# Abstract

Bayes Theorem is not just an obscure artifact of the statistics of probability handed down to us from centuries ago. You can use it now to make decisions that will affect your financial well-being.

A relatively simple Excel-based tool helps you choose the right course of action in the face of uncertain probabilities and inexact test results. It is available for FREE.

What can the Bayesian AI (“Artificial Intelligence”) Tool do for You?

I know you are probably not in the oil business (and neither am I), but the best way to understand an abstract concept is to go through a practical example. The example below has to do with oil exploration and drilling, but the

* Bayesian AI Advisor*can handle any problem where you know: 1) the probability of success if you take action without doing further testing, 2) the cost of further testing and the probability the test results will be reliable, 3)the cost of taking some action, 4) the financial benefit to you if the action is successful, and5) the compensation you expect for taking the risk.

**NOTE: Some of the Figures in this Posting were damaged during automatic porting from Google Knols to WordPress. To access a later version that is easier to read please click here>>> http://tvpclub.blogspot.com/2013/08/a-practical-view-of-bayesian-inference.html <<< advTHANKSance !**

You can download the Bayesian AI Advisor at:

The

* Bayesian AI Advisor *will compute themost likely financial implications of your actions. What if you proceed withoutfurther testing? What if you get a positive test result and proceed? What if you get a negative test result and proceed? Several other examples in very different domains are included later in this Knol, but for now, put on your hard hat and let us imagine we are in the oil business!Example #1- Should we drill here, drill now?

OIL EXPLORATION Case #1 – Screenshot of * Bayesian AI Advisor* INPUT and RESULTS Panels

http://polaris.umuc.edu/~iglickst/swen603/Bayesian%20AI%20Advisor.xls

Situation

Thirty oil wells have been drilled in a particular area and only three of them have yielded commercially-viable quantities of petroleum. The others were “dry holes”. So the chances of a newly-drilled hole striking oil are about10%. You could dojust go ahead and drill without further testing, but that would be relatively expensive and you could turn up a dry hole.Seismic testingisless expensive, but even if you get apositive result, thatdoesn’t guarantee oil will be found. The probability of getting a positive seismic test result in this area is 15%. The testsresults have aprobability of90% of being positive if there is indeed oil at the site. You input the three probabilities (10%, 15%, and 90%) into the

** Bayesian AI Advisor**INPUT panel as indicated above.

The

** Bayesian AI Advisor**uses Bayes Theorem to calculate that there is an

*probability of60%that you will be successful finding oil given a positive seismic test. OK, now that Bayes has spoken, what should you do?*

**inverse**To answer that question you first have to input four pieces of financial data. According to experienced oilmen in the area, it will cost $100K to do a seismic test and $2000K if you decide to drill a well. if you do strike oil, you could get $7000K based on current prices, but you need a 30% return on investment (ROI)to account for the risk and the possibility petroleum prices could go down further. ROI is: { Gain from Investment – Cost of Investment } / {Cost of Investment}

The figure above shows the numbers you would input into the

* Bayesian AI Advisor*for the given situation, and the output you would get. NOTE: These are just made-up values to illustrate the process and may or may not represent the actual financial situation in any real-world situation. If you use this tool to make money, please send me my share. If you lose money, you are on you own :^)

Choices

What to do? 1) Go ahead and drill without seismic testing? 2) Do the seismic testing and, if you get a positive result, go ahead and drill? 3) Forget about it – all these numbers hurt my head!

The

** Bayesian AI Advisor**crunches the numbers you entered and gives you the results and recommendations on what to do!

According to analysis using Bayes Law, in the given situation if you drill without doing the seismic testing, you are most likely to invest $20,000K for each successful oil well, for a net loss of $13,000K and an ROI of -65%. OUCH! Not a good result!

If you go ahead and invest the $100K for the seismic testing of several locations and eventually get a positive result and then drill, you are most likely toinvest $5,000K for each successful oil well, for a net gain of $2,000K and an ROI of +40%. WOW! That would be great!

