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Decision Making under Uncertainty
and Bayesian Probability
Hanan S. Bell PhD
March 4, 2015
What is a decision?
•  Alternatives – something to choose among
•  Outcomes – what you get based on your choice
•  Uncertainty and structure – how the outcomes
are computed
•  Values – what is the worth of the outcomes
Jane’s Party Problem
•  Jane wants to hold a birthday party for her
daughter Sally.
•  Jane is trying to decide whether to hold it
indoors or outdoors.
•  Jane would much prefer an outdoor party if the
weather is sunny, but an outdoor party would be
a disaster if it rains.
Jane’s Values
Indoors Outdoors Sun 50 100 Rain 60 0 •  If the probability of rain is 5/11 (.454), then Jane’s expected value of either
option is 54.5
•  If the probability of rain is .3, then Jane’s expected value of indoors is .3*60+.
7*50=53 and Jane’s expected value of outdoors is .7*100+.3*0=70
•  If the probability of rain is .7, then Jane’s expected value of indoors is .7*60+.
3*50=57 and Jane’s expected value of outdoors is .7*0+.3*100=30
•  Jane maximizes expected value by being outdoors if the probability of rain is
less than .454
But wait – what do you mean by
probability?
What is this expected value?
It will either rain or it won’t and Jane
either gets the good outcome or not.
Subjectivist Bayesian Probability
•  Probability encodes all our information about an
uncertain event
•  It reflects a state of information, not a state of
things
•  It is not necessarily the same thing as frequency
although it often is
•  Different people can have different states of
information—hence different probabilities and
different decisions
Probability in weather and climate
•  Information sources
▫  Statistical history
▫  Current climate conditions
▫  Model results
▫  Experience
•  Note that ensemble results are not necessarily a
complete probability—they are a mean over a set
of varying initial conditions that may or may not
be equally likely—systemic model flaws not
always included.
What is expected value
•  Expected value computed by the sum of multiplying
each potential outcome value by its probability
•  It represents the mean value of the outcomes that
could occur if the decision could be repeated
multiple times
•  Note that the expected value may not be one of the
outcome values that could actually occur in a single
decision
•  Expected value maximizing decision makers expect
to do well over many decisions even though some
turn out badly
Farmer John’s planting problem
•  John is deciding whether to plant his field with
his regular crop or a drought resistant crop
•  The drought resistant seed is more expensive
and has a lower yield in normal rainfall
conditions
•  The drought resistant crop has a lower yield in a
drought than under normal rain, but the regular
seed has no net return in a drought
Climate John’s Values
Seed Type Resistant Regular Normal 50 100 Drought 60 0 •  Note that this is exactly the same problem as Jane’s except that it depends
on intra-seasonal or seasonal climate probabilities as opposed to daily
weather probabilities—same solution
•  But the problem is too simple. John could plant part of his field with each
type of seed (Jane can’t hold half of the party outside and half inside in the
same way.
•  Let x=% of field planed with resistant seed; 1-x= % with regular seed
•  Let p = probability of drought
•  Expected value of planting = p*(60x)+(1-p)*(100(1-x)+50x)
•  To max expected value take derivative of above and set to zero and find
relevant value of x
•  Oops—there is no x in the derivative—the expected value is linear in x
Why doesn’t John hedge his bets?
•  We haven’t really talked about value of outcomes
•  Because we use expected value decision making,
we are saying we are not risk averse
•  Most people are risk averse
•  Von-Neumann-Morgenstern utility transforms
outcome values to a form such that maximizing
expected utility will maximize utility
•  Other things that may impact values
▫  Time preferences
▫  Social preferences (public policy questions)
Other issues
•  Is drought vs normal really the right issue?
•  Perhaps soil moisture, temperature, humidity,
radiation are the right issues
•  In this case we have multiple uncertain variables
with potential probabilities that are continuous,
not binary.
•  These variables are not necessarily independent.
Sequential decisions
•  Frequently, a decision problem involves multiple
sequential decisions that arise as uncertainty
evolves
•  For example, climate change=>there are
decisions now and down the line
•  Formal structures such as decision trees or
influence diagrams can be used to deal with
these situations
Summary
•  Probabilities encode all our information about
an uncertain event.
•  Decisions can be set up in a way that maximizing
expected utility maximizes the expected value of
the outcomes.
•  Decision problems can have very complex
structures.
•  Weather and climate are important uncertain
variables in many questions facing individuals
and societies.