pyBKT

Python implementation of the Bayesian Knowledge Tracing algorithm and variants, estimating student cognitive mastery from problem solving sequences.

    pip install pyBKT

Based on the work of Zachary A. Pardos ([email protected]) and Matthew J. Johnson ([email protected]) @ https://github.com/CAHLR/xBKT. All-platform python adaptation and optimizations by Anirudhan Badrinath ([email protected]). Data helpers and other utility functions written by Frederic Wang ([email protected]). Original Python and boost adaptation of xBKT by Cristian Garay ([email protected]). For implementation details, analysis of runtime and data requirements, and model variant replication testing, refer to:

Badrinath, A., Wang, F., Pardos, Z.A. (2021) pyBKT: An Accessible Python Library of Bayesian Knowledge Tracing Models. In S. Hsiao, & S. Sahebi (Eds.) Proceedings of the 14th International Conference on Educational Data Mining (EDM). Pages 468-474.

Bulut, O., Shin, J., Yildirim-Erbasli, S. N., Gorgun, G., & Pardos, Z. A. (2023). An introduction to Bayesian knowledge tracing with pyBKT. Psych, 5(3), 770-786.

Examples from the paper can be found in pyBKT-examples repo.

pyBKT Quick Start Tutorial

pyBKT Tutorial from LAK Workshop in Google Colab Notebook

Requirements

Python >= 3.5

Supported OS: All platforms! (Yes, Windows too)

Supported model variants

pyBKT can be used to define and fit many BKT variants, including these from the literature:

• Individual student priors, learn rate, guess, and slip [1,2]
• Individual item guess and slip [3,4,5]
• Individual item or resource learn rate [4,5]

1. Pardos, Z. A., Heffernan, N. T. (2010) Modeling Individualization in a Bayesian Networks Implementation of Knowledge Tracing. In P. De Bra, A. Kobsa, D. Chin (Eds.) Proceedings of the 18th International Conference on User Modeling, Adaptation and Personalization (UMAP). Big Island of Hawaii. Pages. Springer. Pages 255-266. [doi]

2. Pardos, Z. A., Heffernan, N. T. (2010) Using HMMs and bagged decision trees to leverage rich features of user and skill from an intelligent tutoring system dataset. In J. Stamper & A. Niculescu-Mizil (Eds.) Proceedings of the KDD Cup Workshop at the 16th ACM Conference on Knowledge Discovery and Data Mining (SIGKDD). Washington, D.C. ACM. Pages 24-35. [kdd cup]

3. Pardos, Z. & Heffernan, N. (2011) KT-IDEM: Introducing Item Difficulty to the Knowledge Tracing Model. In Konstant et al. (eds.) Proceedings of the 20th International Conference on User Modeling, Adaptation and Personalization (UMAP). Girona, Spain. Springer. Pages 243-254. [doi]

4. Pardos, Z. A., Bergner, Y., Seaton, D., Pritchard, D.E. (2013) Adapting Bayesian Knowledge Tracing to a Massive Open Online College Course in edX. In S.K. D’Mello, R.A. Calvo, & A. Olney (Eds.) Proceedings of the 6th International Conference on Educational Data Mining (EDM). Memphis, TN. Pages 137-144. [edm]

5. Xu, Y., Johnson, M. J., Pardos, Z. A. (2015) Scaling cognitive modeling to massive open environments. In Proceedings of the Workshop on Machine Learning for Education at the 32nd International Conference on Machine Learning (ICML). Lille, France. [icml ml4ed]

Installation and setup

This is intended as a quick overview of steps to install and setup and to run pyBKT locally.

We offer both a pure Python port and a Python/C++ extension version of pyBKT for the sake of accessibility and ease of use on any platform. Note that pip, by default, will install the C++/Python version unless the required libraries are not found or there is an error during installation. In the case of such issues, it will revert to the pure Python implementation.

The former pure Python versions does not fit models or scale as quickly or efficiently as the latter (due to nested for loops needed for DP). Here are a few speed comparisons - both on the same machine - that may be useful in deciding which version is more appropriate given the usage (e.g. model fitting is far more demanding than prediction).

