How Prevalent is Transitivity-Failure in Bayesian Confirmation?

It is a well-known fact among researchers working with probability calculus that transitivity can fail for Bayesian confirmation. That is, it is possible that A confirms B and B confirms C while A fails to confirm C, or graphically:

More formally, transitiviy-failure is any case where the following conditions are jointly satisfied:

1. $P(B|A)>P(B)$
2. $P(C|B)>P(C)$
3. $P(C|A)\ngtr P(C)$

Still, there is a growing number of conditions in the literature under which this cannot happen, some of which are surprisingly weak. This raises the question how prevalent the phenomenon of transitivity-failure is: perhaps, Bayesian confirmation is transitive in most cases?

This repo provides the code for a Monte Carlo approximation of the prevalence of transitivity-failure. Results for other inference patterns from non-monotonic reasoning and the logic of conditionals are also provided.

The results are published in the British Journal for the Philosophy of Science:

• https://www.journals.uchicago.edu/doi/10.1086/731830

A penultimate version of the paper can be found here:

• https://github.com/jottemka/transitivity_failure/blob/1cbcee8356a2d79dc0bef647c299de2d73079e83/transitivity.pdf

Results

Two types of prevalence values are calculated:

1. Conjunctive Prevalence reflects how likely it is that the antecedent of an inference pattern is satisfied and the consequent of that pattern is violated.

2. Conditional Prevalence reflects how likely it is that the consequent of an inference pattern is violated if the antecedent of that pattern is satisfied.

The plot above shows how the prevalence values for transitivity stabilize after 10,000 probability distributions. The table below also shows the results for other patterns from non-monotonic reasoning and the logic of conditionals. Notice that $A > B$ abbreviates that $A$ confirms $B$ and $A >_C B$ abbreviates that $A$ confirms $B$ conditional on $C$.

Label	Inference Pattern	Conjunctive Prevalence	Conditional Prevalence
Transitivity	If $A > B$ and $B > C$, then $A > C$	0.089709	0.358918
Conjunctive Transitivity	If $A > B$ and $B > C$, then $A∧B > C$	0.056351	0.225455
Conditional Transitivity	If $A > B$ and $B > C$, then $A >_B C$	0.110268	0.441173
Cumulative Transitivity	If $A > B$ and $A∧B > C$, then $A > C$	0.056489	0.225832
Agglomeration	If $B > A$ and $B > C$, then $B > A∧C$	0.025136	0.100567
Cautious Monotonicity	If $B > A$ and $B > C$, then $B∧C > A$	0.056483	0.225984
Rational Monotonicity	If $B > A$ and $B \ngtr \neg C$, then $B∧C > A$	0.056483	0.225984
Corroboration	If $A > B$ and $C > B$, then $A >_C B$ and $C >_A B$	0.091586	0.366428
Amalgamation	If $A > B$ and $C > B$, then $A∨C > B$	0.025011	0.100067

So, even though transitivity is not generally valid for Bayesian confirmation, transitivity-style inferences are more likely right than wrong. Nevertheless, they are more likely wrong than instances of other inference patterns.

Further Reading

1. https://doi.org/10.1086/288279

2. https://doi.org/10.1086/289089

3. https://link.springer.com/chapter/10.1007/978-94-017-2300-8_12

4. https://www.journals.uchicago.edu/doi/abs/10.1093/bjps/54.4.613

5. https://link.springer.com/article/10.1007/s13194-011-0033-7

6. https://link.springer.com/article/10.1007/s10670-020-00349-7

Virtual Environment Setup

Use the requirements file to create a new environment for this task.

pyenv local 3.11.3
python -m venv .venv
source .venv/bin/activate
pip install --upgrade pip
pip install -r requirements.txt

WindowsOS type the following commands :

Install the virtual environment and the required packages by following commands.

For PowerShell CLI :

python -m venv .venv
.venv\Scripts\Activate.ps1
pip install --upgrade pip
pip install -r requirements.txt

For Git-Bash CLI:

python -m venv .venv
source .venv/Scripts/activate
pip install --upgrade pip
pip install -r requirements.txt