A New Condition for Agglomeration in Bayesian Confirmation

Bayesian confirmation does not generally agglomerate over conjunction. That is, whenever a piece of evidence $E$ confirms two hypotheses $H_1$ and $H_2$ individually, it does not follow that $E$ also confirms them conjunctively. Or more formally, condition 1 below does not entail condition 2:

1. $P(H_1|E)>P(H_1)$ and $P(H_2|E)>P(H_2)$

2. $P(H_1\land H_2|E)>P(H_1\land H_2)$

However, if the following new condition is satisfied, condition 1 does entail condition 2:

3. $P(\neg H_1\land \neg H_2|E)>P(\neg H_1\land \neg H_2)$

I call condition 3 NOR-confirmation. To see how peculiar this new result is, imagine the following dialogue:

Researcher: Mr. Jones, good to see you! I just received your lab results. I have some good and some bad news for you. The bad news is that the results confirm that you have viral pneumonia; and they also confirm that you have bacterial pneumonia.

Jones: Oh dear! So I have both viral and bacterial pneumonia?! That explains why I feel so miserable!

Researcher: Well, that is not quite what I said, Mr. Jones! In any case, the good news is that the results also confirm that you have neither viral nor bacterial pneumonia.

Jones: Wait, didn’t you just tell me the opposite? Do the results confirm that I have viral and bacterial pneumonia or do they confirm that I don’t?!

Researcher: Well, they confirm both, Mr. Jones, albeit in different ways.

Jones: How can this be? Is there something wrong with the lab results?

Researcher: No, I can assure you that our lab results are flawless and absolutely reliable. In fact, it follows that they also confirm that you have viral and bacterial pneumonia at the same time.

The researcher’s utterances sound confusing her inference seems invalid. But the researcher’s inference is valid and everything she says is consistent.

This repo provides the code for the numerical results involving the new NOR-confirmation condition.

The results are published in Philosophy of Science:

• https://www.cambridge.org/core/journals/philosophy-of-science/article/new-condition-for-agglomeration-in-bayesian-confirmation/BF3906892462EEE562594F1F2713C119

A penultimate version of the paper can be found here:

• https://github.com/jottemka/agglomeration_condition/blob/main/agglomeration_ps.pdf

Results

Two types of prevalence values for NOR-confirmation are calculated:

1. Conjunctive Prevalence: proportion of probability functions satisfying conditions 1 and 3 and thus 2.

2. Conditional Prevalence: proportion of probability functions satisfying conditions 3 if they satisfy 1.

These results are shown on the left-hand side of the plot. For comparison, the corresponding values for Simpson's paradox are shown on the right-hand side. The precise values are shown below:

Effect	Conjunctive Prevalence	Conditional Prevalence
NOR-confirmation Effect	0.025101	0.100429
Simpson's Paradox	0.008324	0.033288

Further Reading

1. http://fitelson.org/confirmation/carnap_logical_foundations_of_probability.pdf

2. https://conservancy.umn.edu/server/api/core/bitstreams/659aa39b-5cd4-46d3-8f9a-94e97fafe464/content

3. https://www.tandfonline.com/doi/abs/10.1080/01621459.1972.10482387

4. https://www.cambridge.org/core/journals/philosophy-of-science/article/abs/how-to-confirm-the-conjunction-of-disconfirmed-hypotheses/45E5ECA1BA4293F465BEC18677CBD4BC

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