Bayes Theorem and Inference Reasoning for Project Managers
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The document discusses Bayes' Theorem and how it can help project managers make inferences about project risks and outcomes. Bayes' Theorem expresses the relationship between a hypothesis (A) and a condition (B) that influences the hypothesis. It helps calculate the probability of a hypothesis being true based on prior beliefs and observed evidence. The document provides an example where a project test (A) may be affected by weather conditions (B). Bayes' Theorem and a Bayes grid can help determine the underlying probability of a test passing, based on observations of test outcomes under different weather conditions.
![Bayes’ Theorem and Inference Reasoning for
Project Managers
The Plausible Hypothesis
Project managers often face the task of evaluating the plausibility of an event happening
during the course of a project that would affect project performance. Plausibility is in the
spectrum of “uncertainty to risk”, a spectrum that reaches from “possibility —> plausible
—> probable —> planable”. In this context, project managers and their risk management
brethren hypothesize the plausible from the range of possibilities.
Degree of Plausibility
It’s helpful to think of probability as the “degree of plausibility of a hypothesis”. By this
definition, probability is still quantitatively scaled from 0 to 1. Numbers near 0 mean the
hypothesis is very implausible even if it is a possibility; numbers near 1 mean the
hypothesis is certain enough to be planned in terms of risk response or project
performance affects.
Probabilities are not data
Now, probabilities are not themselves data; they are not measureable artifacts. Thus,
probabilities are subjective and open to many vagaries introduced by bias, opinion, and
personal experience. By extension, plausibility is a subjective evaluation. For this
reason, project managers are led toward “inference reasoning”, also known as “inductive
reasoning”. [Many confuse probability with statistics. Statistics are data obtained by
processing measured observations according to certain processing rules.]
Infer a property
We “infer” something we can’t directly observe by working backward through a
supposition from observed data. That is, given observations of actual outcomes, we draw
an inference as to what the situation, condition, or event must have been to cause those
outcomes to occur.
For the case in hand—plausible hypothesis—we surmise a hypothesis that we can’t
observe directly; we can only observe actual outcomes. For example, we might
hypothesize that a coin is not fair. We can not ‘observe’ an unfair coin [unless it has two
heads or two tails]; we can only observe the outcomes of testing the coin for fairness.
About timing
Now when making an inference there are two time frames involved:
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![• Posterior: The time after estimates are made when observations are taken of actual
outcomes—we call this the posterior time; and
• A priori: The previous—or prior—period when we estimated probabilities based
on estimates or subjective factors.
Reasoning forward in time, as in ‘a priori’ estimates, is deductive; reasoning backward in
time, as in posterior analysis, is inductive and inferential.
In the example of the coin, the a priori estimate—a deduction—was that the coin is not
fair. The posterior data observations either confirm this hypothesis is TRUE or not.
From the confirmation, we draw an inference about the coin.
In short, what we observe may differ from what we expect. This may occur because
effects, events, and conditions may influence outcomes. Thus, when making an
inference, these effects must be accounted for or else we will draw the inference
incorrectly.
Hypothesis and inference
Putting it together, in the a priori timeframe we hypothesize a possible event and estimate
its plausibility. Then, in the posterior timeframe, we make observations of actual
outcomes. The outcomes may be different than hypothesized. We try to draw an
inference about why we observe what we do. And we estimate what adjustments need to
be made to the a priori estimates so that they are more accurate next time.
Thomas Bayes’ Theorizes
An eighteenth century English mathematician by the name of Thomas Bayes was among
the first to think about the plausible hypothesis problem. In doing so, he more or less
invented a different definition of probability—a definition different from the prevailing
conventional definition based on chance. Bayes posited: probability is the degree to
which the truth of a situation—as determined by observation—varies from our
expectation for that situation. You probably recognize Bayes’ idea is the plausibility
definition of probability in slightly different terms.
Bayes was curious about the variance between truth and expectation. To assuage his
curiosity, he worked out the mathematical rules for relating a priori probabilities of a
hypothesis, posterior observations, and effects [conditions, events, or influences] that
would impact the a priori estimates in a way that explained the posterior observations.
Today, this is usually framed as conditional probabilities wherein the probability of one
event is actually dependent upon, or conditioned by, the probability of another event.
The outcome of his investigations was the formulation of Bayes’ Theorem.
