Calculate lower and upper probability bounds

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)

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.

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.

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, "\(\Rightarrow\)"). The first three follow the standard R syntax for logical operators (see base::logical):

Not: ! or -
And: & or && or *
Or: +; if argument solidus = FALSE, also || or | are allowed.

The "if-then" connective is represented by the infix operator >; internally x > y is simply defined as x or not-y.

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)
P(-a + b  |  c && H)
P(!a + b  |  c * H)

or, if argument solidus = FALSE, in the following ways:

P(!a | b  ~  c & H)
P(-a + b  ~  c && H)
P(!a || b  ~  c * H)

It is also possible to use p or Pr or pr 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)
)
#> min max 
#>   0   1 

## Trivial example with inequality constraint
inferP(
  target = P(a | h),
  P(!a | h) >= 0.2
)
#> min max 
#> 0.0 0.8 

#' ## 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
)
#> min max 
#> 0.0 0.3 

## 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
)
#>  min  max 
#> 0.06 0.06 

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

## Cut rule of sequent calculus
inferP(
  target = P(X + Y | I & J),
  P(A & X | I) == 1,
  P(Y | A & J) == 1
)
#> min max 
#>   1   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)
)
#>       min       max 
#> 0.6666667 0.6666667