Probabilistic Belief Contraction

Abstract

Probabilistic belief contraction has been a much neglected topic in the field of probabilistic reasoning. This is due to the difficulty in establishing a reasonable reversal of the effect of Bayesian conditionalization on a probabilistic distribution. We show that indifferent contraction, a solution proposed by Ramer to this problem through a judicious use of the principle of maximum entropy, is a probabilistic version of a full meet contraction. We then propose variations of indifferent contraction, using both the Shannon entropy measure as well as the Hartley entropy measure, with an aim to avoid excessive loss of beliefs that full meet contraction entails.

Appendix: Proofs

Theorem 1

Indifferent contraction satisfies all the postulates of contraction C1 to C6 and CO.

Proof

Let P be the probability distribution that represents the agent’s initial belief state. Let α be a belief of the agent, that is, P(α) = 1. We further assume that α is a non-trivial belief, that is, $\nvdash\alpha$. The result of indifferent contraction of P by α is denoted by P ⁻_α .

Let the cardinality of $\Upomega, [\alpha], [\neg\alpha]$ and the support of P be n, l, m, and l′ respectively. Suppose the initial distribution is as follows:

$$ P=\langle p_{1},\ldots,p_{l},\underbrace{0,\ldots,0}_{m}\rangle. $$

The result of indifferent contraction is then given by the following:

$$ P^{-}_{\alpha}= \langle dp_{1},\ldots, dp_{l}, \underbrace{\frac{1}{h+m},\ldots,\frac{1}{h+m}}_{m}\rangle, $$

where h = 2^H(P) and $d=\frac{h}{h+m}$. From Definition 1 it follows that indifferent contraction satisfies the postulate C1. Clearly 0 < d < 1. For every $\omega\in[\alpha]$, we have P ⁻_α (ω) = d P(ω). The resultant probability of α is given by, $P^{-}_{\alpha}(\alpha) = \sum_{\omega\in[\alpha]} d P(\omega) = d$. Thus P ⁻_α (α) < 1, satisfying postulate C2.

When $\vdash\alpha\leftrightarrow\beta$, then we have [α] = [β] and $[\neg\alpha]=[\neg\beta]$. Indifferent contraction of P by α is such that P ⁻_α (ω) = d P(ω), when $\omega\in[\alpha]$ and similarly, indifferent contraction of P by β is such that P ⁻_β (ω) = d P(ω), when $\omega\in[\beta]$, where $d = \frac{h}{h+m}$, m is the cardinality of [$\neg\alpha$] as well as that of [$\neg\beta$]. Thus we have P ⁻_α (ω) = P ⁻_β (ω), when $\omega\in[\alpha]$ (or when $\omega\in[\beta]$). Also, $P^{-}_{\alpha}(\omega) = \frac{1}{h+m} = P^{-}_{\beta}(\omega)$, for $\omega\in[\neg\alpha]$ (or $\omega\in[\neg\beta]$). Thus we have P ⁻_α (ω) = P ⁻_β (ω), for every $\omega\in\Upomega$, hence P ⁻_α  = P ⁻_β , satisfying the postulate C3.

From Definition 1, it is clear that indifferent contraction satisfies the postulate C4. Now let us consider C5. Suppose P(α) = 1, by Definition 1, conditionalizing P ⁻_α by α gives P. Since expansion is modelled by Bayesian conditionalization, we have (P ⁻_α ) ⁺_α  = P, indifferent contraction satsifies the postulate C5.

As a result of contraction by $\alpha\wedge\beta, [\neg\alpha\vee\neg\beta]\subseteq\|P^{-}_{\alpha\wedge\beta}\|$. Hence $P^{-}_{\alpha\wedge\beta}(\neg\alpha)>0$. Conditionalizing P ⁻_α∧β by $\neg\alpha$ gives a uniform distribution over $[\neg\alpha]$, which is also the result of conditionalizing P ⁻_α by $\neg\alpha$. Hence, indifferent contraction satisfies C6.

The entropy of P ⁻_α , which is given by indifferent contraction, is log(2^H(P) + m). Since log is a monotonic function, log(2^H(P)) < log(2^H(P) + m), where m is the cardinality of $[\neg\alpha]$ and is a positive integer (since we have assumed that α is not a theorem). Thus H(P ⁻_α ) ≥ H(P), satisfying the postulate CO. $\square$

Theorem 2

An indifferent contraction is a full meet contraction.

Proof

Let P be the initial distribution representing the agent’s belief state and $\mathcal{K}$ be the corresponding set of beliefs, given by Eq. 1. Let α be a belief of the agent, that is, P(α) = 1. The result of indifferent contraction of P by α is denoted by P ⁻_α and the corresponding belief set is denoted by $\mathcal{K}_{\alpha}^{-}$.

