## Shogenji’s new measure of Bayesian justification

Proponents of Bayesian epistemology have invented several different quantitative measures of how available evidence bears on different hypotheses. These measures are referred to as measures of evidential support, coherence, confirmation, or justification, depending a bit on the precise significance ascribed to them. Presently, Bayesians disagree over which of these measure is the most useful, but a new paper by Shogenji provides very appealing alternative measure and, with some luck, may even settle this debate.

At first sight, it might appear that the posterior probability $P(h|e)$ of a hypothesis $h$ conditional on observed evidence $e$ is the perfect measure for a Bayesian. However, a hypothesis sometimes has a high posterior probability simply by virtue of a high prior probability, without necessarily being support/confirmed/justified by virtue of the observed evidence. The second obvious candidate is the likelihood $P(e|h)$ of the evidence given the hypothesis, but this too fails to capture what Bayesians mean by support/confirmation/justification. Instead, several more complicated measures have been proposed by different authors. Here’s a sample (from Atkinson and Crupi et al., 2007):

• Carnap (1950)
$\displaystyle D(h,e) = P(h|e) - P(e)$

• Keynes (1921)
$\displaystyle R(h,e) = \log\left( \frac{P(h|e)}{P(h)} \right)$

• Kemeney and Oppenheim (1952)
$\displaystyle K(h,e) = \frac{P(e|h) - P(e|\lnot h)}{P(e|h) + P(e|\lnot h)}$

• Good (1950)
$\displaystyle L(h,e) = \log\left( \frac{P(h|e)}{P(\lnot h|e)} \right)$

• Carnap (1950)
$\displaystyle C(h,e) = P(h \land e) - P(h) P(e)$

• Christensen (1999)
$\displaystyle S(h,e) = P(h|e) - P(h| \lnot e)$

• Nozick (1981)
$\displaystyle N(h,e) = P(e|h) - P(e|\lnot h)$

• Mortimer (1988)
$\displaystyle M(h,e) = P(e|h) - P(e)$

• Finch (1960)
$\displaystyle R'(h,e) = \frac{P(h|e)}{P(h)} - 1$

• Rips (1960)
$\displaystyle G(h,e) = 1 - \frac{P(\lnot h|e)}{P(\lnot h)}$

The above measures are all normalized so that the hypothesis is considered supported, confirmed or justified by the evidence when the measure is positive and not so when the measure is negative. From the above embarrassment of riches, it’s at least clear that there is agreement that support/confirmation/justification is a relation between a hypothesis and available evidence, via a Bayesian probability. Shogenji reasons from a new perspective and arrives at the following measure of justification:

$\displaystyle J(h,e) = 1 - \frac{\log(P(h|e))}{\log(P(h))} = \frac{\log(P(h)) - \log(P(h|e))}{\log(P(h))}$

This measure has the convenient property that, for any two hypotheses that are independent both a priori and a posteriori (i.e. $P(h_1\land h_2) = P(h_1) P(h_2)$ and $P(h_1\land h_2|e) = P(h_1|e) P(h_2|e)$ are both satisfied), the condition $J(h_1,e) = J(h_2,e) = t$ implies $J(h_1 \land h_2, e) = t$. Other relations, where the equalities are replaced by inequalities, also hold. Up to monotonic transformations, Shogenji’s measure is furthermore unique in having this property.

It’s interesting to apply this measure to a modified version of Linda the feminist bank teller. In its original form, it concerns the Bayesian posterior probability and is taken to illustrates incorrect but intuitive probabilistic reasoning. From Wikipedia:

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which is more probable?

1. Linda is a bank teller.
2. Linda is a bank teller and is active in the feminist movement.

Clearly, $P(h_b|e) \geq P(h_b \land h_f|e)$, with $e$ denoting the presented information about Linda, $h_b$ the hypothesis that she is a bank teller, and $h_f$ the hypothesis that she is active in the feminist movement. But what if the question is changed to concern degree of support/confirmation/justification instead of posterior probability (as discussed by Crupi et al., 2007b)? One then have an interesting test case for competing measures! Let

$\displaystyle J_b = \frac{\log(P(h_b)) - \log(P(h_b|e))}{\log(P(h_b))}$

be the degree of justification for the hypothesis that Linda is a bank teller. Then

$\displaystyle J_{bf} = \frac{\log(P(h_b)) - \log(P(h_b|e)) + \log(P(h_f|h_b)) - \log(P(h_f|h_b\land e))}{\log(P(h_b)) + \log(P(h_f|h_b))}$

or, equivalently,

$\displaystyle J_{bf} = \frac{J_b \log(P(h_b))}{\log(P(h_b)) + \log(P(h_f|h_b))} + \frac{\log(P(h_f|h_b)) - \log(P(h_f|h_b\land e))}{\log(P(h_b)) + \log(P(h_f|h_b))}$

Assuming for simplicity that the hypotheses $h_b$ and $h_f$ are independent both a priori and on the available evidence,

$\displaystyle J_{bf} = \frac{J_b \log(P(h_b))}{\log(P(h_b)) + \log(P(h_f))} + \frac{\log(P(h_f)) - \log(P(h_f|e))}{\log(P(h_b)) + \log(P(h_f))}$

or, equivalently,

$\displaystyle J_{bf} = \frac{J_b \log(P(h_b)) + J_f \log(P(h_f))}{\log(P(h_b)) + \log(P(h_f))}$

The degree of justification $J_{bf}$ in the conjunction of both hypotheses is seen to be an interpolation of the degrees of justification for the individual hypotheses. Since it is intuitively felt that the information about Linda confirms the hypothesis $h_f$ fairly strongly while being at best irrelevant for $h_b$, it is natural that some of this degree of confirmation is retained for the conjunction $h_b\land h_f$ and that $J_{bf} > J_b$. This illustrates the unique way in which Shogenji’s measure deals with the problem of irrelevant conjunction.

Shogenji’s article and Atkinson’s commentary are recommended for further discussion.

References

D. Atkinson. Confirmation and justification. A commentary on Shogenji’s measure. Synthese (online first), DOI: 10.1007/s11229-009-9696-4

V. Crupi, K. Tentori, and M. Gonzalez. On Bayesian Measures of Evidential Support: Theoretical and Empirical Issues. Philosophy of Science 74:229 (2007) [Available here.]

V. Crupi, B. Fitelson and K. Tentori. Probability, Confirmation, and the Conjunction Fallacy. Thinking & Reasoning 14:182 (2007b) [Also available at the Phil-Sci Archive and here.]

T. Shogenji. The degree of epistemic justification and the conjunction fallacy. Synthese (online first), DOI: 10.1007/s11229-009-9699-1