Drilling problem

Drilling problem with Wolfram.md

Mill's method of induction with ChatGPT and Wolfram plug-in

User prompt: Do you know Mill's methods of induction?

Wolfram GPT: Yes, I am familiar with Mill's methods of induction. These are a set of methods proposed by philosopher John Stuart Mill to identify causal relationships. They are used in inductive reasoning to analyze observations and derive general principles. Mill described five methods in his work A System of Logic:

1. The Method of Agreement: Identifies a common factor present in all cases where the phenomenon occurs.
2. The Method of Difference: Identifies a factor that is present when the phenomenon occurs and absent when it does not, under otherwise similar conditions.
3. The Joint Method of Agreement and Difference: Combines the first two methods to strengthen causal inference.
4. The Method of Residues: Explains the remaining part of a phenomenon by assigning it to known causes after accounting for other parts.
5. The Method of Concomitant Variations: Establishes a causal relationship by observing how changes in one factor are associated with changes in another.

These methods aim to systematically investigate cause-and-effect relationships and are foundational in experimental and observational science. If you would like, I can delve into specific examples or applications of these methods!

The classic food poisoning problem

User prompt: please explain Mill's methods using the classic food poisoning example.

Sure! Here's an example using the classic food poisoning problem. Suppose several people ate a meal together and later some of them got sick. We want to figure out which food item might have caused the illness.

Data

Suppose several people ate a meal together and later some of them got sick. We want to figure out which food item might have caused the illness. We record what each person ate and whether they became sick:

Person	Ate Chicken	Ate Salad	Ate Ice Cream	Became Sick
A	Yes	Yes	Yes	Yes
B	Yes	No	Yes	Yes
C	No	Yes	Yes	No
D	Yes	Yes	No	Yes
E	No	No	Yes	No

Analysis

Using Mill's Methods, we proceed:

1. Method of Agreement:

We look for a common factor in all cases where people became sick.

• Among A, B, and D (those who got sick), all ate chicken. This suggests chicken might be the cause.

2. Method of Difference:

Compare cases where people became sick (A, B, D) with those who didn't (C, E).

• Person C did not eat chicken and did not get sick.
• Person E also did not eat chicken and did not get sick.
• This reinforces the idea that chicken could be the cause.

3. Joint Method of Agreement and Difference:

Combining the above, chicken appears to be the common factor in sickness and absent when there is no sickness. This strengthens our conclusion.

4. Method of Residues:

If we know for certain that salad and ice cream do not cause sickness (e.g., from prior testing), the remaining cause is likely chicken.

5. Method of Concomitant Variations:

If we had quantitative data (e.g., the amount of chicken consumed and the severity of sickness), we might find that more chicken consumption corresponds to worse symptoms, further linking chicken to the illness.

Conclusion

Based on the analysis, chicken is the most likely cause of food poisoning in this case. Would you like a more detailed breakdown or an analysis using numerical data?

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A harder problem

User prompt: This case was very straightforward; can you make up a more difficult one?

Wolfram GPT:

Let’s complicate the scenario with more data and additional food items. The goal is to apply Mill's Methods to figure out the likely cause of food poisoning when the situation isn't as clear-cut. Suppose 10 people attended a dinner party, and the following foods were served: chicken, salad, ice cream, shrimp, and bread. Each person ate a subset of the foods, and some got sick afterward. Here's the table:

Person	Chicken	Salad	Ice Cream	Shrimp	Bread	Sick
A	Yes	Yes	Yes	No	Yes	Yes
B	No	Yes	Yes	Yes	No	Yes
C	Yes	No	No	Yes	Yes	No
D	Yes	Yes	Yes	Yes	Yes	Yes
E	No	No	Yes	Yes	Yes	No
F	Yes	No	Yes	Yes	No	Yes
G	No	Yes	No	Yes	Yes	No
H	Yes	Yes	No	Yes	Yes	Yes
I	Yes	No	Yes	No	Yes	No
J	No	Yes	Yes	Yes	Yes	Yes

Analysis

Method of Agreement:

Look at all the people who got sick: A, B, D, F, H, and J. The foods they ate:

• Chicken: A, D, F, H
• Salad: A, B, D, H, J
• Ice Cream: A, B, D, F, J
• Shrimp: B, D, F, H, J
• Bread: A, D, H, J

No single food item was eaten by all sick individuals. However:

• Ice cream was eaten by 5 out of the 6 sick individuals (A, B, D, F, J).
• Shrimp was eaten by 5 out of the 6 sick individuals (B, D, F, H, J).

Method of Difference:

Compare cases of sickness with cases where people didn’t get sick (C, E, G, I):

• Ice Cream: Only 1 out of 4 healthy individuals (E) ate ice cream. This makes ice cream suspicious.
• Shrimp: 3 out of 4 healthy individuals (C, E, G) ate shrimp. This makes shrimp less likely to be the cause.

Joint Method of Agreement and Difference:

Ice cream shows up more consistently with sickness and less consistently with health. This strengthens the suspicion of ice cream being the culprit.

Method of Residues:

If we have prior knowledge ruling out salad, bread, and shrimp as causes, ice cream remains as the most plausible explanation.

Method of Concomitant Variations:

If we had data on how much ice cream each person consumed and the severity of symptoms, we might find a positive correlation between the two.

Conclusion

Based on the analysis, ice cream is the most likely cause of the food poisoning. The presence of shrimp in many sick cases initially made it a suspect, but the difference analysis cleared it. This example illustrates how combining the methods helps resolve ambiguities in complex datasets.

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An Oil and Gas well drilling application

User prompt: the table below indicates which of the wells (first column) were problematic during drilling due to loss of circulation (last column); can you use Mill's method to tell me which of the geophysical methods in the other column is the most predictive of drilling problems?

