calcNoisyAndTable: Calculate the conditional probability table for a Noisy-And...

In ralmond/CPTtools: Tools for Creating Conditional Probability Tables

Calculate the conditional probability table for a Noisy-And or Noisy-Min distribution

Description

Calculates the conditional probability table for a noisy-and distribution. This follows a logical model where all inputs must be true for the output to be true; however, some "noise" is allowed that produces random deviations from the pure logic. The noisy-min is a generalization in which all variables are ordered, and the weakest of the parent variables drives the conditional probabilities of the child variable.

Usage

calcNoisyAndTable(skillLevels, obsLevels = c("True", "False"),
                  bypass = rep(0, length(skillLevels)), noSlip=1,
                  thresholds = sapply(skillLevels, function(states) states[1]))
calcNoisyAndFrame(skillLevels, obsLevels = c("True", "False"),
                  bypass = rep(0, length(skillLevels)), noSlip=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".
noSlip	A scalar value between 0 and 1. This represents the probability that the output will be true when all of the inputs are true (e.g., 1 - the probability that an examinee will make a careless error).
bypass	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 met, even if it is not met.
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-and distribution assumes that both the input and output variables are binary. Basically, the output should be true if all of the inputs are true. Let S_k = 1 if the kth input is true, and let r_k be the bypass parameter corresponding to the kth input variable. (If the S_k's represent a skill, then r_k represents the probability that an examinee who lacks that skill will bypass the need for that skill in solving the problem.) Then the probability of the true state for the output variable will be:

\Pr(X=True|{\bf S}) = r_0 \prod_k r_k^{1-S_k},

where r_0 (the noSlip parameter) is the probability that the output will be true even when all of the inputs are true.

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.)

The noisy-min is a generalization

Value

For calcNoisyAndTable, 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 calcNoisyAndFrame 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 DINA and NIDA models, but uses a slightly different parameterization. In particular, if the noSlip parameter is omitted, it is a noisy input deterministic and-gate (NIDA), and if the bypass parameters are omitted, it is similar to a deterministic input noisy and-gate (DINA), except is lacks a guessing 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 and table
and <- calcNoisyAndFrame(list(c("True","False"),c("True","False")),
                                 c("Right","Wrong"))

## DINA, logical-and except that is allows for a small chance of slipping.
dina <- calcNoisyAndFrame(list(c("True","False"),c("True","False")),
                          noSlip=.9)

##NIDA, logical-and except that inputs can randomly be bypassed
nida <- calcNoisyAndFrame(list(c("True","False"),c("True","False")),
                          bypass=c(.3,.4))

##Full Noisy And distribution
noisyAnd <- calcNoisyAndFrame(list(c("True","False"),c("True","False")),
                          noSlip=.9,bypass=c(.3,.4))

thresh <- calcNoisyAndFrame(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.