Our friend Bayes -- Part III

In this Part III [of IV] of our series on Bayes we complete the Bayes Grid. If you missed Part I or Part II, you can click to catch up.

Bayes' Grid
Here's where we left off in Part II. 'A' is an independent probabilistic event, in this case the weather, and we have empirical observations that give us the probability of good weather, 'A+', as 0.6 .  We are seeking information about the project test results, B, for which we have one project observation: the conditional situation of  'B+' when 'A+' is present, 90% probable.  And again, this is not an intersection of two events--good weather and good results happening in the same timeframe--it's a dependency: good results because of good weather.

Now, as we said in Part II, without another independent observation, we can go no farther. 

New observation
Let's assume that in the course of testing, the test manager observes that given bad weather conditions, 'B+ | A-', the B+ success rate is 2%, thus showing that even given the condition of "the weather is not favorable", there are positive test results. 

Take note: the 2% success of 'B+ | A-' may be erroroneous results.  In other words, the test may be designed to fail if the weather is not good [ie, test results are dependent on weather which is our theme for this example].  Or, there may be a misunderstanding of cause and effect.  In any event, again we return to Bayes' equation:

P(B+ | A-) = P(B+ and A-) * P(A-) = 0.02

Solving for the intersection, we find

P(B+ and A-) = 0.008

Using the result from above and the grid math to compute 'B- and A-' = 0.392, we now have this grid:

Grid results

The computed figures in the light blue column adjacent to the test results arise from the grid math that requires all columns and rows to add.

We also can validate this grid: the dark blue cells sum to 1.0.  They sum to their counterparts in the light blue by column and row, and the light blue columns and rows sum to 1.0.  All space has been accounted for.

We now have a result for the elusive B+: We see that 54.8% of the time the test results will be good, and 54% of the time good results will coincide--that is, intersect--with good weather.  In fact, if the weather is good, as it is 60% of the time, then we forecast 90% test success.

To be continued
In the final Part, we'll address a few tricks and traps in this method, and provide some insight to the what the grid is telling the risk manager.

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Customer Satisfaction

Here's a quotation on customer satisfaction that says a lot for my way of thinking:

If the customer is not satisfied, he may not want to pay for our efforts. If the customer is not successful, he may not be able to pay. If he is not more successful than he already was why should he pay?

A quote from Niels Malotaux

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Our friend Bayes -- Part II

In Part I of this series, we developed the idea that Thomas Bayes was a rebel in his time, looking at probability problems in a different light, specifically from the proposition of dependencies between probabilistic events.

In Part I we posed the project situation of 'A' and 'B', where 'A' is a probabilistic event--in our example 'A' is the weather--and 'B' is another probabilistic event, the results of tests. We hypothesized that 'B' had a dependency on 'A', but not the other way 'round.

Bayes' Grid

The Figure below is a Bayes' Grid for this situation. 'A+' is good weather, and 'B+' is a good test result. 'A' is independent of 'B', but 'B' has dependencies on 'A'. The notation, 'B+ | A' means a good test result given any conditions of the weather, whereas 'B+ | A+' [shown in another figure] means a good test result given the condition of good weather. 'B+ and A+'  means a good test result when at the same time the weather is good. Note the former is a dependency and the latter is a intersection of two conditions; they are not the same.

  

The blue cells all contain probabilities; some will be from empirical observations, and others will be calculated to fill in the blanks. The dark blue cells are 'unions' of specific conditions of 'A' and 'B'. The light blue cells are probabilities of either 'A' or 'B'.

Grid Math

There are a few basic math rules that govern Bayes' Grid.

• The dark blue space [4 cells] is every condition of 'A' and 'B', so the numbers in this 'space' must sum 1.0, representing the total 'A' and 'B' union
• The light blue row just under the 'A' is every condition of 'A', so this row must sum to 1.0
• The light blue column just adjacent to 'B' is every condition of 'B' so this column must sum to 1.0
• The dark blue columns or rows must sum to their light blue counter parts

Now, we are not going to guess or rely on a hunch to fill out this grid. Only empirical observations and calculations based on those observations will be used.

Empirical Data

First, let's say the empirical observations of the weather are that 60% of the time it is good and 40% of the time it is bad. Going forward, using the empirical observations, we can say that our 'confidence' of good weather is 60%-or-less. We can begin to fill in the grid, as shown below.

In spite of the intersections of A and B shown on the grid, it's very rare for the project to observe them. More commonly, observations are made of conditional results.  Suppose we observe that given good weather, 90% of the test results are good. This is a conditional statement of the form P(B+ | A+) which is read: "probability of B+ given the condition of A+".  Now, the situation of 'B+ | A+' per se is not shown on the grid.  What is shown is 'B+ and A+'.  However, our friend Bayes gave us this equation:

P(B+ | A+) * P(A+) = P (B+ and A+)  = 0.9 * 0.6 = 0.54

Take note: B+ is not 90%; in fact, we don't know yet what B+ is.  However, we know the value of 'B+ and A+' is 0.54 because of Bayes' equation given above.