Well	Curvature	Diffraction imaging	Coherence	AVAz	VVAZ	Problem
A	TRUE	TRUE	FALSE	TRUE	TRUE	TRUE
B	TRUE	TRUE	TRUE	FALSE	TRUE	TRUE
C	FALSE	FALSE	TRUE	TRUE	FALSE	FALSE
D	FALSE	TRUE	FALSE	FALSE	TRUE	FALSE
E	TRUE	FALSE	FALSE	TRUE	TRUE	FALSE
F	TRUE	TRUE	FALSE	FALSE	FALSE	TRUE
G	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE

Wolfram GPT: Using the table you provided, I can analyze the data using Mill's methods to determine which factor (Curvature, Diffraction, Coherence, AVAz, or VVAZ) is most predictive of problematic wells. Here's the step-by-step breakdown:

Step 1: Method of Agreement

For the wells that are problematic (A, B, F, G), the associated factors are:

Well	Curvature	Diffraction	Coherence	AVAz	VVAZ
A	TRUE	TRUE	FALSE	TRUE	TRUE
B	TRUE	TRUE	TRUE	FALSE	TRUE
F	TRUE	TRUE	FALSE	FALSE	FALSE
G	TRUE	TRUE	TRUE	TRUE	TRUE

Common factors among problematic wells:

• Curvature: TRUE for all wells (A, B, F, G).
• Diffraction: TRUE for all wells (A, B, F, G).

Other columns (Coherence, AVAz, VVAZ) vary among the problematic wells and are less likely to predict problems.

Step 2: Method of Difference

Compare problematic wells (A, B, F, G) with non-problematic wells (C, D, E):

Well	Curvature	Diffraction	Coherence	AVAz	VVAZ	Problem
C	FALSE	FALSE	TRUE	TRUE	FALSE	FALSE
D	FALSE	TRUE	FALSE	FALSE	TRUE	FALSE
E	TRUE	FALSE	FALSE	TRUE	TRUE	FALSE

• Curvature: TRUE for all problematic wells, but FALSE for at least one non-problematic well (C, D).
• Diffraction: TRUE for all problematic wells, but FALSE for at least one non-problematic well (C, E).

Neither Curvature nor Diffraction alone distinguishes problematic wells perfectly.

Step 3: Joint Method of Agreement and Difference

Curvature and Diffraction together show strong agreement with problematic wells, but not uniquely.

• Coherence: TRUE for some problematic wells (B, G), but also for non-problematic well (C). This makes it less predictive.
• AVAz: FALSE for some problematic wells (B, F) and TRUE for others (A, G), so it is not consistent.
• VVAZ: TRUE for some problematic wells (A, B, G), but also for non-problematic wells (E, D).

Step 4: Conclusion

The most predictive combination of factors seems to be Curvature and Diffraction. They are both TRUE for all problematic wells, though not uniquely different from non-problematic wells. For higher confidence, more data or further statistical analysis might be required. Would you like me to formalize this into a visualization or perform a deeper

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User prompt: What if I told you that all methods - Curvature, Diffraction, Coherence, AVAz, and VVAZ - have robust theoretical basis for predicting this partiucluar drilling problem?

Wolfram GPT: If all the methods (Curvature, Diffraction, Coherence, AVAz, and VVAZ) have a theoretical basis for predicting the drilling problem, then we need to identify the method(s) most predictive in this dataset based on their empirical performance. Mill's methods provide qualitative insights, but we might also quantify the relationships using statistical techniques to identify the strongest predictor(s). Let’s proceed systematically.

Updated Analysis Using Mill's Methods

1. Method of Agreement:

Identify factors that are consistently TRUE for all problematic wells (A, B, F, G):

• Curvature: TRUE for all problematic wells.
• Diffraction: TRUE for all problematic wells.
• Coherence: TRUE for B and G but FALSE for A and F → inconsistent.
• AVAz: TRUE for A and G but FALSE for B and F → inconsistent.
• VVAZ: TRUE for A, B, and G but FALSE for F → inconsistent.

From the Method of Agreement, Curvature and Diffraction are the most consistent predictors.

2. Method of Difference:

Compare the problematic wells (A, B, F, G) with non-problematic wells (C, D, E):

Factor	Problematic Wells (A, B, F, G)	Non-Problematic Wells (C, D, E)
Curvature	TRUE	FALSE for C, D; TRUE for E
Diffraction	TRUE	FALSE for C, E; TRUE for D
Coherence	TRUE for B, G; FALSE for A, F	TRUE for C; FALSE for D, E
AVAz	TRUE for A, G; FALSE for B, F	TRUE for C, E; FALSE for D
VVAZ	TRUE for A, B, G; FALSE for F	TRUE for D, E; FALSE for C

• Curvature and Diffraction appear more often as TRUE for problematic wells and FALSE for non-problematic ones, although there are exceptions.

3. Joint Method of Agreement and Difference:

Combining Agreement and Difference strengthens the case for Curvature and Diffraction, as they align most strongly with problematic wells and differ most consistently from non-problematic wells.

Quantitative Analysis

If all the methods have theoretical justification, then we might consider a quantitative approach to determine their relative predictive power. For example:

1. Logistic Regression: Fit a model predicting the binary outcome "Problematic" based on the five factors. The regression coefficients would indicate which factors contribute most strongly.

2. Predictive Scoring: Calculate sensitivity (how often a method is TRUE for problematic wells) and specificity (how often a method is FALSE for non-problematic wells).

Would you like me to use a logistic regression model or compute predictive scores to quantify which factor is the best predictor empirically? I can generate this analysis computationally if you'd like.

User prompt: no, thanks, I've done that on my own already!