The

** Bayesian AI Advisor**is pretty sure what you should do in this case:

*Test first. If Test is Positive, do the Procedure.*

Proceed with the seismic testing and, if you get a positive result, go ahead and drill and get rich. If you get a negative test result you are advised to abandon the plan to drill at that point. Do not throw good money after the “bad” $100K you spent on the seismic testing. Move to another spot and continue seismic testing and hope for better test results.

NOTE: Later in this Knol, I will show you EXAMPLES of what happenswhen oil prices go up or down and other variables change. Given different circumstances, the

** Bayesian AI Advisor**is capable of recommending that you proceed to drill without testing or that you abandon plans to get into the oil business in that area entirely.

Background – What is Bayes Theorem?

If you really want to know all the intimate statistical probabalistic stuff, just Google “Bayes Theorem” or “Bayesian” and you will get lots of links replete with mathematical symbols in all their glory. (Don’t worry if you can’treally understand all those symbols -perhaps your daughter could play them on her flute :^)

Here is All You Really Need to Know about Bayes

The Rev. Thomas Bayes was a Presbyterian minister and mathematician who lived in the 1700’s. His great contribution to mathematics was the concept of “

** inverse**probability”. He came up with it at a time when only “

**forward**probabilty” was generally known.

**Forward**Probability is Easy

**Forward**probability has to do with making predictions based on previous knowledge. Say you know the following about*Poupon University*: 1) There are a total of3000 students, 2)1200 are Liberal Arts majors, 3) 60 areMath majors, and 4) 600 are Engineeringmajors.

If you go toCommons and pick a student at random, what is the probability you’ll picka Liberal Artsmajor? A Math major? An Engineer? (Assuming, of course, that Engineers are as likely as, say English majors, to take a break fromtheir studies and to go to the Commons :^)

Thatis easy using

**forward**probability: 1)Liberal Arts: 1200/3000 = 0.4 or a40%chanceof picking a Liberal Arts major. 2) Math: 60/3000 = 0.02 or a 2% chance of picking a Math major. 3) Engineering: 600/3000 = 0.2 or a20% chance you’ll pick an Engineer.

**Probability is Hard**

*Inverse*What Bayes figured out,

** inverse**probability, is harder. (An English major once told methat “

*backward*poets rhyme

*inverse*” but that is a different matter :^)

Hey, here’s a student wearing a Bayes TheoremT-Shirt! Based on the facts given above, what is the probability he or she is a Liberal Arts major? A Math major? an Engineer? The formula on his shirt can help you do the

** inverse**probability to figure that out.

T-Shirt with Bayes Theorem in Mathematical Symbols

But first you needsome additional facts: a) Based on sales by the college store,about 300 students out of the 3000 on campus, or 10%, owna BayesT-shirt. b) 100% ofthe Math majors own them (of course :^). c)30% of the engineeers.d) Only 1% of the Liberal Arts majors.e)You are in luck, today is April 7, the day Rev. Thomas Bayes died 1761, so everybody who owns a Bayes T-Shirt will be wearing one!

If you pick a student

**who is wearing aBayes T-Shirt**, at random, what is the**inverse**probability that he or she is a Liberal Arts major? A Math major? An Engineer? Why don’t you guess right now – pleasewrite your answers down so you will be properly amazed when you find out how wrong (or right) you were!

