Coherence and Reliability: The Case of Overlapping Testimonies

Abstract

A measure of coherence is said to be reliability conducive if and only if a higher degree of coherence (as measured) among testimonies implies a higher probability that the witnesses are reliable. Recently, it has been proved that several coherence measures proposed in the literature are reliability conducive in scenarios of equivalent testimonies (Olsson and Schubert 2007; Schubert, to appear). My aim is to investigate which coherence measures turn out to be reliability conducive in the more general scenario where the testimonies do not have to be equivalent. It is shown that four measures are reliability conducive in the present scenario, all of which are ordinally equivalent to the Shogenji measure. I take that to be an argument for the Shogenji measure being a fruitful explication of coherence.

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Notes

  1. 1.

    Henceforth I will, for ease of exposition, use the terms ‘reliable’ and ‘unreliable’ instead of ‘fully reliable’ and ‘fully unreliable’.

  2. 2.

    As will be seen below, I use, for technical reasons, a slightly different notion of reliability conduciveness than the one Olsson and I used in our (2007).

  3. 3.

    This modification has also the following nice consequence. If the only ceteris paribus clause is P(R i ) = P′(R i ), it needs to be accompanied by a clause restricting the probability that an unreliable witness will give a certain testimony. For example, in (Olsson and Schubert 2007), it was stipulated that the probability that an unreliable witness will give a certain testimony is equal to the probability that the content of the testimony is true, i.e. P(E i R i ) = P(A i ). Given the present understanding of the ceteris paribus condition no such extra clause is necessary.

  4. 4.

    It was first proposed, in a slightly different context, by Keynes (1921, p. 151). See Angere (2008, p. 6).

  5. 5.

    Fuller versions of these counter-arguments are to be found in my Schubert (to appear). The arguments that I try to meet below are the ones I consider the strongest. There are a few additional arguments in Fitelson (2003), which I do not attempt to meet here, for lack of space. The main additional argument that need to be met is the so-called “subset problem”, i.e. the idea that the degree of coherence of a set should not solely depend on the probability of the entire set and the probabilities of its members, but also on the probabilities of its proper subsets with at least two members. The subset problem is a complicated issue which cannot be dealt with here. In any case it does not, of course, pose a problem in the present case of two testimonies (though it might do so in cases involving more testimonies).

  6. 6.

    Since C d , C Po and C Le are ordinally equivalent with C Sh in the present context, the argument applies to them as well. Meijs gives a similar argument against C r , which of course also applies to all ordinally equivalent measures (Meijs 2006, p. 28).

  7. 7.

    For the general formula for the Shogenji measure is C Sh (A 1,…,A n ) = P(A 1,…,A n )/(P(A 1) × ⋯ × P(A n )). This is equal to P(A)1−n in a case of completely overlapping testimonies, and hence the Shogenji measure is an increasing function of the number of testimonies, and a decreasing function of the probability of what the witnesses agree upon, in a case of completely overlapping testimonies. Hence it has the most characteristic features of striking agreement.

  8. 8.

    Lewis (1946, p. 246), BonJour (1985, p. 148).

  9. 9.

    Alternatively, one could see this as an indication that the Shogenji measure indeed has a high degree of similarity to notable coherence theorists’ pre-theoretical concept of coherence. On that interpretation, the Shogenji measure scores a hit according to the first criterion, too.

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Appendix

Appendix

Proof of observation 1

$$ \begin{aligned} P(R_{1} \left| {E_{1} ,E_{2} } \right.) = {\frac{{P(R_{1} ,E_{1} ,E_{2} )}}{{P(E_{1} ,E_{2} )}}} \end{aligned}$$

