What are we going to do?


This app refers to the presentation 'Probabilistic Knowledge Structures' by Heller and Wickelmaier. It will demonstrate parameter estimation in probabilistic knowledge structures:

  • If you want to be reminded of the three estimation procedures, proceed to the tab 'Estimation methods' above (For a more comprehensive overview, see the presentation by Heller and Wickelmaier)
  • If you are already confident in your knowledge about parameter estimation in probabilistic knowledge structures, you can directly proceed to 'Application' on the left and see how changes to the observed data and the assumed knowledge structure will affect parameter estimation
  • In the end, you can test your knowledge with a small quiz

The three estimation methods


With a given knowledge structure and observed response patterns, there are three methods to estimate the parameters of a basic local independence model (BLIM):

  • Maximum Likelihood (ML) Estimation
  • Minimum Discrepancy (MD) Estimation
  • Minimum Discrepancy ML Estimation (MDML)

Method Principle Pros Cons
ML estimates parameters that maximize the probability of the observed data
  • driven by the likelihood of the data
  • (approximately) unbiased estimates
  • iterative (EM algorithm)
  • might inflate error rates for good fit
MD assumes that any response pattern is generated by the knowledge state closest to it
  • computationally efficient (explicit estimators)
  • avoids inflating the error rates
  • ignores the likelihood of the data
  • estimates not unbiased
MDML ML estimation under certain MD restrictions
  • minimizes the expected number of response errors
  • maximizes the likelihood under this constraint
  • reference for quantifying the amount of fit obtained by inflating error rates
  • estimates not unbiased

For this example, we will be working with a knowledge domain of four items: Q = {a,b,c,d}


1) Set the observed response frequencies of ...



2) Set your knowledge structure 𝒦

You have selected the following knowledge structure:

3) Calculate your parameter estimates

The error rate estimates for each item are:


The estimates of the state probabilities are:

4) Calculated expected frequencies with each parameter set


5) Simulate response patterns with each parameter set

To which of the three parameter estimation methods do these statements apply?

Your results