Bayes Theorem is shownin mathematical symbols on the T-Shirt above. It translates to mathematical English as follows:

** Probability of A given B equals Probability of B given A multiplied by the Probability of A and divided by the Probability of B **

In case you did not understand that translation, here it is in plain Englishusing the Bayes T-shirt and Engineering students as an example:

** The Probability a Student is an Engineer given that he or she is wearing aBayes T-Shirt,is equal to theProbability of wearing a Bayes T-Shirt given a student is anEngineer multiplied by the Probability a Randomly-selectedStudent is an Engineer, alldivided by the Probability a Randomly-selected student is wearinga Bayes T-Shirt. **

Let’ s run the numbers:

**P(Wearing Bayes T-Shirt given Engineer)**=30% [from a fewparagraphs above]**P(Engineer) = 20%**[from the forward probability stuff]**P(Wearing Bayes T-Shirt) = 10%**[from a few paragraphs above]** P(Engineer given Wearing Bayes T-Shirt) = P(Wearing Bayes T-Shirt given Engineer) x P(Engineer) / P(Wearing Bayes T-Shirt) = 30 x 20 / 10 = 60% **

So,

*given*that the student you pick at random is actually wearinga Bayes T-Shirt, there is a 60% chance he or she is an Engineer! WOW! What did you guess? 60%isahigher percentage than I would have guessed becauseonly 20% of the students on campus are Engineers. So, using the “test” that a student is wearing a Bayes T-Shirt*increases*your chances of picking an Engineer by a factor of three. WOW, the power of Bayes Theorem is impressive.

If you run the numbers for Math majors you get 100x 2 / 10 = 20%. There is a 20% chancehe or she is aMath major. WOW! A surprising result because only 2% of the students on campus are Math majors! So using the “test” that a student is wearing a Bayes T-Shirt

*increases*your chances of picking a Math majorby a factor of ten. WOW, the power of Bayes Theorem is impressive.

If you run the numbers for Liberal Arts majors you get: 1x 40 / 10 = 4%. There is only a 4% chancehe or she is aLiberal Arts major, surprising because they make up 40% of the student body. Bayes predicts you are ten times

*less*likely to pick a Liberal Arts major using the Bayes T-Shirt test, a negative result. (You could use that test to*avoid*Liberal Arts majors -or they could use the same test to avoid Engineers and Math majors:^)

So, in summary,if a student happens to be wearing a Bayes T-Shirt on this campus, there is an 80% chance he or she is in Math or Engineeringeven though only 22% of the students are in those twomajors.

Bayesian Controversy

According to http://psychology.wikia.com/wiki/Bayesian_probability there is considerable controversy between “Bayesians” and “Frequentists” as to the true meaning and interpretation of “probability”.

**Interpretation**

*Bayesian*“In the philosophy of mathematics**Bayesianism**is the tenet that the mathematical theory of probability is applicable to the degree to which a person believes a proposition . Bayesians also hold that Bayes’ theorem can be used as the basis for a rule for updating beliefs in the light of new information —such updating is known as*Bayesian inference . *In this sense, Bayesianism is an application of the probability calculus and a probability interpretation of the term*probable*, or —as it is usually put —an*interpretation of probability*.”

**Frequentist**Interpretation

“A quite different interpretation of the term*probable*has been developed by frequentists . In this interpretation, what are*probable*are not propositions entertained by believers, but events considered as members of collectives to which the tools of statistical analysis can be applied.”

The

**frequentists**demand that probability statements be based on hard data derived from actual observation and experiments. On the other hand,** Bayesians**allow each person to assign different Bayesian probabilities to the same proposition.

“Although there is no reason why different interpretations (senses) of a word cannot be used in different contexts, there is a history of antagonism between Bayesians and frequentists, with the latter often rejecting the Bayesian interpretation as ill-grounded. The groups have also disagreed about which of the two senses reflects what is commonly meant by the term ‘probable’.”An interesting example would be if ten coin tosses resulted in seven heads and three tails. A**frequentist**wouldsay the probability is 70/30 heads unless and until further tosses proved otherwise. A** Bayesian**would consider the situation from a larger prospective. Was the coin provided by a trusted person or some stranger in a bar? Is there reason to believe the coin is fair or loaded? Based on that consideration, one

**might assign a probability of 50/50, since ten tosses with a 70/30 result is statistically possible with a fair coin. Another**

*Bayesian***might conclude, from the situation, that the coin is probably loaded and assign a 70/30 probability. Yet another might split the difference and assign 60/40, …**