Now, since the grid has to add in every direction, we also know that the second number in the A+ column is 0.06, P(B- and A+).

However, we can go no farther until we obtain another independent emprical observation.
 

To be continued

In the next posting in this series, we will examine how the project risk manager uses the rest of the grid to estimate other conditional situations.

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Our friend Bayes -- Part I

Our friend Bayes, Thomas Bayes, late of the 18th century, an Englishman, was a mathematician and a pastor who's curiosity led him to ponder the nature of random events.

There was already a body of knowledge about probabilities by his time, so curious Bayes went at probability in a different way. Until Bayes came along, probability was a matter of frequency:

"How many times did an event happen/how many times could an event happen". In other words, "actual/opportunity".

To apply this definition in practice, certain, or "calibrated", information is needed about the opportunity, and of course actual outcomes are needed, often several trials of actual outcomes.

Bayes' Insight
Recognizing the practicalities of obtaining the requisite information, brother Bayes decided, more or less, to look backward from actual observations to ascertain and understand conditions that influenced the actual outcomes, and might influence future outcomes.

So Bayes developed his own definition of probability that is not frequency and trials oriented, but it does require an actual observation. Bayes’ definition of probability, somewhat paraphrased, is that probability is...

The ratio of expected value before an event happens to the actual observed value at the time the event happens.

This way of looking at probability is really a bet on an outcome based on [mostly subjective] evaluations of circumstances that might lead to that outcome. It's a ratio of values, rather than a frequency ratio.

Bayes' Theorem
He developed a widely known explanation of his ideas [first published after his death] that have become known as Bayes' Theorem. Used quantitatively [rather qualitatively as Bayes himself reasoned], Bayesian reasoning begins with an observation, hypothesis, or "guess" and works backward through a set of mathematical functions to arrive at the underlying probabilities.

To use his theorem, information about two probabilistic events is needed:

One event, call it 'A', must be independent of outcomes, but otherwise has some influence over outcomes. For example, 'A' could be the weather. The weather seems to go its own way most of the time. Specifically 'good weather' is the event 'A+', and 'bad weather' is the event 'A-'. 

The second event, call it 'B', is hypothesized to have some dependency on 'A'. [This is Bayes' 'bet' on the future value] For example, project test results in some cases could be weather dependent. Specifically, 'B+' is the event 'good test result' and 'B-' is a bad test result;  test results could depend on the weather, but not the other way 'round.

Project Questions
Now situation we have described raises some interesting questions:

• What is the likelihood of B+, given A+? 
• What are the prospects for B+ if A+ doesn't happen? 
• Is there a way to estimate the likelihood of B+ or B- given any condition of A? 
• Can we validate that B indeed depends on A?

Bayes' Grid
Curious Bayes [or those who came after him] realized that a "Bayes' Grid", a 2x2 matrix, could help sort out functional relationships between the 'A' space and the 'B' space. Bayes' Grid is a device that simplifies the reasoning, provides a visualization of the relationships, and avoids dealing directly with equations of probabilities.

Since there's a lot detail behind Bayes' Grid, we'll take up those details in Part II of this series.

Photo credit: Wikipedia

Facts and the future

"Facts do not cease to exist because they are ignored"

Aldous Huxley

Huxley's point is a good one: inconvenient truths, to paraphrase a popular idea, are like the tree falling in the forrest: there is still a sound, even if nobody is around to hear it.

But they are only facts if they are in the past. To quote David Hulett: "There are no facts about the future, only estimates"

So beware: the past holds the facts; the future holds only estimates, and estimates are only known within a range of uncertainty.

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Sausage making

It seems like the popular media and the spread of real-time networking has exposed sausage making [i.e. "the process"] to many new initiates that had no idea "that's how it's done". To many, the details of "getting there" are disconcerting, even disgusting. Unfortunately, many get caught up by the drama of the process and overlook the value of the results.

Projects are not immune: many stakeholders are exposed to project processes like never before. Dashboards, elaborate workflow engines, tweets from embedded associates, and all other manner of project detail is now 'out there'.

The key to success in the more transparent environment is the same key as before: focus on results and accomplishments. Be sure that value is only earned--and credit given--for results, not for process and effort.

In the end, the process will be forgotten; even heroic efforst will be forgotten, but results--delivered to users, customers, and stakeholders--will be a permanent memorial to the success of the project.

Photo: wickenden/flickr

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Favorite quote: Time

"There is no undo button for our oceans of time"

Tom Pike
"Rethink, Retool, Results"

Projects are one-time endeavors; there's only one change to get them right!

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