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![Bayes’ Theorem defined
Bayes’ Theorem expresses a relationship between a hypothesis and a condition [event, or
circumstance] that influences the hypothesis. In the examples that follow, the hypothesis
is labeled A, and the influencing condition is labeled B. The theorem uses a construct of
the form ‘A | B’ meaning ‘A given the presence of B’, or ‘A given B’. The general
formulation of his rule is:
Probability ( A | B ) = Probability ( B | A ) x Probability (A) / Probability ( B )
Where the posterior results—A | B—a bit different from our expectation. Thus, A
depends on B, but B does not depend on A.
For project management purposes, it’s enough to understand that the left side of the
formula is the posterior outcomes, the hypothesis ‘A’ modified by the presence of ‘B’.
And, on the right side of the formula, Probability ( B | A ) is the ‘likelihood’ of B being
TRUE at the same time A is TRUE. Multiplying the likelihood by P(A) then gives us the
likelihood of B and A being TRUE for all possibilities of A. That is: “Probability ( B | A )
x Probability (A)” is actually the probability of A and B being TRUE at the same time,
giving this equality that will come in handy later:
Probability ( B | A ) x Probability (A) = P (A and B)
Finally, on the right side, Probability (B) normalizes the probability of A and B being
TRUE at the same time to the probability that B is actually TRUE.
Some identities
Rewrite the equation above and note the symmetry:
• Probability (A) = P (A and B) / P (B | A)
• Probability ( B ) x Probability ( A | B ) = Probability ( B | A ) x Probability (A)
And with a little reasoning, you can also write:
• Probability (A and B) = Probability (B and A).
These identities will used when we form a Bayes’ Grid to evaluate project situations.
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![sum to the probability of A~| B. The left white column must sum to the probability of
B~.
Applying observations to the grid
Next we run some tests and write down our observations. Because there are two
variables, A and B, we need two sets of independent observations to solve all the
relationships.
First observation: We observe the probability of passing a test under good conditions of
the weather, B~, is 75%, that is P( A~ | B~ ). But since we know the weather has some
influence, we also know that 75% is not P(A~).
B, on the other hand, is a set of conditions, like the weather, that we can independently
measure and estimate. Let’s say that in this example the probability of B~, good weather,
being present is 65%. Note: the statistics of B are not the second observation we need
because the observation we want is a posterior interaction between A and B.
Here is the grid as we know it from what we have observed about B:
We can calculate some of the cells from Bayes’ Theorem and the first A | B observation:
P ( A~ | B ~ ) = 0.75 = P ( A~ and B~ ) / P(B~)
P ( A~ | B ) = 0.75 = P ( A~ and B~ ) / 0.65
Solving for P ( A~ and B~ ):
0.65 x 0.75 = 0.4875 = P ( A~ and B~ ).
We then solve for the other value for the white grid cell in the first column that must sum
to 0.65. [We could also use the equation: P ( A^ | B~ ) = 0.25 = P ( A^ and B~ ) / 0.65]
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Bayes Theorem and Inference Reasoning for Project Managers
- 1. Bayes’ Theorem and Inference Reasoning for Project Managers John C. Goodpasture, PMP Managing Principal Square Peg Consulting LLC www.sqpegconsulting.com www.johngoodpasture.com Page 1 of 8 Copyright by John C Goodpasture, © 2010
- 2. Bayes’ Theorem and Inference Reasoning for Project Managers The Plausible Hypothesis Project managers often face the task of evaluating the plausibility of an event happening during the course of a project that would affect project performance. Plausibility is in the spectrum of “uncertainty to risk”, a spectrum that reaches from “possibility —> plausible —> probable —> planable”. In this context, project managers and their risk management brethren hypothesize the plausible from the range of possibilities. Degree of Plausibility It’s helpful to think of probability as the “degree of plausibility of a hypothesis”. By this definition, probability is still quantitatively scaled from 0 to 1. Numbers near 0 mean the hypothesis is very implausible even if it is a possibility; numbers near 1 mean the hypothesis is certain enough to be planned in terms of risk response or project performance affects. Probabilities are not data Now, probabilities are not themselves data; they are not measureable artifacts. Thus, probabilities are subjective and open to many vagaries introduced by bias, opinion, and personal experience. By extension, plausibility is a subjective evaluation. For this reason, project managers are led toward “inference reasoning”, also known as “inductive reasoning”. [Many confuse probability with statistics. Statistics are data obtained by processing measured observations according to certain processing rules.] Infer a property We “infer” something we can’t directly observe by working backward through a supposition from observed data. That is, given observations of actual outcomes, we draw an inference as to what the situation, condition, or event must have been to cause those outcomes to occur. For the case in hand—plausible hypothesis—we surmise a hypothesis that we can’t observe directly; we can only observe actual outcomes. For example, we might hypothesize that a coin is not fair. We can not ‘observe’ an unfair coin [unless it has two heads or two tails]; we can only observe the outcomes of testing the coin for fairness. About timing Now when making an inference there are two time frames involved: Page 2 of 8 Copyright by John C Goodpasture, © 2010