Let: \( \begin{aligned} P\left( {A_{i} } \right) = & a_{i} \\ P(A_{1} ,A_{2} ) = & a_{1,2} \\ P(E_{i} ) = & e_{i} \\ P(R_{i} \left| E \right._{i} ) = & m_{i} \\ P(R_{1} ,E_{1} ,E_{2} ) = & P(R_{1} ,R_{2} ,E_{1} ,E_{2} ,A_{1} ,A_{2} ) + P(R_{1} ,\neg R_{2} ,E_{1} ,E_{2} ,A_{1} ,A_{2} ) \\ & + P(R_{1} ,\neg R_{2} ,E_{1} ,E_{2} ,A_{1} ,\neg A_{2} ) \\ P(\neg R_{1} ,E_{1} ,E_{2} ) = & P(\neg R_{1} ,R_{2} ,E_{1} ,E_{2} ,A_{1} ,A_{2} ) + P(\neg R_{1} ,R_{2} ,E_{1} ,E_{2} ,\neg A_{1} ,A_{2} ) \\ & + P(\neg R_{1} ,\neg R_{2} ,E_{1} ,E_{2} ,A_{1} ,A_{2} ) + P(\neg R_{1} ,\neg R_{2} ,E_{1} ,E_{2} ,A_{1} ,\neg A_{2} ) \\ & + P(\neg R_{1} ,\neg R_{2} ,E_{1} ,E_{2} ,\neg A_{1} ,A_{2} ) + P(\neg R_{1} ,\neg R_{2} ,E_{1} ,E_{2} ,\neg A_{1} ,\neg A_{2} ) \\ \end{aligned} \) Let R 1 be a random variable taking R 1 and ¬R 1 as its values (and likewise for R 2, E 1, E 2, A 1 and A 2). Then:

$$ \begin{aligned} {\user2{R}}_{1}, {\user2{R}}_{2}, {\user2{E}}_{1}, {\user2{E}}_{2}, {\user2{A}}_{1}, {\user2{A}}_{2} = & \left( {{\user2{E}}_{1}, {\user2{R}}_{2}, {\user2{A}}_{2}, {\user2{E}}_{2} \left| {{\user2{R}}_{1} } \right., {\user2{A}}_{1} } \right)\left( {{\user2{R}}_{1}, {\user2{A}}_{1} } \right) \\ = & \left( {{\user2{E}}_{1} \left| {{\user2{R}}_{1} } \right., {\user2{A}}_{1} } \right)\left( {{\user2{R}}_{2}, {\user2{A}}_{2}, {\user2{E}}_{2} \left| {{\user2{R}}_{1} } \right., {\user2{A}}_{1} } \right)\left( {{\user2{R}}_{1}, {\user2{A}}_{1} } \right)\quad ({\text{5}}) \\ = & \left( {{\user2{E}}_{1} \left| {{\user2{R}}_{1} } \right., {\user2{A}}_{1} } \right)\left( {{\user2{R}}_{2} , {\user2{A}}_{2}, {\user2{E}}_{2}, {\user2{R}}_{1}, {\user2{A}}_{1} } \right) \\ = & \left( {{\user2{E}}_{1} \left| {{\user2{R}}_{1} } \right., {\user2{A}}_{1} } \right)\left( {{\user2{E}}_{2}, {\user2{R}}_{1}, {\user2{A}}_{1} \left| {{\user2{R}}_{2}, {\user2{A}}_{2} ,} \right.} \right)\left( {{\user2{R}}_{2}, {\user2{A}}_{2} } \right) \\ = & \left( {{\user2{E}}_{1} \left| {{\user2{R}}_{1} } \right., {\user2{A}}_{1} } \right)\left( {{\user2{E}}_{2} \left| {{\user2{R}}_{2}, {\user2{A}}_{2} ,} \right.} \right)\left( {{\user2{R}}_{1}, {\user2{A}}_{1} \left| {{\user2{R}}_{2}, {\user2{A}}_{2} } \right.} \right)\left( {{\user2{R}}_{2}, {\user2{A}}_{2} } \right)\quad ({\text{5}}) \\ = & \left( {{\user2{E}}_{1}, {\user2{R}}_{1}, {\user2{A}}_{1} } \right)\left( {{\user2{E}}_{2}, {\user2{R}}_{2}, {\user2{A}}_{2} } \right){\frac{{\left( {{\user2{R}}_{1}, {\user2{A}}_{1}, {\user2{R}}_{2}, {\user2{A}}_{2} } \right)}}{{\left( {{\user2{R}}_{1}, {\user2{A}}_{1} } \right)\left( {{\user2{R}}_{2}, {\user2{A}}_{2} } \right)}}} \\ = & \left( {{\user2{E}}_{1}, {\user2{R}}_{1}, {\user2{A}}_{1} } \right)\left( {{\user2{E}}_{2}, {\user2{R}}_{2}, {\user2{A}}_{2} } \right){\frac{{\left( {{\user2{A}}_{1}, {\user2{A}}_{2} } \right)}}{{\left( {{\user2{A}}_{1} } \right)\left( {{\user2{A}}_{2} } \right)}}}\quad ({\text{6}}) \\ = & \left( {{\user2{A}}_{1} \left| {{\user2{E}}_{1}, {\user2{R}}_{1} } \right.} \right)\left( {{\user2{A}}_{2} \left| {{\user2{E}}_{2}, {\user2{R}}_{2} } \right.} \right){\frac{{\left( {{\user2{A}}_{1}, {\user2{A}}_{2} } \right)}}{{\left( {{\user2{A}}_{1} } \right)\left( {{\user2{A}}_{2} } \right)}}}\left( {{\user2{R}}_{1} \left| {{\user2{E}}_{1} } \right.} \right)\left( {{\user2{E}}_{1} } \right)\left( {{\user2{R}}_{2} \left| {{\user2{E}}_{2} } \right.} \right)\left( {{\user2{E}}_{2} } \right) \\ \end{aligned} $$

