Measuring the overall incoherence of credence functions

Abstract

Many philosophers hold that the probability axioms constitute norms of rationality governing degrees of belief. This view, known as subjective Bayesianism, has been widely criticized for being too idealized. It is claimed that the norms on degrees of belief postulated by subjective Bayesianism cannot be followed by human agents, and hence have no normative force for beings like us. This problem is especially pressing since the standard framework of subjective Bayesianism only allows us to distinguish between two kinds of credence functions—coherent ones that obey the probability axioms perfectly, and incoherent ones that don’t. An attractive response to this problem is to extend the framework of subjective Bayesianism in such a way that we can measure differences between incoherent credence functions. This lets us explain how the Bayesian ideals can be approximated by humans. I argue that we should look for a measure that captures what I call the ‘overall degree of incoherence’ of a credence function. I then examine various incoherence measures that have been proposed in the literature, and evaluate whether they are suitable for measuring overall incoherence. The competitors are a qualitative measure that relies on finding coherent subsets of incoherent credence functions, a class of quantitative measures that measure incoherence in terms of normalized Dutch book loss, and a class of distance measures that determine the distance to the closest coherent credence function. I argue that one particular Dutch book measure and a corresponding distance measure are particularly well suited for capturing the overall degree of incoherence of a credence function.

Appendix: SSK’s neutral/sum and neutral/max measures

In a series of papers, SSK have developed a class of measures of degrees of incoherence based on Dutch books (2000, 2002a, 2002b). In this appendix, I focus on the neutral/sum measure, to give a more detailed formal exposition of my arguments in Sect. 6.2. To see how this Dutch book measure works, suppose there is an agent who has a credence function P that assigns credences to a set of propositions $\{\hbox {A}_{1},\ldots , \hbox {A}_{\mathrm{n}}\}$.¹⁹ We can represent a bet on or against one of these propositions according to the agent’s credences in the following way:

$$\begin{aligned} \hbox {Bet: }\upalpha \left( {\hbox {I}\left( {\hbox {A}_\mathrm{i} } \right) -\hbox {P}\left( {\hbox {A}_\mathrm{i} } \right) } \right) \end{aligned}$$

In this case, $\hbox {I}(\hbox {A}_{\mathrm{i}})$ is the indicator function of $\hbox {A}_{\mathrm{i}}$, which assigns a value of 1 if $\hbox {A}_{\mathrm{i}}$ is true and a value of 0 if $\hbox {A}_\mathrm{i}$ is false. $\hbox {P}(\hbox {A}_\mathrm{i})$ is the credence the agent assigns to the proposition $\hbox {A}_\mathrm{i}$. The coefficient $\upalpha$ determines both the size of the bet, as well as whether the agent is betting on or against $\hbox {A}_\mathrm{i}$. If $\upalpha >0$, then the agent bets on the truth of $\hbox {A}_\mathrm{i}$, whereas if $\upalpha <0$, the agents bets against the truth of $\hbox {A}_\mathrm{i}$.

In the following, it will be assumed that an agent who assigns a precise credence to a proposition thereby evaluates as fair the bet on and the bet against that proposition at the price that is fixed by her credence.

An agent is incoherent if there is a collection of gambles she evaluates as fair that together guarantee a loss. Formally, we can represent this as follows: Let $\hbox {A}_1,\ldots ,\hbox {A}_\mathrm{n}$ be the propositions that some agent assigns credences to, let Cr be the agent’s credence function, which may or may not be probabilistically coherent, and let S be the set of possible world states. If there is some choice of coefficients $\upalpha _1, \ldots ,\upalpha _\mathrm{n}$, such that the sum of the payoffs of the bets on or against $\hbox {A}_1,\ldots ,\hbox { A}_\mathrm{n}$ is negative for every world state $\hbox {s}\in \hbox {S}$, then the agent is vulnerable to a Dutch book. Thus, there is a Dutch book iff ²⁰

$$\begin{aligned} \mathop {\sup }\limits _{s\in S} \sum _{i=1}^n {\alpha _i } (I_{A_i } (s)-Cr(A_i ))<0 \end{aligned}$$

This formula tells us how to determine whether a Dutch book can be made against an agent who has a given credence function in a given set of propositions. We can capture the guaranteed loss an agent faces from a collection of gambles of the form $\hbox {Y}_\mathrm{i} =\upalpha \left( {\hbox {I}\left( {\hbox {A}_\mathrm{i}}\right) -\hbox {Cr}\left( {\hbox {A}_\mathrm{i}}\right) } \right)$ as follows:²¹

$$\begin{aligned} G(Y)=-\min \left\{ 0,\mathop {\sup }\limits _{s\in S} \sum _{i=1}^n {\alpha _i } (I_{A_i } (s)-Cr(A_i ))\right\} \end{aligned}$$

