This is the final article in my series on Bayesian Argumentation. To understand this essay, read the introductory article for definition of key concepts and terminology.
Relevance is Not Absolute
Relevance exists in the context of the subject’s other prior beliefs. For example, if the subject believes that ($\bar{𝐶}$) the car is out of gas, and also ($\bar{B}$) the battery is dead, then both of these are good reasons to believe ($\bar{A}$) the car won’t start.
Yet neither is relevant on its own by our definition of relevance! Given that the car is out of gas, it makes no difference whether the battery is dead or not: the car won’t start anyway. In other words, ($\bar{B}$) the battery is dead is irrelevant to ($\bar{A}$) the car won’t start given ($\bar{𝐶}$) the car is out of gas.
But if the subject believes (𝐶) the car has gas, then ($\bar{B}$) the battery is dead will probably be relevant. When accepting one premise causes another premise to become relevant, we say that the premises are corelevant. If a premise is corelevant with some unexpressed premise, we can say that the premise is conditionally relevant.
Definition of Conditional Relevance
To define corelevance mathematically, we need to first define the conditional relevance of 𝐵 to 𝐴 given 𝐶, $R(A,B \vert C)$:
$$ R(A,B|C) = P(A|B,C) - P(A|\bar{B},C) $$
Definition of Corelevant
Then 𝐵 and 𝐶 are corelevant to 𝐴 if:
$$ R(A,B|C) ≠ R(A,B|\bar{C}) $$
Quantifying Corelevance
We can measure the magnitude of the correlevance as the difference:
$$ CR(A;B,C) = R(A,B|C) - R(A,B|\bar{C}) $$
It’s easy show that co-relevance is symmetrical (proof).
$$ CR(A;B,C) = CR(A;C,B) $$
Warrants
This idea of unexpressed beliefs that justify an argument evokes the idea of the warrant from the field of argumentation theory.
In every argument there is an unstated claim that this premise supports this conclusion. This doesn’t need to be stated because it’s implied by the fact that the argument was made. After asserting people are wearing jackets in support of the conclusion it’s probably cold outside, I don’t need to add, pedantically, “and you see, if people are wearing jackets it must be cold outside”.
This unexpressed premise that justifies the inferential leap from premise to conclusion is called the warrant.
The warrant doesn’t have to be a logical formula such as “if people are wearing jackets it must be cold outside”. It can be based on any kind of inferential rule (deductive, inductive, intuitive) or argumentation scheme (authority, analogy, example) – whatever justifies the inference in the mind of the arguer. Some academics use different terms for these concepts: our terminology is influenced by the influential Toulmin model, except we prefer the traditional terms premise and conclusion over grounds and claim. More precise definitions of our terms are given in the Deliberati Argument Model.
Bayesian Warrants
What is the warrant in a Bayesian model?
The warrant clearly has to do with the subject’s prior beliefs, because a Bayesian agent’s priors are precisely what justify, in their mind, any inferential leap from premise to conclusion.
For example, if our subject is more likely to believe that (𝐴) it is going to rain today if they believe that (𝐵) the sky is cloudy than if they do not, then there clearly exists a warrant justifying, in the subject’s mind, the inferential leap from 𝐵 to 𝐴.
But why does this warrant exist in the subject’s mind? What is the inferential rule that actually justifies the inference? Is it a deductive inference? Inductive? Gut feeling?
In the Toulmin Model, the warrant would be a rule along the lines of a cloudy sky is a sign of rain. In a Bayesian model, there are only corelevant claims that the subject must believe for the premise to be relevant. Every corelevant phrase helps, to paraphrase Toulmin’s words, to “justify the inferential leap from premise to conclusion”. So it’s best to think of all the warrant being the entire set of relevant prior beliefs.
If the prior beliefs of our subject are represented by the probability measure $P$, then we can say that, in the mind of the subject, a warrant exists justifying the inference from premise 𝐵 to conclusion 𝐴 iff:
$$ P(A|B) ≠ P(A|\bar{B}) $$
If the warrant exists, we say 𝐵 is relevant to 𝐴.
Counterfactual Relevance
Unfortunately, this definition of conditional relevance above still doesn’t fully capture the the common notion of “relevance”, because we can almost always find some second premise that makes the premise conditionally relevant. For example, the premise (𝐻) The car has a hood ornament may not seem relevant to (𝐴) the car will start, but it is conditionally relevant given the premise (𝑀) The car is powered by a magical hood ornament.
Of course, 𝑀 is pretty implausible – $P(M)$ may be infinitesimally small. But other more plausible corelevant premises may have small probabilities. For example, if the subject just filled the car with gas, they will be quite certain that (𝐺) the car has gas and thus $P(\bar{G})$ might be infinitesimally small. So in both cases we have corelevant premises with small prior probabilities, but a car running out of gas is something that is likely to actually happen in many similar scenarios, even if not this particular one.
Accounting for the difference in relevance in these two cases takes us into the metaphysical realm of modal logic, possible worlds, counterfactuals, and other difficult epistemological questions, that we won’t try to answer here.
Summary
So we’ve rounded out our reconciliation of Bayesianism and argumentation theory by defining the warrant as those prior beliefs that make the premise relevant. If you’ve read this far, I am impressed! I hope this has been useful.
In the next article I summarize all the definitions and mathematical formulas introduced in this series.
Proofs
Proof 1
Symmetry of Corelevance
$$ CR(A;B,C) = CR(A;C,B) $$
Proof:
$$ \begin{aligned} CR(A;B,C) &= R(A,B \vert C) - R(A,B \vert \bar{C}) \cr &= ( P(A \vert B,C) - P(A \vert \bar{B},C) ) \cr &\space\space\space\space- ( P(A \vert B,\bar{C}) - P(A \vert \bar{B},\bar{C}) ) \cr &= ( P(A \vert B,C) - P(A \vert B,\bar{C}) ) \cr &\space\space\space\space- ( P(A \vert \bar{B},C) - P(A \vert \bar{B},\bar{C}) ) \cr &= ( P(A \vert C,B) - P(A \vert \bar{C},B) \cr &\space\space\space\space- ( P(A \vert C,\bar{B}) - P(A \vert \bar{C},\bar{B}) ) \cr &= R(A,C \vert B) - R(A,C \vert \bar{B}) \cr &= CR(A;C,B) \cr \end{aligned} $$