Instruction and Periodic Accountability Assessment (IPAA) model.
We need to support a verison of the MML algorithm for temporal models.
Assume we have
X[,,t-1] - f(X[,,t-1],U[,,t],CGMP[t],Z[,,t],deltaT) -> X[,,t]
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h(X[,,t-1],Q[t-1],EMP[t-1],Z[,,t]) h(X[,,t],Q[t],EMP[t],Z[,,t])
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V V
Y[,,t-1] Y[,,t]
X[,,t] is a multivariate latent variable. Dimensions are persons,
competency and time.
Y[,,t] is a multivariate observable variable (may be missing values).
Dimensions are person, observable, and time.
Z[,,t] is a multivariate collection of background variables.
Dimensions are person, variable and time. Note that some of these
variables may be constant over the time period involved.
f is the competency growth process.
f0 is a special process which takes only a set of parameters as inputs
and gives the initial distribuiton of the latent variables X[,,0].
CGMP[] is a list of Proficiency Model parameters. The parameters at
time 0 have special interpretation (and probably a different
structure).
To support Bayesian estimation, we will define a proficiency model
prior distribution
CGMP[t] ~ g(CGMHP) for t>0
CGMP[0] ~ g0(CGMHP)
U is the "action" taken at each time point.
As a simplifying assumption, we will assume that the process is time
invariant, except for the constants U which act as a driver function.
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h is the evidence model.
It gives the distribuiton of the observable Y, as a function of the
latent variable X, a set of parameters, and a constant Q which
describes the assessment design at each time point. This setup
essentially allows for a different Q-Matrix describing the design at
each time point.
EMP[t] ~ c(EMHP)
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The variables Q, U are decision variables and are set by the
curriculum designers.
Three classes of problem arise in this framework:
1) Filtering/Forecasting --- Given a set of observables Y, and
decision variables Q and U, and parameters, EMP, CGMP, estimate
X[t]. This problem is "filtering" if Y[t] is observed and
forecasting if Y[t] is not yet observed.
2) Parameter estimation. Given a set of observables Y, and decision
variables Q and U, estimate EMP and CGMP.
Note that given a realization of X, EMP[t] and CGMP[t] are all rendered
independent. This supports EM and MCMC like algorithms over this
space.
3) Decision optimization. Find U and Q to maximize some reward
generally subject to some constraints.
Note that we must generally by able to sample and calculate
likelihoods (up to a constant) from f and h (as well as c and g).