In order to normalize the guaranteed loss to be able to measure an agent’s degree of incoherence, we can divide the guaranteed loss by the sum of the coefficients of the individual bets. This normalization is called the “neutral/sum” normalization by SSK. We can thus compute the rate of loss H(Y):

$$\begin{aligned} H(Y)=\frac{G(Y)}{\sum \limits _{i=1}^n {\left| {\mathop \alpha \nolimits _i } \right| } } \end{aligned}$$

The degree of incoherence can be determined for a set of propositions and a credence function over these propositions by choosing the coefficients $\upalpha _{1},\ldots , \upalpha _{\mathrm{n}}$ in such a way that H(Y) is maximized. To maximize H(Y), it may be necessary not to include certain propositions in the Dutch book, which can be achieved by setting the relevant coefficients $\upalpha _\mathrm{i}$ to 0.

We can illustrate how the measure works with an example. Suppose an agent has credences in two propositions, q and $\sim \hbox {q}$. Her credence assignment $f$ is incoherent, since she assigns $f\left( \hbox {q} \right) =0.5$ and $f({\sim }\hbox {q})=0.6$. In order to measure her rate of incoherence, we will first set up two bets with her, one for each proposition, and sum them in order to determine their combined payoff: $Y=\alpha _1 (I_q (s)-0.5)+\alpha _2 (I_{\lnot q} (s)-0.6)$

Since we can either be in a world where q is true or in a world where q is false, we can get two values for Y:

• If q, then $Y=\alpha _1 0.5-\alpha _2 0.6$

• If $\sim$q, then $Y=\alpha _2 0.4-\alpha _1 0.5$

Thus, we can calculate G(Y) as follows, where $\upalpha _{1}$, $\upalpha _2 >0$: ²²

if $\upalpha _2 \ge 1.25 \upalpha _1$ or $\upalpha _1 \ge 1.2 \upalpha _2$, then the second term in the braces in the G(Y) equation²³ is positive, which means that $G(Y)=0$

otherwise, $G(Y)=-\mathop {\sup }\limits _{s\in S} \{\alpha _1 (I_q (s)-0.5)+\alpha _2 (I_{\lnot q} (s)-0.6)\}$

Thus, when a Dutch book can be made, (i.e. when $\hbox {G}\left( \hbox {Y} \right) >0)$ we can measure the rate of incoherence by choosing the coefficients in such a way that H(Y) is maximized:

$$\begin{aligned} H(Y)=\frac{\mathop {-\sup }\limits _{s\in S} \{\alpha _1 (I_q (s)-0.5)+\alpha _2 (I_{\lnot q} (s)-0.6)\}}{\left| {\alpha _1 } \right| +\left| {\alpha _2 } \right| } \end{aligned}$$

The rate of incoherence is maximized in this example if we choose $\upalpha _1 =\upalpha _2$, which results in a rate of incoherence of 0.05.

I will now move on to the problems with the measure discussed in Sect. 6.2. The first example involved a comparison of the following two credence functions:

$$\begin{aligned} \begin{array}{lll} f\left( \hbox {p} \right) =0.6&{}\quad \quad &{} g\left( \hbox {p} \right) =0.6 \\ f\left( {{\sim }\hbox {p}} \right) =0.6&{}\quad \quad &{} g\left( {{\sim }\hbox {p}} \right) =0.6 \\ f\left( \hbox {q} \right) =0.5, &{}\quad \quad &{} g\left( \hbox {q} \right) =0.6 \\ f\left( {{\sim }\hbox {q}} \right) =0.5&{}\quad \quad &{} g\left( {{\sim }\hbox {q}} \right) =0.6 \\ \end{array} \end{aligned}$$

We noted that intuitively, $g$ is overall more incoherent than $f$. However, this is not the result we get from the neutral/sum measure. According to this measure, the agent would be equally incoherent in both cases. Here’s how that result comes about. First, consider the case in which the agent adopts $f$. The formula to calculate the degree of incoherence is the following (as the reader can easily verify, including bets on q and $\sim$q couldn’t possibly lead to a higher guaranteed loss, so they can and should be left out):

$$\begin{aligned} H(Y)=\frac{\mathop {-\sup }\limits _{s\in S} \{\alpha _1 (I_p (s)-0.6)+\alpha _2 (I_{\lnot p} (s)-0.6)\}}{\left| {\alpha _1 } \right| +\left| {\alpha _2 } \right| } \end{aligned}$$