*Bayesian*Of course, if the coin is tossed another hundred times, both the

**frequentists**and the** Bayesians**may change their assigned probabilities. It may turn out that the

**frequentist**70/30 was closer to the truth -or- that the

*50/50 was a better call.*

**Bayesian**Examples #1, #2, and #3- Variations on Oil Exploration Cases

At the top of this Knol I gave EXAMPLE #1 where the

** Bayesian AI Advisor**gave what might be called the”common sense” strategy: “Test first and if the test is positive, drill.” Well,as I suggested, if the situation changes, the

**is fully capable of giving different advice.**

*Bayesian AI Advisor*OIL EXPLORATION Cases #1, #2 and #3-

Screenshots of * Bayesian AI Advisor* INPUT and RESULTS Panels

Oil Case #1 – Test Here First, Drill Here if Positive

The leftmost image, with the #1 label, is our old friend Case #1, where the

** Bayesian AI Advisor**suggested we should “test first and then drill if the test was positive”, with a predicted ROI of40%.

Oil Case #2 – Do NotDrill Here -HELL NO- Don’t Even Test

The middle image is case #2 where, as indicated by the red box, all we changed was the value of petroleum. It went down from $7000K to $6000K, a reduction of 15%. (As those of us who lived through the decade from 2000 to 2009 can attest, petroleum can go up and down by much higher percentages!) Now, the

** Bayesian AI Advisor**says:

*Hopeless venture. (Can you reduce expected ROI?).*

WOW, how the outlook has worsened! According to analysis using Bayes Law, in the given situation if you drill without doing the seismic testing, you are most likely to invest $20,000K for each successful oil well, for a net loss of $14,000K and an ROI of -70%. OUCH! Not a good result!

If you go ahead and invest the $100K for the seismic testing of several locations and eventually get a positive result and then drill, you are most likely toinvest $5,000K for each successful oil well, for a net gain of $1,000K and an ROI of +20%. That looks pretty good, but we need an ROI of at least 30% to cover the risk of enticing investors to put up the money, given the volatility of oil prices.

Unless we can convince the investors to reduce their expected ROI to 20%, we are out of luck and we shouldneither test nor drill at this site.

Oil Case #3 – Drill Here, Drill Now – No need to Test

The rightmost image is case#3 where we’ve moved our exploration to a different area of the oil patch where, as indicated by the red box,the chances of striking oil withouttesting are much better. Notice how the Bayesian

* inverse*probability has gone up from 60% to nearly 85%.In that case, according to the

**, there is:**

*Bayesian AI Advisor**No need to Test. Go ahead with Procedure.*

WOW, we can make money even if we don’t test before drilling. Our net gain would be a bit more if we test first and drill only if the test is positive, BUT, the ROI would be a bit less in that case. So, since ROI is more important that simple gain, we are better off drilling without testing! (Darned clever that

** Bayesian AI Advisor**!)

Examples #4, #5, and #6- Targeted Marketing

Guess what, the

** Bayesian AI Advisor**loves targeted marketing problems!.

Say we are marketing something fairly expensive, with a correspondingly high profit margin. Suppose it would appeal only to a highly specialized audience. For example, something that only mathematically and technologically-oriented collegestudents would buy. Indeed, what if ownership of a Bayes T-Shirt turned out to be an excellent “test indicator” to qualify a prospect? If we could get hold of a list of students who bought those shirts -or just go on campus and approach anyone wearing one- we could restrict our sales pitches to them.If the Bayesian

* Inverse*probability was high enough for this select group, we could afford to lure them to our sales pitch with the promise ofa gift or a free meal.

Bad Reputation of “Free Bait” Not Always Justified

Of course you are familiar with this type of targeted marketing approach. Offer a free trial subscription to a magazine. Give alower monthly price for cell phone service or cable-TV fora year and jack it up after the customer is hooked. Offera free vacation toa time share property, etc. The key to the success of these plans is to qualify the targets and then offer them the “free bait”. You better qualify them well because the “free bait” can be costly to you if all you attract are people who have no intention of buying and are just looking for a free meal or gift!

