Pinference (version 0.2.6)

inferP: Calculate lower and upper probability bounds

Description

inferP() calculates the minimum and maximum allowed values of the probability for a propositional-logic expression conditional on another one, given numerical or equality constraints for the conditional probabilities for other propositional-logic expressions.

Usage

inferP(target, ..., solidus = TRUE)

Value

A vector of min and max values for the target probability, or NA if the constraints are mutually contradictory. If min and max are 0 and 1 then the constraints do not restrict the target probability in any way.

Arguments

target

The target probability expression (see Details).

...

Probability constraints (see Details).

solidus

logical. If TRUE (default), the symbol | is used to introduce the conditional in the probability; in this case any use of || for the 'or'-connective will lead to an error. If FALSE, the symbol ~ is used to introduce the conditional; in this case the symbols |, || can be used for the 'or`-connective.

Details

The function takes as first argument the probability for a logical expression, conditional on another expression, and as subsequent (optional) arguments the constraints on the probabilities for other logical expressions. Propositional logic is intended here.

The function uses the lpSolve::lp() function from the lpSolve package.

Logical expressions

A propositional-logic expression is a combination of atomic propositions by means of logical connectives. Atomic propositions can have any name that satisfies R syntax for object names. Examples:

a
A
hypothesis1
coin.lands.tails
coin_lands_heads
`tomorrow it rains` # note the backticks

Available logical connectives are "not" (negation, "\(\lnot\)"), "and" (conjunction, "\(\land\)"), "or" (disjunction, "\(\lor\)"), "if-then" (implication, "\(\Rightarrow\)") which internally is simply defined as "x or not-y". Two kinds of notation are available:

Arithmetic notation: should be used with argument solidus = TRUE:

• Not: -

• And: *

• Or: +

• If-then: >

Logical notation (see base::logical): should be used with the argument solidus = FALSE:

• Not: !

• And: & or &&

• Or: | or ||

• If-then: two-argument function ifthen()

The two kinds of notation can actually be mixed (only | and || are not allowed if solidus = TRUE), but beware of very unusual connective precedence in that case; for example, !a + b is interpreted as !(a + b). If you use mixed notation you should explicitly group with parentheses to ensure the desired connective precedence. A warning is issued when ! is used for negation together with +, *, because the grouping is dangerously counter-intuitive in this case.

Examples of logical expressions:

a
a & b
(a + hypothesis1) & -A
red.ball & ((a > -b) + c)

Probabilities of logical expressions

The probability of an expression \(X\) conditional on an expression \(Y\)in entered with syntax similar to the common mathematical notation \(\mathrm{P}(X \vert Y)\). The solidus "|" is used to separate the conditional (note that in usual R syntax such symbol stands for logical "or" instead). If the argument solidus = FALSE is given in the function, then the tilde "~" is used instead of the solidus (note that in usual R syntax such symbol introduces a formula instead). For instance

\(\mathrm{P}(\lnot a \lor b \:\vert\: c \land H)\)

can be entered in the following ways, among others (extra spaces added just for clarity):

P(-a + b  |  c * H)

or, if argument solidus = FALSE:

P(!a | b  ~  c & H)
P(!a || b  ~  c && H)

It is also possible to use Pr, p, pr, F, f, T (for "truth"), or t instead of P.

Probability constraints

Each probability constraint can have one of these four forms:

P(X | Z) = [number between 0 and 1]
P(X | Z) = P(Y | Z)
P(X | Z) = P(Y | Z) * [positive number]
P(X | Z) = P(Y | Z) / [positive number]

where X, Y, Z are logical expressions. Note that the conditionals on the left and right sides must be the same. Inequalities <= >= are also allowed instead of equalities.

See the accompanying vignette for more interesting examples.

References

T. Hailperin: Best Possible Inequalities for the Probability of a Logical Function of Events. Am. Math. Monthly 72(4):343, 1965 doi:10.1080/00029890.1965.11970533.

T. Hailperin: Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications. Associated University Presses, 1996 https://archive.org/details/hailperin1996-Sentential_probability_logic/.

Examples

## No constraints
inferP(
  target = P(a | h)
)

## Trivial example with inequality constraint
inferP(
  target = P(a | h),
  P(-a | h) >= 0.2
)

#' ## The probability of an "and" is always less
## than the probabilities of the and-ed propositions:
inferP(
  target = P(a & b | h),
  P(a | h) == 0.3,
  P(b | h) == 0.6
)

## P(a & b | h) is completely determined
## by P(a | h) and P(b | a & h):
inferP(
    target = P(a & b | h),
    P(a | h) == 0.3,
    P(b | a & h) == 0.2
)

## Logical implication (modus ponens)
inferP(
  target = P(b | I),
  P(a | I) == 1,
  P(a > b | I) == 1
)

## Cut rule of sequent calculus
inferP(
  target = P(X + Y | I & J),
  P(A & X | I) == 1,
  P(Y | A & J) == 1
)

## Solution to the Monty Hall problem (see accompanying vignette):
inferP(
    target = P(car2  |  you1 & host3 & I),
    ##
    P(car1 & car2  |  I) == 0,
    P(car1 & car3  |  I) == 0,
    P(car2 & car3  |  I) == 0,
    P(car1 + car2 + car3  |  I) == 1,
    P(host1 & host2 | I) == 0,
    P(host1 & host3 | I) == 0,
    P(host2 & host3 | I) == 0,
    P(host1 + host2 + host3  |  I) == 1,
    P(host1  |  you1 & I) == 0,
    P(host2  |  car2 & I) == 0,
    P(host3  |  car3 & I) == 0,
    P(car1  |  I) == P(car2  |  I),
    P(car2  |  I) == P(car3  |  I),
    P(car1  |  you1 & I) == P(car2  |  you1 & I),
    P(car2  |  you1 & I) == P(car3  |  you1 & I),
    P(host2  |  you1 & car1 & I) == P(host3  |  you1 & car1 & I)
)