As before, the agent’s rate of loss is maximized when we set $\upalpha _{1 }=\upalpha _{2} >$ 0. We can thus simplify the calculation as follows:

$$\begin{aligned} H(Y)=\frac{0.2a_1 }{2a_1 }=0.1 \end{aligned}$$

Thus, if the agent adopts $f$ as her credence function, her rate of loss is 0.1. Let us now compare this to what happens if the agent adopts $g$. If her credence function is $g$, we can calculate the rate of loss as follows:

$$\begin{aligned}&H(Y)\\&\quad =\!\frac{\mathop {\!-\!\sup }\limits _{s\in S} \{\alpha _1 (I_p (s)-0.6)+\alpha _2 (I_{\lnot p} (s)-0.6)+\alpha _3 (I_q (s)-0.6)+\alpha _4 (I_{\lnot q} (s)-0.6)\}}{\left| {\alpha _1 } \right| +\left| {\alpha _2 } \right| \!+\!\left| {a_3 } \right| +\left| {a_4 } \right| } \end{aligned}$$

If we try to find values for $\upalpha _{1}-\upalpha _{4}$ that maximize H(Y), we get an interesting result. There is no way of picking values for $\upalpha _{1}-\upalpha _{4}$ that lead to a higher rate of incoherence than 0.1. Rather, we get exactly the same rate of incoherence we had before as long as we choose $\upalpha _{1}=\upalpha _{2 }$and $\upalpha _{3}=\upalpha _{4}$, and we choose $\upalpha _1 >0$ and/or $\upalpha _3 >0$. However, this result is in tension with the intuition that an agent who adopts $g$ is more incoherent than an agent who adopts $f$. We can see more easily how this result arises if we strip our example down to the essential parts. In the calculation of the rate of loss of $g$, we are essentially combining two normalized Dutch books of the same kind into one, by adding the numerators and adding the denominators of each one. The two normalized Dutch books are the same as the one Dutch book we made against the agent who adopts $f$. Thus, to make it simple, the calculation for the rate of loss of the agent who adopts $g$ goes as follows:

$$\begin{aligned} H(Y)=\frac{0.2a_1 +0.2a_3 }{2a_1 +2a_3 } \end{aligned}$$

If two fractions are combined in such a way that the numerators and the denominators are added, the value of the resulting fraction is always in between or equal to the two original fractions. Thus, if we combine two normalized Dutch books with the same rate of loss in the neutral/sum measure, the resulting rate of loss is the same as before.²⁴

Moreover, it can even be beneficial when employing the neutral/sum measure not to make certain Dutch books at all in order to maximize the rate of loss. Remember that we are allowed to choose $\upalpha _\mathrm{i}$ = 0 if necessary to maximize the rate of loss. In a case in which there are two Dutch books that can be made against an agent, but one of them leads to a greater rate of loss on its own, the total rate of loss can be maximized by setting the relevant coefficients to 0. For example, suppose an agent has the credence function $f$, that is defined as follows: $f\left( \hbox {p} \right) =0.6,f\left( {{\sim }\hbox {p}} \right) =0.5,f\left( \hbox {T} \right) =0.9$. An agent who adopts $f$ can be Dutch booked in two ways: on her incoherent credences in the partition $\left\{ {\hbox {p}, {\sim }\hbox {p}} \right\}$, and on her non-zero credence in a contradiction. In this case, the rate of loss comes down to:

$$\begin{aligned} H(Y)=\frac{0.1a_1 +0.1a_3}{2a_1 +a_3} \end{aligned}$$

The rate of loss in this case reaches its maximum value of 0.1 if we set $\upalpha _{1}$ = 0. This amounts to only Dutch booking the agent on her credence in the tautology, but refraining from Dutch booking her on her incoherent credences in the partition $\left\{ {\hbox {p}, {\sim }\hbox {p}} \right\}$.

This feature of the measure’s normalization is the source of the swamping problem. Since only the worst Dutch book determines the agent’s degree of incoherence, incoherencies in other parts of the credence function get swamped and are not reflected in the agent’s degree of incoherence. This also gives rise to the problematic example in Sect. 6.2, in which the measure orders two credence functions in a way that seems intuitively exactly backwards, which makes the measure unsuitable to evaluate reasoning processes.

In order to determine the degree of incoherence of a credence function according to the neutral/max measure instead, we must make the following adjustment: instead of normalizing the Dutch book loss by dividing the betting loss by the sum of the stakes of the individual bets, we must normalize it by dividing the Dutch book loss by the stakes of the largest bet included in the gamble. Hence, the rate of loss that must be maximized becomes:

$$\begin{aligned} H(Y)=\frac{G(Y)}{\max \{\left| {\alpha _1 } \right| ,\ldots ,\left| {\alpha _n } \right| \}} \end{aligned}$$

I hope this appendix helpfully supplements the informal presentation of the arguments in the main body of the paper, and I will leave it to the reader to verify the results in the remaining sections of the main text.