQUENTIAL DECISION TREES
by
Cirthu.r
William A. Martin
April 1971
528/71
JUL
19
^^]l^,
ABSTRACT
The goal of a sequential decision problem is to identify one object
selected from
object.
a
knovm set of objects by applying a series of tests to the
The outcomes of tests already performed can be used to select the
next test.
Each test may be applied at most once.
Given the a priori probabi-
lities of the objects, the cost of each test, and the probabilities of the
outcomes of each test given the object to vjhlch it is applied, an algorithm
is given for constructing the tree of tests v;hich minimizes the average
cost of identifying an object.
It is assumed that the test results do not
depend on the order in which the tests are applied.
The algorithm uses a
branch and bound procedure to construct partial decision trees.
A lov7er
bound on the cost of completing a partial tree is taken as the minimum
cost of a set of tests V7hich will provide information about the objects
equal to their entropy, under the assum.ption that the tests are independent.
632653
A Lower Bound
We are given a set X of objects and a series of K tests.
has outcomes in the set Y
.
We are also given the cost C
the joint probability distribution p(>:, y
and tests 1 to k produce outcomes y
to y
,
.
.
.
The k
test
of each test and
that object X occurs
,y, )
•
The entropy H(X) of the set of objects before any tests are performed
is defined to be
Since each term in the sum is > 0, it is clear that the entropy is
zero if and only if the probability of some object is equal to 1.
There-
fore, if the entropy is not zero after several tests have been applied it
is not possible that the object has been identified.
Similarly, if the
average decrease in entropy produced from several tests is not at least
equal to the initial entropy then it is not possible for the objects to
be identified in every case.
The decrease in entropy from applying test Y, is
1(X;Y
)
= H(X) - H(xIy
)
=
-V
^'^1
S
p(>:,y,
,
•
•
•
,y^) log p(x)
\
^
X.Y-..,Y^
S-^
X,Y^....,\^
p^'^-'^'i
V
^°2 p(^'Iyi)
PCxIy
)
- 2
Similarly,
the decrease in entropy from applying
H(X) - HCXJY.Y
)
=
1 J
-Z2
Y Y
tests Y
and Y
p(x,y^,...,y^,) log p(x)
Y
"*'2_/
=
tv.'O
p(x,y ,...,y
)
log p(x|y y
)
2^
^
pU.yi>--.>yK)
1^2-:^—
2.
.
p(>-'»yi.---.yK>
^^^-YuWir
+
I(X;YJ + I(X;Y^|y^)
But I(X;Y.|y.) < I(X;Y.)
since
EpCy.)p(y.)
V7hich
using the inequality log
l(X;Y.|Y^) - l(X;Y.) <
(
Y^
2.
V7
<
(u - 1)
becomes
pCy^pCyJ
p(x,y,....y,)
-J^^^r
)
- 1 =
is
i
(Note that equality occurs when pCy.jy.) = p(y.!)p(y.).)
We use the above vrell-knovm result to compute a lower bound on the cost
of testing the objects.
The result says that each test yields at most an
entropy equal to that Which results when it is applied first.
tests the benefit of the doubt and compute these entropies.
costs of the tests
vje
We give the
Using the
then compute the cost per unit of entropy hopefully
to be received from each test and rank the tests in the order of increasing
cost per unit.
of tests
v.-'hich
objects.
Finally, using this ranking, we form the
lox-?est
cost set
could possibly yield entropy equal to that of the set of
A fractional part of one test may be included in this set in order
to get the exact amount of entropy needed.
then include the
VJe
sair.e
fraction
of its cost in the cost of this set, which is the lower bound in question.
Branch and Bound
Armed V7ith this lower bound we can employ a branch and bound algorithm
like that described by Reinwald and Soland (2)
decision trees.
.
We form a tree of partial
The root node of the tree of partial decision trees
corresponds to no tests applied.
A node is a complete node of
a decision
tree if no tests remain which have not been applied in reaching that node,
if one object has probability one at that node, or if branches extend from
that node.
A node of the tree of partial decision trees
is a terminal node
if the decision tree at that node contains no incomplete nodes.
The ave-
rage cost of the decision tree of each non-terminal node is bounded by the
costs of the tests applied so far plus the lovjer bounds on the cost of
I
- 4
each of the sub-decision trees v^hich has
xiot
been completed.
When the cost of a terminal node is less. than the lower bounds on all
the non-terninal nodes, this terminal node represents the optimum tree.
When this is not the case the non-terminal node V7hich contains the partial
decision tree V7ith the lovrest cost bound is extended by adding a branch
for each test not yet applied.
It is possible that some uncertainty vjill remain after all of the tests
are applied.
In this case, more than one order of applying the tests vill
yield the same results.
This situation can be detected by evaluatlns the
result of applying all of the tests in a given order.
If the reinaining un-
certainty is small it may be desirable to find the best tree which reduces
the entropy below some threshold.
.
References
1.
Gallager, H.G., "Information Tlieory and Reliable Communication",
Wiley (1968).
2.
Reinwaldj
L_.
and Soland R.
Tables to Opt'
Processing
2.
Reinv7ald, L.
Tj
;!
.
,
"Conversion of Limited-Entry Decision
Computer Programs I:
and Soland R.
,
"Conversion of Limited-Entr>' Decision
Tables to Optimum Computer Programs II:
Requirement".
Minimum Average
JACM (July 1966)
JACM (October 1967).
Minimum Storage
Date Due
N**"
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M.I.T. Alfred P. Sloan
Mklk
school of Management
Nos.523-71-Working Papers.
Nos. 532-71 Nos. 523-71 to Nos. 532-71
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