- 3. • Posterior: The time after estimates are made when observations are taken of actual outcomes—we call this the posterior time; and • A priori: The previous—or prior—period when we estimated probabilities based on estimates or subjective factors. Reasoning forward in time, as in ‘a priori’ estimates, is deductive; reasoning backward in time, as in posterior analysis, is inductive and inferential. In the example of the coin, the a priori estimate—a deduction—was that the coin is not fair. The posterior data observations either confirm this hypothesis is TRUE or not. From the confirmation, we draw an inference about the coin. In short, what we observe may differ from what we expect. This may occur because effects, events, and conditions may influence outcomes. Thus, when making an inference, these effects must be accounted for or else we will draw the inference incorrectly. Hypothesis and inference Putting it together, in the a priori timeframe we hypothesize a possible event and estimate its plausibility. Then, in the posterior timeframe, we make observations of actual outcomes. The outcomes may be different than hypothesized. We try to draw an inference about why we observe what we do. And we estimate what adjustments need to be made to the a priori estimates so that they are more accurate next time. Thomas Bayes’ Theorizes An eighteenth century English mathematician by the name of Thomas Bayes was among the first to think about the plausible hypothesis problem. In doing so, he more or less invented a different definition of probability—a definition different from the prevailing conventional definition based on chance. Bayes posited: probability is the degree to which the truth of a situation—as determined by observation—varies from our expectation for that situation. You probably recognize Bayes’ idea is the plausibility definition of probability in slightly different terms. Bayes was curious about the variance between truth and expectation. To assuage his curiosity, he worked out the mathematical rules for relating a priori probabilities of a hypothesis, posterior observations, and effects [conditions, events, or influences] that would impact the a priori estimates in a way that explained the posterior observations. Today, this is usually framed as conditional probabilities wherein the probability of one event is actually dependent upon, or conditioned by, the probability of another event. The outcome of his investigations was the formulation of Bayes’ Theorem. Page 3 of 8 Copyright by John C Goodpasture, © 2010
- 4. Bayes’ Theorem defined Bayes’ Theorem expresses a relationship between a hypothesis and a condition [event, or circumstance] that influences the hypothesis. In the examples that follow, the hypothesis is labeled A, and the influencing condition is labeled B. The theorem uses a construct of the form ‘A | B’ meaning ‘A given the presence of B’, or ‘A given B’. The general formulation of his rule is: Probability ( A | B ) = Probability ( B | A ) x Probability (A) / Probability ( B ) Where the posterior results—A | B—a bit different from our expectation. Thus, A depends on B, but B does not depend on A. For project management purposes, it’s enough to understand that the left side of the formula is the posterior outcomes, the hypothesis ‘A’ modified by the presence of ‘B’. And, on the right side of the formula, Probability ( B | A ) is the ‘likelihood’ of B being TRUE at the same time A is TRUE. Multiplying the likelihood by P(A) then gives us the likelihood of B and A being TRUE for all possibilities of A. That is: “Probability ( B | A ) x Probability (A)” is actually the probability of A and B being TRUE at the same time, giving this equality that will come in handy later: Probability ( B | A ) x Probability (A) = P (A and B) Finally, on the right side, Probability (B) normalizes the probability of A and B being TRUE at the same time to the probability that B is actually TRUE. Some identities Rewrite the equation above and note the symmetry: • Probability (A) = P (A and B) / P (B | A) • Probability ( B ) x Probability ( A | B ) = Probability ( B | A ) x Probability (A) And with a little reasoning, you can also write: • Probability (A and B) = Probability (B and A). These identities will used when we form a Bayes’ Grid to evaluate project situations. Page 4 of 8 Copyright by John C Goodpasture, © 2010
