calcNoisyOrTable: Calculate the conditional probability table for a Noisy-Or...

In ralmond/CPTtools: Tools for Creating Conditional Probability Tables

Calculate the conditional probability table for a Noisy-Or distribution

Description

Calculates the conditional probability table for a noisy-and distribution. This follows a logical model where at least one inputs must be true for the output to be true; however, some "noise" is allowed that produces random deviations from the pure logic.

Usage

calcNoisyOrTable(skillLevels, obsLevels = c("True", "False"),
                 suppression = rep(0, length(skillLevels)), noGuess = 1,
                 thresholds = sapply(skillLevels, function(states) states[1]))
calcNoisyOrFrame(skillLevels, obsLevels = c("True", "False"),
                 suppression = rep(0, length(skillLevels)), noGuess = 1,
                 thresholds = sapply(skillLevels, function(states) states[1]))

Arguments

skillLevels	A list of character vectors giving names of levels for each of the condition variables.
obsLevels	A character vector giving names of levels for the output variables from highest to lowest. As a special case, can also be a vector of integers. Its length should be 2, and the first value is considered to be logically equivalent to "true".
suppression	A vector of the same length as skillLevels. For each parent variable, this represents the probability that the process will act as if that input condition is not met, even if it is met.
noGuess	A scalar value between 0 and 1. This represents the probability that the the output will be false even when all of the inputs are false (e.g., 1-guessing probability).
thresholds	If the input variables have more than two states, values that are equal to or higher than this threshold are considered true. It is assumed that the states of the variables are ordered from highest to lowest.

Details

The noisy-or distribution assumes that both the input and output variables are binary. Basically, the output should be true if any of the inputs are true. Let S_k = 1 if the kth input is true, and let q_k be the suppression parameter corresponding to the kth input variable. (If the S_k's represent a skill, then q_k represents the probability that an examinee who has that skill will fail to correctly apply it.) Then the probability of the true state for the output variable will be:

\Pr(X=True|{\bf S}) = 1 - q_0 \prod_k q_k^{1-S_k},

where q_0 (the noGuess parameter) is the probability that the output will be false even when all of the inputs are false.

It is assumed that all variables are ordered from highest to lowest state, so that the first state corresponds to "true" the others to false. If the input variable has more than two states, then it can be reduced to a binary variable by using the threshold argument. Any values which are equal to or higher than the threshold for that variable are assumed to be true. (In this case, higher means closer to the the beginning of the list of possible values.)

Value

For calcNoisyOrTable, a matrix whose rows correspond configurations of the parent variable states (skillLevels) and whose columns correspond to obsLevels. Each row of the table is a probability distribution, so the whole matrix is a conditional probability table. The order of the parent rows is the same as is produced by applying expand.grid to skillLevels.

For calcNoisyOrFrame a data frame with additional columns corresponding to the entries in skillLevels giving the parent value for each row.

Note

This is related to the DINO and NIDO models, but uses a slightly different parameterization. In particular, if the noSlip parameter is omitted, it is a noisy input deterministic and-gate (NIDO), and if the bypass parameters are omitted, it is similar to a deterministic input noisy and-gate (DINO), except is lacks a slip parameter.

Author(s)

Russell Almond

References

Almond, R.G., Mislevy, R.J., Steinberg, L.S., Yan, D. and Williamson, D.M. (2015) Bayesian Networks in Educational Assessment. Springer. Chapter 8.

Pearl, J. (1988) Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann.

Diez, F. J. (1993) Parameter adjustment in Bayes networks. The generalized noisy OR-gate. In Heckerman and Mamdani (eds) Uncertainty in Artificial Intelligence 93. Morgan Kaufmann. 99–105.

Srinivas, S. (1993) A generalization of the Noisy-Or model, the generalized noisy OR-gate. In Heckerman and Mamdani (eds) Uncertainty in Artificial Intelligence 93. Morgan Kaufmann. 208–215.

See Also

calcDSTable, calcDNTable, calcDPCTable, expand.grid, calcNoisyOrTable

Examples

## Logical or table
or <- calcNoisyOrFrame(list(c("True","False"),c("True","False")),
                                 c("Right","Wrong"))


## DINO, logical-or except that is allows for a small chance of slipping.
dino <- calcNoisyOrFrame(list(c("True","False"),c("True","False")),
                          noGuess=.9)


##NIDO, logical-or except that inputs can randomly be bypassed
nido <- calcNoisyOrFrame(list(c("True","False"),c("True","False")),
                          suppression=c(.3,.4))

##Full Noisy Or distribution
noisyOr <- calcNoisyOrFrame(list(c("True","False"),c("True","False")),
                          noGuess=.9,suppression=c(.3,.4))


thresh <- calcNoisyOrFrame(list(c("H","M","L"),c("H","M","L")),
                                 c("Right","Wrong"),
                                 threshold=c("M","H"))

ralmond/CPTtools documentation built on Dec. 27, 2024, 7:15 a.m.