Although some instances of targeted marketinghave given the genrea bad reputation,there is nothing wrong, in principle, with offering something of value to qualified people to get them to listen to your sales pitch. For example, when Iinterviewed seniors on a college campus for possible jobs as engineers at the company where I worked, I was allowed to invite up to 25% of the top candidates for an expense-paid trip to our facility – if they met certain GPA and other qualifications.

The key is to make sure that the qualifying “test” is effective enough to justify the “free bait”.

TARGETED MARKETING Cases #4, #5 and #6-

Screenshots of * Bayesian AI Advisor* INPUT and RESULTS Panels

Marketing Case #4- Test the Person, Offer Free Bait if Qualified

The leftmost panel shows a successful situation where the “Prior” Probability is very low – only about

**five**out of**one-thousand**people would want our product and be able to afford it. That is only0.5%.

Assume we have developed some type of “qualifying test” that identifies about

**ten**out of**one-thousand**people as targets. That is only1%.(This “test” might bea prospect list we could buy from a market data company that tracks interests and buying habits and disposable income of specific types of people. Alternatively, we might send agents toyacht clubs or professional societies or whatever type of affinity group that tends to have people who qualify for our product. Our agents wouldglad-hand people and identify likely prospects. Another possibility would be to have our current customers recommend their associates with similar interests and income.)

Notice that these two probabilities 0.5 and 1%are very much lower than the probabilities we used for the oil exploration cases. Targeted marketingmakes sense only when the targets are a very select population and our product has a high markup. (If the targets were a higher percentage of the general population, we would use the mass media such as the internet, TV, and print publications.)

The third probability we need to input is

**conditional**probability, the probability thatsomeone who would buy our product would pass our qualifying test. We couldsurvey current customers using the “test” we have devised and determine what percentage of them would pass the “test”. In this case, let us assume85% of our current customers would pass the test.

We input these three probabilities and the

** Bayesian AI Advisor**calculates the

**probability that we will be successful in selling our product to a person who passed the test. In this case that comes out to be42.5%.**

*inverse*Now we need the four essential financial inputs. Let us assumewe will have to spend about $1 per peson for the qualifying test. We’ll offer qualified prospects a gift or meal that costs us about $15 each. If they buy our product or service we’ll get a gross return of about $600 and weneed an ROI of 15% to justify the risk.

In this case, withsuch a small percentage of the population as our audience, itwould be foolish to invite everybody for the “free bait” – we would lose big! According to the

** Bayesian AI Advisor**,an untargeted approach would cost us about $3000 for each person who buys, anet loss of $2400 and a ROI of -80%. We expected results like that which is why we intend to do targeted marketing.

The

** Bayesian AI Advisor**is pretty sure what you should do in this case:

*Test first. If Test is Positive, do the Procedure.*

Proceed with the qualification testing and, if you get a positive result, go ahead offer them the free gift or meal andget rich.

OK, according to the

** Bayesian AI Advisor**, a targeted campaign would cost us about $506 for each eventual sale, andnet us $94 each foran ROI of over 18%, so that is definitely the way to go. If we can hook and net hundreds or thousands of people, we can get very rich.

Marketing Case #5 – Illustrating the Sensitivity to Test Quality

What if the conditional probability is 75% rather than 85%? That means the test quality is not as good as we’d like and only about 75% of our current customers would pass.The red box in the middle panelindicates the change in test qualityassumptions. With such a low percentage of the population interested in our product, we will have to do lots of testing to qualify a prospect, so the quality and cost to generate quantities ofcustomers will be enormous. Indeed, the

** Bayesian AI Advisor**calculates this as a deal breaker – we would make only $27 per saleand have anROI of less than 5%, which does not justify the risk.