A probabilistic contraction function is said to be a full meet contraction (Gärdenfors 1988) when it satisfies the postulates C1 to C5 and the associated contracted belief set $\mathcal{K}^{-}_{\alpha}$ is such that $[\mathcal{K}^{-}_{\alpha}] = [\mathcal{K}]\cup[\neg\alpha]$.

Theorem 1 shows that indifferent contraction satisfies postulates C1 to C5. From Eq. 1, we get that $\mathcal{K}^{-}_{\alpha} = \{\beta\in\mathcal{L} : P^{-}_{\alpha}(\beta) = 1\}$, that is, $\mathcal{K}^{-}_{\alpha} = \{\beta\in\mathcal{L} : \|P^{-}_{\alpha}\|\subseteq[\beta]\}$. Since, as a result of indifferent contraction, every $\neg\alpha$-world is assigned positive probability, we have $[\neg\alpha]\subseteq\|P^{-}_{\alpha}\|$. Since $\|P^{-}_{\alpha}\| = \|P\| \cup [\neg\alpha]$, it follows that $[\mathcal{K}^{-}_{\alpha}] = [\mathcal{K}]\cup[\neg\alpha]$. Hence indifferent contraction is a full meet contraction. $\square$

Theorem 3

A submaximal entropy contraction is a full meet contraction.

Proof

Let P be the initial distribution representing the belief state of the agent and α be a belief, that is P(α) = 1. Submaximal entropy contraction results in reducing the probability associated with the α-worlds proportionally and all the $\neg\alpha$-worlds are assigned equal non-zero probability.

A probabilistic contraction function is said to be a full meet contraction (Gärdenfors 1988) when it satisfies the postulates C1 to C5 and the associated contracted belief set $\mathcal{K}^{-}_{\alpha}$ is such that $[\mathcal{K}^{-}_{\alpha}] = [\mathcal{K}]\cup[\neg\alpha]$.

Submaximal entropy contraction satisfies postulates C1 and C5 by definition (Definition 2). It is evident from Definition 2, that submaximal entropy contraction satisfies postulate C4. The contracted probability distribution, P ⁻_α is a uniform distribution over the set $[\neg\alpha]$ and assigns non-zero probability to every $\neg\alpha$-world. Thus P ⁻_α (α) < 1, satisfying postulate C2.

When $\vdash\alpha\leftrightarrow\beta$, then [α] = [β] and $[\neg\alpha] = [\neg\beta]$. As a result of contraction by α, probability associated with every α-world is scaled by a non-zero value d, determined by the minimization problem and the mass 1 − d is spread uniformly over $[\neg\alpha]$. The same happens when contracting by β. Hence submaximal entropy contraction satisfies the postulate C3.

The support of contracted probability distribution P ⁻_α is such that $\|P^{-}_{\alpha}\| = \|P\|\cup[\neg\alpha]$. This is the same as in the case of indifferent contraction. Hence along same lines as Theorem 2, submaximal entropy contraction is a full meet contraction. $\square$

Theorem 4

A minimal contraction based on a selection function $\mathcal{S}$ is a maxichoice contraction.

Proof

Let P be the initial distribution representing the agent’s belief state and $\mathcal{K}$ be the corresponding set of beliefs, given by Eq. 1. Let α be a belief of the agent, that is, P(α) = 1. The result of minimal contraction of P by α is denoted by P ⁻_α and the corresponding belief set is denoted by $\mathcal{K}_{\alpha}^{-}$.

A probabilistic contraction function is said to be a maxichoice contraction when it satisfies the postulates C1 to C5 and the associated contracted belief set $\mathcal{K}_{\alpha}^{-}$ is such that $[\mathcal{K}_{\alpha}^{-}] = [\mathcal{K}]\cup \{\omega\}$ for some $\omega\in[\neg\alpha]$.

Minimal contraction results in a probability distribution where the probabilities associated with the α-worlds are reduced proportionally to a very small value. Minimal contraction, we assume, is equipped with some mechanism to choose a world from the set $[\neg\alpha]$. The chosen world is assigned non-zero probability which is close to 1. From Definition 4 it follows that minimal contraction satisfies postulates C1 and C4. For every $\omega\in[\alpha]$, we have P ⁻_α (ω) = d P(ω), where d is very small. The resultant probability of α is given by, $P^{-}_{\alpha}(\alpha) = \sum_{\omega\in[\alpha]} d P(\omega) = d$. Thus P ⁻_α (α) < 1, satisfying postulate C2.

