PnodeLink: Accesses the link function associated with a Pnode

In ralmond/Peanut: Parameterized Bayesian Networks, Abstract Classes

Accesses the link function associated with a Pnode

Description

In constructing a conditional probability table using the discrete partial credit framework (see calcDPCTable), the effective thetas for each row of the table is converted into a vector of probabilities using the link function. The function PnodeLink accesses the link function associated with a Pnode.

Usage

PnodeLink(node)
PnodeLink(node) <- value

Arguments

node	A Pnode object.
value	The name of a link function or function object which can serve as the link function.

Details

Following the framework laid out in Almond (2015), the function calcDPCTable calculates a conditional probability table using the following steps:

1. Each set of parent variable states is converted to a set of continuous values called effective thetas (see PnodeParentTvals). These are built into an array, eTheta, using expand.grid where each column represents a parent variable and each row a possible configuration of parents.

2. For each state of the node except the last, the set of effective thetas is filtered using the local Q-matrix, PnodeQ(node) = Q. Thus, the actual effect thetas for state s is eTheta[,Q[s,]].

3. For each state of the node except the last, the corresponding rule is applied to the effective thetas to get a single effective theta for each row of the table. This step is essentially calls the expression: do.call(rules[[s]], list(eThetas[,Q[s,]]), PnodeAlphas(node)[[s]], PnodeBetas(node)[[s]]).

4. The resulting set of effective thetas are converted into conditional probabilities using the link function.

A link function is a function of three arguments. The first is a matrix of effective theta values with number of rows equal to the number of rows of the conditional probability matrix and number of columns equal to the number of states of node minus one (ordered from highest to lowest). The second is an optional link scale, the third is a set of names for the states which is used to give column names to the output matrix. The second and third both default to NULL.

Currently two link functions are partialCredit and gradedResponse. Note that the function gradedResponse assumes that the effective thetas in each row are in increasing order. This puts certain restrictions on the parameter values. Generally, this can only be guaranteed if each state of the variable uses the same combination rules (see PnodeRules(node)), slope parameters (see PnodeAlphas(node)) and Q-matrix (see PnodeQ(node)). Also, the intercepts (see PnodeBetas(node)) should be in decreasing order. The partialCredit model has fewer restrictions.

The value of PnodeLinkScale(node) is fed to the link function. Currently, this is unused; but the DiBello-normal model (see calcDNTable) uses it. So the link scale parameter is for future expansion.

Value

A character scalar giving the name of a combination function or a combination function object.

Note that the setter form may destructively modify the Pnode object (this depends on the implementation).

Note

The functions PnodeLink and PnodeLink<- are abstract generic functions, and need specific implementations. See the PNetica-package for an example.

A third normal link function, which would use the scale parameter, is planned but not yet implemented.

Author(s)

Russell Almond

References

Almond, R. G. (2015) An IRT-based Parameterization for Conditional Probability Tables. Paper presented at the 2015 Bayesian Application Workshop at the Uncertainty in Artificial Intelligence Conference.

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

See Also

Pnode, PnodeQ, PnodeRules PnodeLinkScale, PnodeLnAlphas, PnodeBetas, BuildTable, PnodeParentTvals, maxCPTParam calcDPCTable, mapDPC Compensatory, OffsetConjunctive

Examples

## Not run: 
library(PNetica) ## Requires implementation
sess <- NeticaSession()
startSession(sess)

tNet <- CreateNetwork("TestNet",session=sess)

theta1 <- NewDiscreteNode(tNet,"theta1",
                         c("VH","High","Mid","Low","VL"))
PnodeStateValues(theta1) <- effectiveThetas(PnodeNumStates(theta1))
PnodeProbs(theta1) <- rep(1/PnodeNumStates(theta1),PnodeNumStates(theta1))
theta2 <- NewDiscreteNode(tNet,"theta2",
                         c("VH","High","Mid","Low","VL"))
PnodeStateValues(theta2) <- effectiveThetas(PnodeNumStates(theta2))
PnodeProbs(theta2) <- rep(1/PnodeNumStates(theta2),PnodeNumStates(theta2))

partial3 <- NewDiscreteNode(tNet,"partial3",
                            c("FullCredit","PartialCredit","NoCredit"))
PnodeParents(partial3) <- list(theta1,theta2)

## Usual way to set link is in constructor
partial3 <- Pnode(partial3,rules="Compensatory", link="gradedResponse")
PnodePriorWeight(partial3) <- 10
PnodeBetas(partial3) <- list(FullCredit=1,PartialCredit=0)
BuildTable(partial3)

## increasing intercepts for both transitions
PnodeLink(partial3) <- "partialCredit"
## Full Credit is still rarer than partial credit under the partial
## credit model
PnodeBetas(partial3) <- list(FullCredit=0,PartialCredit=0)
BuildTable(partial3)

## Can use different slopes with partial credit
## Make Skill 1 more important for the transition to ParitalCredit
## And Skill 2 more important for the transition to FullCredit
PnodeLnAlphas(partial3) <- list(FullCredit=c(-.25,.25),
                                PartialCredit=c(.25,-.25))
BuildTable(partial3)

## Can also use Q-matrix to select skills
## Set up so that first skill only needed for first transition, second
## skill for second transition; Adjust alphas to match
PnodeQ(partial3) <- matrix(c(TRUE,TRUE,
                             TRUE,FALSE), 2,2, byrow=TRUE)
PnodeLnAlphas(partial3) <- list(FullCredit=c(-.25,.25),
                                PartialCredit=0)
BuildTable(partial3)

DeleteNetwork(tNet)

## End(Not run)

ralmond/Peanut documentation built on Sept. 19, 2023, 8:27 a.m.