- 5. An example The set-up Let us define an “event space” A as having event A~ and the counter-event A^. The presence of A^ means A~ did not occur. Similarly, we define an event space for B in the same way. To put it into a project context, let’s say that A~ is a passed test, and A^ is the same test failed. Let’s define B~ as influencing condition present for the test, and B^ means the influencing condition is missing. If the test is outdoors, B could be some aspect of the weather. Presumably A is affected by B, but there is some possibility that A could pass even without B. Of course, B—the weather—is not affected by A, the project test. As project managers we would like to know how likely it is that a test will pass; that is, we want to know the P (A~), but we can’t observe this directly because B~ or B^ is present and influences the test results. Thus, we can only draw an inference about A~ from the observations of A in the presence of B. However, there is a tool that can help; it is called Bayes’ Grid. Bayes’ Grid To employ Bayes’ Theorem to find P(A~) we form a grid of A and B where we can put down some of the observable data about A and B, and then calculate the other information not available from observations. The grid below has the cells labeled with the elements from Bayes’ Theorem with weather in the two vertical columns and test performance in the two horizontal rows: The test results (A) are conditioned on the weather (B) in this example. P(A~| B) is read as “probability of a passing test given any condition of the weather”. Other cells are read similarly. The cross points in the grid in the white cells are probability intersections. ‘A~ and B~’ in the upper left is the probability of a successful test and the influencing conditions present. Since the white grid represents the entire space of A and B, the grid must sum to 1. The grid must also sum up and down and left and right. For instance the top white row must Page 5 of 8 Copyright by John C Goodpasture, © 2010
- 6. sum to the probability of A~| B. The left white column must sum to the probability of B~. Applying observations to the grid Next we run some tests and write down our observations. Because there are two variables, A and B, we need two sets of independent observations to solve all the relationships. First observation: We observe the probability of passing a test under good conditions of the weather, B~, is 75%, that is P( A~ | B~ ). But since we know the weather has some influence, we also know that 75% is not P(A~). B, on the other hand, is a set of conditions, like the weather, that we can independently measure and estimate. Let’s say that in this example the probability of B~, good weather, being present is 65%. Note: the statistics of B are not the second observation we need because the observation we want is a posterior interaction between A and B. Here is the grid as we know it from what we have observed about B: We can calculate some of the cells from Bayes’ Theorem and the first A | B observation: P ( A~ | B ~ ) = 0.75 = P ( A~ and B~ ) / P(B~) P ( A~ | B ) = 0.75 = P ( A~ and B~ ) / 0.65 Solving for P ( A~ and B~ ): 0.65 x 0.75 = 0.4875 = P ( A~ and B~ ). We then solve for the other value for the white grid cell in the first column that must sum to 0.65. [We could also use the equation: P ( A^ | B~ ) = 0.25 = P ( A^ and B~ ) / 0.65] Page 6 of 8 Copyright by John C Goodpasture, © 2010
- 7. Now, we need to find the other values of the grid, and for this we will need a second independent observation: For convenience, x and y are shown to make it easier to write what we need to know: Top row: X = 0.4875 + Y Bottom row: 1-X = 0.1625 + 0.35-Y, simplifying: X = .4875 + Y Two Unknowns So, we have two unknowns and only one equation. We know Y > 0 and < 0.35 because the sum of the four white cells = 1.0. This means X is between .4875 and .8375, and ‘1 – X’ is between 0.5125 and 0.1625. Any value of Y that satisfies the equation with X will be a possible valid inference. We could guess at the second equation by guessing a value for X and Y that satisfies the equation. But guessing carries no credibility. The best way to resolve this is with actual observations from the project outcomes. We already have an observation of test results when the weather is good. If we now take test measurements when the weather is bad, we then have a second independent set of observations that fulfill P (A~ | B^). Suppose we observe that P (A~ | B^) is 40%, meaning there is some test success even when the weather is bad. We can now calculate the Y value in the grid: P (A~ | B^) = P (A~ and B^) / P (B^) Rearranging the equation and filling in the known values: 0.4 x 0.35 = P (A~ and B^) = 0.14 Page 7 of 8 Copyright by John C Goodpasture, © 2010
- 8. Take note that the white cells add top and bottom, left and right, to their respective shaded cells. Take note that the sum of all four of the white cells added together is 1. This means that the entire event space is accounted for in the grid. Hypothesis: A~ From the grid we now see that the value of the hypothesis, A~, regardless of the weather, is 0.6275. Our observations were 0.75 when the weather was good and 0.4 when the weather was bad. Our inference is that the underlying hypothesis is 0.6275. Summary Bayes’ Theorem provides the project manager information in the form of probabilities about the performance of one project activity when it is conditioned upon the performance of another. There are some required prerequisites: A must depend on B, but B must be independent of A. And, there must be two independent sets of observations of the posterior performance of the interaction of A and B. Attributes not observed may be calculated using Bayes Theorem. A Bayes grid provides assistance in the calculations. +++++++++++++++++++ To read more: johngoodpasture.com sqpegconsulting.com Page 8 of 8 Copyright by John C Goodpasture, © 2010