Marketing Case #6 – Illustrating Error Detectionin the Bayesian AI Advisor

What if we enter probabilies that don’t make sence? Well, the

** Bayesian AI Advisor**has some error detection capabilities, as illustrated by the red boxes in the rightmost panel.

We have made two errors! The first red “Error” denotes that P(+) < P(S) x P(+|S), which is not possible. The second red “Error” lights up if the combination of P(S), P(+), and P(+|S) causes P(S|+) > 0.999. We have entered probabilities that caused the

** Bayesian AI Advisor**to calculate a value of 121.4%.Probabilities cannot go over 100%.

The

** Bayesian AI Advisor**also checks that the Gross Benefit if Successful is greater than the Cost of Procedure and that the Cost of Procedure is greater than the Cost of Test. The little red “OK” indicates that those values are not in error.

Example # 7 – Medical TestingHighly Accurate Tests but Ultra-Low “Probability of Success”

Medical testing illutrates another aspect of the value of analysis based on Bayesian

* inverse*probability. Generally, medical tests, such as tests for early detection of serious diseases or use of illegal drugs in the workplace are very good (with 95% to 99.99%accuracy) but the targets are rare in the population. Of course it is good that only a very small percentage of people have these serious diseases or use illegal drugs in the workplace, but, as you will see based on Bayesian analysis, that leads to a fairly high level of false positives.

MEDICAL TESTING – Case #7

Assume one in ten- thousand people in a given population is in the initial stages of some serious disease. Early testing could detect the disease in time to take preventive action. P(S) = 0.01% expresses that

**one in a ten-thousand**probability.

Assume further that there is a test that can detect the disease in its early stages with a 99.99% probability. That is, if a person has the disease and takes the test, there is a 99.99% probability the test will come back positive. However, the probability of a positive test is 0.2%, which means that, while99.8% of the population will get a negative result, 0.2% will get a positive.

The computed Bayes

* inverse*probability is 5%which means that only one in twenty people who get a positive test result will actually have the disease. There will be about nineteen false positives for every ten-thousand people tested. All twentypeople will have to be called back in for more intensive and intrusive (and expensive) re-testing using other techniques to assure they do not have the disease. For every person who has the disease, nineteen will be alarmed and inconvenienced unnecessarily.

If we assume the cost of the initial testing is about $1 per person, which is quite low for any kind of medical test, and that the cost of follow-up testing will be about $500 for each person identified as possibly having the disease,the cost of finding that one person in ten-thousand will be over $20,000 dollars! If we have to testone-million people,we will find about one-hundred with the disease and unnecessarily alarm and inconveniencenearly two-thousand, and spend over $2M! If we have to test two-hundred million people, it will cost over $400M!

Testing for Illegal Drugs in the Workplace

The

* Bayesian AI Advisor*could be used to determine the cost of doing screening tests in the workplace to identify employees who may be using hard drugs. They are putting themselves and their co-workers in jeopardy through their actions and risking liability actions against their employer if a co-worker or a member of the public is affected by their actions.

However, even if the tests are highly accurate and correctly catch 99% of the employees who have used illegal drugs recently, they are likely to have “false positives” for even more innocent employees. This willnecessitate follow-up testing that may be more intrusive and alarming, as well as very expensive. The

* Bayesian AI Advisor*will compute the likely costs of such a screening program. Management will have to determine if the elimination of various types of drugs among employees is worth the overall expense and disruption.

For example, it may well be worth the expense and disruption of innocent employee’s lives to do complete screening for airline pilots, bus drivers, employees in high-security jobs where drug use may expose them to blackmail, and others whose actions could cost lives and expose employers to gigantic liability claims. On the other hand, employees in less critical jobs might be screened only if evidence comes up that they are behaving strangely at work, are in financial stress, are having domestic troubles, etc.

Sensitivity Analysis – How Robust Are Your Results?Why Do Sensitivity Analysis?