Suppose α and β are two beliefs such that $\vdash \alpha\leftrightarrow\beta$, then $[\neg\alpha]=[\neg\beta]$. The selection function $\mathcal{S}$ is such that $\mathcal{S}([\neg\alpha]) = \mathcal{S}([\neg\beta])$. That is, the same possible world is chosen from the set $[\neg\alpha]$($=[\neg\beta]$) to be assigned positive probability, and hence the result of contraction by α is same as contraction by β. Hence, minimal contraction satisfies C3. By Definition 4, minimal contraction satisfies the postulate C5.

As a result of minimal contraction, we have $||P^{-}_{\alpha}|| = ||P||\cup\{\omega\}$, where $\omega\in[\neg\alpha]$. From Equation 1, it is clear that $[\mathcal{K}_{\alpha}^{-}] = [\mathcal{K}]\cup \{\omega\}$ for some $\omega\in[\neg\alpha]$, as required. $\square$

Theorem 5

A preferential contraction function is a transitively relational partial meet contraction function.

Proof

Let P be the initial distribution representing the agent’s belief state and $\mathcal{K}$ be the corresponding set of beliefs, given by Eq. 1. Let α be a belief of the agent, that is, P(α) = 1. The result of preferential contraction of P by α is denoted by P ⁻_α and the corresponding belief set is denoted by $\mathcal{K}_{\alpha}^{-}$. Let $\sqsubseteq$ be a total preorder relation on $\Upomega$. We need to show that preferential contraction satisfies the postulates C1 to C6. Let the cardinality of $\Upomega, [\alpha], [\neg\alpha]$ and the support of P be n, l, m, and l′ respectively. Suppose the initial distribution is as follows:

$$ P=\langle p_{1},\ldots,p_{l},\underbrace{0,\ldots,0}_{m}\rangle. $$

Let the cardinality of $[\neg\alpha]$ and $min_{\sqsubseteq}[\neg\alpha]$ be m and m′ respectively. The probability distribution resulting from preferential contraction is as follows:

$$ P^{-}_{\alpha} = \langle d p_{1},\ldots, d p_{l}, \underbrace{\underbrace{\frac{1}{h+m'},\ldots,\frac{1}{h+m'}}_{m'}, 0,\ldots, 0}_{m}\rangle. $$

where h = 2^H(P) and $d=\frac{h}{h+m'}$. From Definition 5, it is clear that preferential contraction satisfies C1. The resultant probability of α is P ⁻_α (α) = d P(α) = d, which is less than 1. Hence preferential contraction satisfies C2. When $\vdash\alpha\leftrightarrow\beta$, then [α] = [β] and $[\neg\alpha] = [\neg\beta]$. Therefore we have $min_{\sqsubseteq}[\neg\alpha] = min_{\sqsubseteq}[\neg\beta]$. We have

$$ P^{-}_{\alpha}(\omega) = P^{-}_{\beta}(\omega) = \left\{ \begin{array}{ll} d P(\omega), & {\text{if}} \; \omega\in[\alpha] = [\beta] \\ \frac{1}{h+m'}, & {\text{if}} \; \omega\in min_{\sqsubseteq}[\neg\alpha] = min_{\sqsubseteq}[\neg\beta] \\ 0, & {\text{otherwise}}. \end{array}\right. $$

Thus preferential contraction satisfies C3. From Definition 5, it is evident that preferential contraction satisfies postulates C4 and C5. When contracting by α∧β, all the worlds in $min_{\sqsubseteq}[\neg\alpha\vee\neg\beta]$ have uniform probability, and similarly when contracting by α, all the worlds in $min_{\sqsubseteq}[\neg\alpha]$ have uniform probability. Suppose that α and β are two beliefs such that $P^{-}_{\alpha\wedge\beta}(\neg\alpha)>0$. Then the total preorder relation $\sqsubseteq$ is such that $min_{\sqsubseteq}[\neg\alpha\vee\neg\beta] \cap [\neg\alpha]\neq\emptyset$. We have $min_{\sqsubseteq}[\neg\alpha] = min_{\sqsubseteq}[\neg\alpha\vee\neg\beta] \cap [\neg\alpha]$. Thus, conditionalizing P ⁻_α∧β by $\neg\alpha$ and conditionalizing P ⁻_α by $\neg\alpha$ both result in a uniform distribution over $min_{\sqsubseteq}[\neg\alpha\vee\neg\beta] \cap [\neg\alpha]$. Hence preferential contraction satisfies C6. Thus, preferential contraction is a transitively relational partial meet contraction. $\square$