Hence:

$$ \begin{aligned} P(R_{1} ,E_{1} ,E_{2} ) = & P(R_{1} ,R_{2} ,E_{1} ,E_{2} ,A_{1} ,A_{2} ) + P(R_{1} ,\neg R_{2} ,E_{1} ,E_{2} ,A_{1} ,A_{2} ) + P(R_{1} ,\neg R_{2} ,E_{1} ,E_{2} ,A_{1} ,\neg A_{2} ) \\ = & e_{1} e_{2} m_{1} m_{2} {\frac{{a_{1,2} }}{{a_{1} a_{2} }}} + e_{1} e_{2} m_{1} \left( {1 - m_{2} } \right) \\ P(\neg R_{1} ,E_{1} ,E_{2} ) = & P(\neg R_{1} ,R_{2} ,E_{1} ,E_{2} ,A_{1} ,A_{2} ) + P(\neg R_{1} ,R_{2} ,E_{1} ,E_{2} ,\neg A_{1} ,A_{2} ) \\ & + P(\neg R_{1} ,\neg R_{2} ,E_{1} ,E_{2} ,A_{1} ,A_{2} ) + P(\neg R_{1} ,\neg R_{2} ,E_{1} ,E_{2} ,A_{1} ,\neg A_{2} ) \\ & + P(\neg R_{1} ,\neg R_{2} ,E_{1} ,E_{2} ,\neg A_{1} ,A_{2} ) + P(\neg R_{1} ,\neg R_{2} ,E_{1} ,E_{2} ,\neg A_{1} ,\neg A_{2} ) \\ = & e_{1} e_{2} \left( {1 - m_{1} } \right)m_{2} + e_{1} e_{2} \left( {1 - m_{1} } \right)\left( {1 - m_{2} } \right) \\ \end{aligned} $$

Hence:

$$ \begin{aligned} {\frac{{P(R_{1} ,E_{1} ,E_{2} )}}{{P(E_{1} ,E_{2} )}}} = & {\frac{{e_{1} e_{2} m_{1} m_{2} {\frac{{a_{1,2} }}{{a_{1} a_{2} }}} + e_{1} e_{2} m_{1} \left( {1 - m_{2} } \right)}}{{e_{1} e_{2} m_{1} m_{2} {\frac{{a_{1,2} }}{{a_{1} a_{2} }}} + e_{1} e_{2} m_{1} \left( {1 - m_{2} } \right) + e_{1} e_{2} \left( {1 - m_{1} } \right)m_{2} + e_{1} e_{2} \left( {1 - m_{1} } \right)\left( {1 - m_{2} } \right)}}} \\ = & {\frac{{{\frac{{a_{1,2} }}{{a_{1} a_{2} }}} + {\frac{{\left( {1 - m_{2} } \right)}}{{m_{2} }}}}}{{{\frac{{a_{1,2} }}{{a_{1} a_{2} }}} + {\frac{{\left( {1 - m_{2} } \right)}}{{m_{2} }}} + {\frac{{\left( {1 - m_{1} } \right)}}{{m_{1} }}} + {\frac{{\left( {1 - m_{1} } \right)\left( {1 - m_{2} } \right)}}{{m_{1} m_{2} }}}}}} \\ \end{aligned} $$

