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Let $P$ be a probability distribution over a finite Boolean algebra $\mathfrak{B}$, and fix a parameter $t_{P} \in (\frac{2}{3}, 1)$. Define the `revision function of $P$', $R_{P}: \mathfrak{B}\setminus\{\bot\} \rightarrow \mathbb{P}(\mathfrak{B})$ as follows, (assuming that $P$ assigns positive probability to all events other than the null event $\bot$)

$R_{P}(X) = \{Y \in \mathfrak{B}|P(Y|X) \geq t\}$

In words, the revision function of $P$ takes an event $X$ in $\mathfrak{B}$ and returns the set of all events whose conditional probability given $X$ (according to $P$) is at least $t_{p}$.

I want to find general method for constructing, for any given $P$ and any $t_{P} \in (\frac{2}{3}, 1)$, another probability function $P^{*}$ with a corresponding parameter $t_{P^{*}} \in (\frac{2}{3}, 1)$ such that $R_{P^{*}} = R_{P}$. In other words, I want a method that constructs, for any probability function and fixed parameter in the specified range, another function that (for some choice of parameter in the specified range) gives rise to the same revision function as the original function.

Importantly, the method should not take $P$ itself (or $t_{P}$) as an input -- it should be a function of $R_{P}$ alone. In essence, I want a method that takes a revision function obtained from a probability function/parameter, and constructs another probability function/parameter pair that also corresponds to the same revision function, and does so without `looking at' the function/parameter that the revision function initially came from.

Let $P$ be a probability distribution over a finite Boolean algebra $\mathfrak{B}$, and fix a parameter $t_{P} \in (\frac{2}{3}, 1)$. Define the `revision function of $P$', $R_{P}: \mathfrak{B}\setminus\{\bot\} \rightarrow \mathbb{P}(\mathfrak{B})$ as follows, (assuming that $P$ assigns positive probability to all events other than the null event $\bot$)

$R_{P}(X) = \{Y \in \mathfrak{B}|P(Y|X) \geq t\}$

In words, the revision function of $P$ takes an event $X$ in $\mathfrak{B}$ and returns the set of all events whose conditional probability given $X$ (according to $P$) is at least $t_{p}$.

I want to find general method for constructing, for any given $P$ and any $t_{P} \in (\frac{2}{3}, 1)$, another probability function $P^{*}$ with a corresponding parameter $t_{P^{*}} \in (\frac{2}{3}, 1)$ such that $R_{P^{*}} = R_{P}$. In other words, I want a method that constructs, for any probability function and fixed parameter in the specified range, another function that (for some choice of parameter in the specified range) gives rise to the same revision function as the original function.

Importantly, the method should not take $P$ itself (or $t_{P}$) as an input -- it should be a function of $R_{P}$ alone. In essence, I want a function that takes a revision function obtained from a probability distribution/parameter, and constructs another probability distribution/parameter pair that also corresponds to the same revision function, and does so without `looking at' the function/parameter that the revision function initially came from.

Let $P$ be a probability distribution over a finite Boolean algebra $\mathfrak{B}$, and fix a parameter $t_{P} \in (\frac{2}{3}, 1)$. Define the `revision function of $P$', $R_{P}: \mathfrak{B}\setminus\{\bot\} \rightarrow \mathbb{P}(\mathfrak{B})$ as follows, (assuming that $P$ assigns positive probability to all events other than the null event $\bot$)

$R_{P}(X) = \{Y \in \mathfrak{B}|P(Y|X) \geq t\}$

In words, the revision function of $P$ takes an event $X$ in $\mathfrak{B}$ and returns the set of all events whose conditional probability given $X$ (according to $P$) is at least $t_{p}$.

I want to find general method for constructing, for any given $P$ and any $t_{P} \in (\frac{2}{3}, 1)$, another probability function $P^{*}$ with a corresponding parameter $t_{P^{*}} \in (\frac{2}{3}, 1)$ such that $R_{P^{*}} = R_{P}$. In other words, I want a method that constructs, for any probability function and fixed parameter in the specified range, another function that (for some choice of parameter in the specified range) gives rise to the same revision function as the original function.

Let $P$ be a probability distribution over a finite Boolean algebra $\mathfrak{B}$, and fix a parameter $t_{P} \in (\frac{2}{3}, 1)$. Define the `revision function of $P$', $R_{P}: \mathfrak{B}\setminus\{\bot\} \rightarrow \mathbb{P}(\mathfrak{B})$ as follows, (assuming that $P$ assigns positive probability to all events other than the null event $\bot$)

$R_{P}(X) = \{Y \in \mathfrak{B}|P(Y|X) \geq t\}$

In words, the revision function of $P$ takes an event $X$ in $\mathfrak{B}$ and returns the set of all events whose conditional probability given $X$ (according to $P$) is at least $t_{p}$.

I want to find general method for constructing, for any given $P$ and any $t_{P} \in (\frac{2}{3}, 1)$, another probability function $P^{*}$ with a corresponding parameter $t_{P^{*}} \in (\frac{2}{3}, 1)$ such that $R_{P^{*}} = R_{P}$. In other words, I want a method that constructs, for any probability function and fixed parameter in the specified range, another function that (for some choice of parameter in the specified range) gives rise to the same revision function as the original function.

Importantly, the method should not take $P$ itself (or $t_{P}$) as an input -- it should be a function of $R_{P}$ alone. In essence, I want a method that takes a revision function obtained from a probability function/parameter, and constructs another probability function/parameter pair that also corresponds to the same revision function, and does so without `looking at' the function/parameter that the revision function initially came from.