Suppose elements $a_1,\dots ,a_k$ of a boolean algebra A are assigned probabilities ${\pi }_1,\dots {\pi }_k$. Boole asked to determine the possible probabilities of some other element $a\in A$, given the assignment $a_i\mapsto {\pi }_i,(i=1,\dots ,k)$. De Finetti's solution of Boole's problem yields a closed interval $I_a\subseteq \mathrm{[}\mathrm{0},\mathrm{1}\mathrm{]}$ such that the set of possible probabilities of a coincides with $I_a$. $I_a$ is nonempty iff the assignment is consistent in de Finetti's sense. Now suppose the probability of $a_k$ undergoes a small perturbation ${\pi }_k\mapsto {\pi }_k+\mathrm{d}{\pi }_k$. We study the resulting modification of $I_a$. For instance, we prove that the one-sided derivatives $\partial \mathrm{length}(I_a)/\partial {\pi }_k^{\pm }$ always have a rational, (generally non-integer) value. In the particular case when each probability ${\pi }_i$ is rational and A is presented as a Lindenbaum algebra, all these derivatives are Turing computable. Their existence domain is decidable by the Tarski-Seidenberg algorithm for polynomial equations and inequalities in real closed fields. Our results build on de Finetti's consistency notion and his solution of Boole's problem, and extend Hailperin's polyhedral methods for combining bounds on probabilities.

假设元素 ${\mathrm{一种}}_{\mathrm{1个}}，\dots ，{\mathrm{一种}}_{ķ}$布尔代数A的概率被分配了${\pi }_{\mathrm{1个}}，\dots {\pi }_{ķ}$。布尔要求确定其他要素的可能概率$\mathrm{一种}\in \mathrm{一种}$给定任务 ${\mathrm{一种}}_{\mathrm{一世}}\mapsto {\pi }_{\mathrm{一世}}，（\mathrm{一世}=\mathrm{1个}，\dots ，ķ）$。De Finetti对布尔问题的解决方案产生了一个封闭的区间${\mathrm{一世}}_{\mathrm{一种}}\subseteq \mathrm{[}\mathrm{0}，\mathrm{1个}\mathrm{]}$使得所述一组可能的概率的一个与一致${\mathrm{一世}}_{\mathrm{一种}}$。 ${\mathrm{一世}}_{\mathrm{一种}}$如果赋值在de Finetti的意义上是一致的，则为非空。现在假设${\mathrm{一种}}_{ķ}$ 受到一点微扰 ${\pi }_{ķ}\mapsto {\pi }_{ķ}+\mathrm{d}{\pi }_{ķ}$。我们研究了对${\mathrm{一世}}_{\mathrm{一种}}$。例如，我们证明了单边导数$\partial \mathrm{长度}（{\mathrm{一世}}_{\mathrm{一种}}）/\partial {\pi }_{ķ}^{\pm }$总是具有一个合理的（通常是非整数）值。在特定情况下，当每个概率${\pi }_{\mathrm{一世}}$是有理的，A表示为Lindenbaum代数，所有这些导数都是图灵可计算的。它们的存在域可通过Tarski-Seidenberg算法确定多项式方程组和实数封闭域中的不等式。我们的结果建立在de Finetti的一致性概念及其对Boole问题的解决方案的基础上，并扩展了Hailperin的多面体方法以结合概率边界。