It should be clear that within the limits of the scenario, P(R 1|E 1, E 2) is a strictly increasing function of P(A 1, A 2)/(P(A 1)P(A 2)). Note that the same holds for P(R 2|E 1, E 2).

Proof of observation 5

In order to see that C d , C Po and C Le are not reliability conducive, let us consider the sets of hypotheses {F 1, F 2} and {G 1, G 2} with the following probability distributions:

$$ P(F_{1} ) = P(F_{2} ) = 0,1 $$
$$ P(F_{1} ,F_{2} ) = 0,025 $$
$$ P(G_{1} ) = P(G_{2} ) = 0,5 $$
$$ P(G_{1} ,G_{2} ) = 0,5 $$

As we have seen, a coherence measure must be an increasing function of D(A 1, A 2) =  P(A 1, A 2)/(P(A 1)P(A 2)) in order to be informative and reliability conducive in this scenario. Now D(F 1, F 2) = 2, 5 and D(G 1, G 2) = 2. Hence in order to be reliability conducive, a measure of coherence must yield C(F 1, F 2) ≥ C(G 1, G 2).

Let us first consider C d :

$$ \begin{aligned} C_{d} \left( {A_{ 1} , A_{ 2} } \right) =\, & \left( {\left( {(P(A_{1} ) + P(A_{2} )} \right){\frac{{P(A_{1} ,A_{2} )}}{{P(A_{1} )P(A_{2} )}}} - P(A_{1} ) - P(A_{2} )} \right)/2 \\ C_{d} (F_{1} ,F_{2} ) =\, & 0,15 \\ C_{d} (G_{1} ,G_{2} ) =\, & 0,5 \\ \end{aligned} $$

Hence we see that C d is not reliability conducive in the scenario at hand.

Next let us consider C Po:

$$ \begin{aligned} C_{\text{Po}} (A_{1} ,A_{2} ) = \,& {\frac{{\left( {C_{\text{Sh}} - 1} \right) \times [2 + \left( {{\frac{{P\left( {A_{1} } \right)}}{{P\left( {A_{2} } \right)}}} + {\frac{{P\left( {A_{2} } \right)}}{{P\left( {A_{1} } \right)}}}} \right)P(A_{1} A_{2} )]}}{{2\left( {C_{\text{Sh}} + 1} \right)}}} \\ C_{\text{Po}} (F_{1} ,F_{2} ) \approx \,& 0,44 \\ C_{\text{Po}} (G_{1} ,G_{2} ) =\, & 0,5 \\ \end{aligned} $$

Thus C Po is not reliability conducive in the scenario at hand either.

Finally, let us consider C Le:

$$ \begin{aligned} C_{\text{Le}} (A_{1} ,A_{2} ) = & \left( {P(A_{1} \left| {A_{2} } \right.) + P(A_{2} \left| {A_{1} } \right.) - P(A_{1} ) - P(A_{2} )} \right)/2 \\ C_{\text{Le}} (F_{1} ,F_{2} ) = & 0,15 \\ C_{\text{Le}} (G_{1} ,G_{2} ) = & 0,5 \\ \end{aligned} $$

Hence C Le is not reliability conducive in the scenario at hand.

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Schubert, S. Coherence and Reliability: The Case of Overlapping Testimonies. Erkenn 74, 263–275 (2011). https://doi.org/10.1007/s10670-010-9246-y

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Keywords

  • Prior Probability
  • Coherence Measure
  • High Posterior Probability
  • Present Scenario
  • Equivalent Measure