In probabilistic logic, the Fréchet inequalities, also known as the Boole–Fréchet inequalities, are rules implicit in the work of George Boole[1][2] and explicitly derived by Maurice Fréchet[3][4] that govern the combination of probabilities about logical propositions or events logically linked together in conjunctions (AND operations) or disjunctions (OR operations) as in Boolean expressions or fault or event trees common in risk assessments, engineering design and artificial intelligence.
These inequalities can be considered rules about how to bound calculations involving probabilities without assuming independence or, indeed, without making any dependence assumptions whatsoever.
If Ai are logical propositions or events, the Fréchet inequalities are where P( ) denotes the probability of an event or proposition.
For example, if A is "has lung cancer", and B is "has mesothelioma", then A & B is "has both lung cancer and mesothelioma", and A ∨ B is "has lung cancer or mesothelioma or both diseases", and the inequalities relate the risks of these events.
Note that logical conjunctions are denoted in various ways in different fields, including AND, &, ∧ and graphical AND-gates.
Logical disjunctions are likewise denoted in various ways, including OR, |, ∨, and graphical OR-gates.
Likewise, the probability of the disjunction A ∨ B is surely in the interval
When the marginal probabilities are very small (or large), the Fréchet intervals are strongly asymmetric about the analogous results under independence.
If A and B are independent, however, the probability of A & B is 6×10−12 which is, comparatively, very close to the lower limit (zero) of the Fréchet interval.
Similarly, the probability of A ∨ B is 4.999994×10−6, which is very close to the upper limit of the Fréchet interval.
This is what justifies the rare-event approximation[5] often used in reliability theory.
The best-possible nature of these bounds follows from observing that they are realized by some dependency between the events A and B.
Comparable bounds on the disjunction are similarly derived.
When the input probabilities are themselves interval ranges, the Fréchet formulas still work as a probability bounds analysis.
Hailperin[2] considered the problem of evaluating probabilistic Boolean expressions involving many events in complex conjunctions and disjunctions.
Some[6][7] have suggested using the inequalities in various applications of artificial intelligence and have extended the rules to account for various assumptions about the dependence among the events.
The inequalities can also be generalized to other logical operations, including even modus ponens.
[6][8] When the input probabilities are characterized by probability distributions, analogous operations that generalize logical and arithmetic convolutions without assumptions about the dependence between the inputs can be defined based on the related notion of Fréchet bounds.
[7][9][10] Similar bounds hold also in quantum mechanics in the case of separable quantum systems and that entangled states violate these bounds.
In particular, we focus on a composite quantum system AB made by two finite subsystems denoted as A and B.
A density matrix on a composite system is separable if there exist
It is evident that structurally the above inequalities are analogues of the classical Fréchet bounds for the logical conjunction.
are restricted to be diagonal, we obtain the classical Fréchet bounds.
The upper bound is known in Quantum Mechanics as reduction criterion for density matrices; it was first proven by[12] and independently formulated by.
are all diagonal, we obtain the classical Fréchet bounds.
To show that, consider again the previous numerical example:
It is worth to point out that entangled states violate the above Fréchet bounds.
Consider for instance the entangled density matrix (which is not separable):
Entangled states are not separable and it can easily be verified that
Another example of violation of probabilistic bounds is provided by the famous Bell's inequality: entangled states exhibit a form of stochastic dependence stronger than the strongest classical dependence: and in fact they violate Fréchet like bounds.