The results and recommendations of the

* Bayesian AI Advisor*are, of course, dependent upon the probabilities and cost factors you input. There are two reasons to do sensitivity analysis and the

**provides graphical aids for both:**

*Bayesian AI Advisor*The inputs are “just estimates”. What if they are off by 10%? 20%? – or more?The

* Bayesian AI Advisor*graphically indicates what would happen to your ROI ifthe true value was lower than you estimated, down to half what you input. It also shows what the ROI would be if any of your input variables were double what you input. (If you can’t estimate a variable closer than a factor of two, you are really not ready to do anything that will put real money at risk!)

If the

* Bayesian AI Advisor*indicates you missed your ROI goal you may be able to make changes to achieve it.The most obvious change is to reduce your desired ROI. For example, if you expected to achieve 30% and the

**indicates you will only meet 20%, perhaps you should settle for that.**

*Bayesian AI Advisor*If reducing minimum acceptable ROI is a deal-breaker for your investors, then you need to consider more difficult changes. These include: 1)Improving the qualification test to tighten the limits, 2)Changing the drilling location or market sector to an area where you are more likely to strike oil and/or find customers, 3)Modifying your product so it has more appeal, 4) Raising the price to gain more benefit from each sale, 5) Reducing the price to increase volume of sales, 6) Reducing the cost of the qualification tests, etc.. The

* Bayesian AI Advisor*graphically indicates which of these changes are most likely to have the best pay-off for you.

Example of Sensitivity Analysis – Oil Exploration Case #2

The figure shows your financial factors (see Oil Exploration Case #2 subsection above): Your minimum acceptable ROI is 30% (indicated by the horizontal dashed line) . The most likely achieved ROI is indicated as the point where the other curves cross. It is about 20%.

Here is how to use the sensitivity analysis graph.

**Benefit if Successful:**The blue line marked with diamonds represents changes to the Benefit if Successful. If you can increase that by 10%, indicated as where it crosses the ROI line, you will be at 30% ROI. That means waiting for oil prices to go up or reducing your refining or transport costs, etc. In the marketing case, it means raising your prices, but that may be self-defeating if sales decline as a result, or reducing your prices to gain volume, but that too may be self-defeating if profits decrease as a result of a smaller margin per item sold.

**Procedure Cost:**Alternatively, according to the green line marked with triangles, if you reduce your Procedure Cost by about 15%, you meet the ROI. That means reducing the cost of drilling that well. In the targeted marketing case, perhaps you can just serve coffee and donuts rather than a full meal.

**Test Cost:**The third possibility is reducing your Test Cost, as indicated by the pink line with squares, but that would require reducing the Test Cost by over 20%. Perhaps you can get the seismic test company to lower their price? Perhaps the market research company will sell you that list of hot prospects for less money?

The figure shows your probability factors: Again, your minimum acceptable ROI is 30% (indicated by the horizontal dashed line) and the achieved ROI, indicated as the point where the other curves cross) is about 20%.

Here is how to use the sensitivity analysis graph.

**Probability of Success:**The black line marked with stary squaresrepresents changes to P(S), Probability of Success. If you could improve that by about 10% you would meet your ROI goal. P(S) is determined by where you intend to drill or the market segment you are addresing, etc., so you would have to move to a different area or market segment to improve that value.

**Probability of Positive Test:**Alternatively, according to the blue line marked with circles, if you reduce your Probability of Positive Test by about 10%, you meet the ROI. That would require making the qualification test tighter, so it discriminates the chances of success more sharply and has fewer “false positives”.

** Probability of Positive Test given Success: **The third possibility is increasing your Probability of Positive Test

*given*Success by about 10%. This requires tighter classification of your previous customers in the targeted marketing case or of the geological assumptions in an oil exploration case.

Please Let Me Know How this Tool Has Helped You

I’d appreciate it if you would Comment on this Knol and let me and others know your experiences with the

** Bayesian AI Advisor**, ideas for improvement, etc. You may also reach me by email at Ira@techie.com .advTHANKSance!