Test Description	pyBKT (Python)	pyBKT (C++)
synthetic data, model fit (500 students)	~1m55s	~1.5s
synthetic data, model fit (5000 students)	~1h30m	~45s
cross validated cognitive tutor data	~4m10s	~3s
synthetic data, predict onestep (500 students)	~2s	~0.8s
synthetic data, predict onestep (5000 students)	~2m15s	~35s

Installing Dependencies for Fast C++ Inferencing (Optional - for OS X and Linux)

Note: this section is not applicable for Windows as running the Python/C++ version is cumbersome and untested. For Windows, we only offer the slower, pure Python version of pyBKT (it will be installed automatically).

Linux

If you have a C++ compiler already installed, pip will install pyBKT with fast C++ inferencing. C++ compilers are already installed on nearly all Linux distributions. If it is not installed on your machine, type sudo apt install gcc g++ if using Debian based distributions. Otherwise, whichever package manager is appropriately suited to your distribution (dnf, pacman, etc.). Without a compiler, pip will install pyBKT without C++ speed optimizations.

Mac

The latest version of Python is necessary for OS X. If homebrew is installed, run the following commands to download the necessary dependencies:

    brew install libomp

Installing pyBKT

You can simply run:

    pip install pyBKT

Alternatively, if pip poses some problems, you can clone the repository as such and then run the setup.py script manually.

    git clone https://github.com/CAHLR/pyBKT.git
    cd pyBKT
    python3 setup.py install

Preparing Data and Running Model

The following serves as a mini-tutorial for how to get started with pyBKT. There is more information available at the Colab notebook listed at the top of the README.

Input and Output Data

The accepted input formats are Pandas DataFrames and data files of type csv (comma separated) or tsv (tab separated). pyBKT will automatically infer which delimiter to use in the case that it is passed a data file. Since column names mapping meaning to each field in the data (i.e. skill name, correct/incorrect) vary per data source, you may need to specify a mapping from your data file's column names to pyBKT's expected column names. In many cases with Cognitive Tutor and Assistments datasets, pyBKT will be able to automatically infer column name mappings, but in the case that it is unable to, it will raise an exception. Note that the correctness is given by -1 (no response), 0 (incorrect), or 1 (correct).

Creating and Training Models

The process of creating and training models in pyBKT resemble that of SciKit Learn. pyBKT provides easy methods of fetching online datasets and to fit on a combination or all skills available in any particular dataset.

from pyBKT.models import Model

# Initialize the model with an optional seed
model = Model(seed = 42, num_fits = 1)

# Fetch Assistments and CognitiveTutor data (optional - if you have your own dataset, that's fine too!)
model.fetch_dataset('https://raw.githubusercontent.com/CAHLR/pyBKT-examples/master/data/as.csv', '.')
model.fetch_dataset('https://raw.githubusercontent.com/CAHLR/pyBKT-examples/master/data/ct.csv', '.')

# Train a simple BKT model on all skills in the CT dataset
model.fit(data_path = 'ct.csv')

# Train a simple BKT model on one skill in the CT dataset
# Note that calling fit deletes any previous trained BKT model!
model.fit(data_path = 'ct.csv', skills = "Plot imperfect radical")

# Train a simple BKT model on multiple skills in the CT dataset
model.fit(data_path = 'ct.csv', skills = ["Plot imperfect radical",
                                          "Plot pi"])

# Train a multiguess and slip BKT model on multiple skills in the
# CT dataset. Note: if you are not using CognitiveTutor or Assistments
# data, you may need to provide a column mapping for the guess/slip
# classes to use (i.e. if the column name is gsclasses, you would
# specify multigs = 'gsclasses' or specify a defaults dictionary
# defaults = {'multigs': 'gsclasses'}).
model.fit(data_path = 'ct.csv', skills = ["Plot imperfect radical",
                                          "Plot pi"],
                                multigs = True)

# We can combine multiple model variants.
model.fit(data_path = 'ct.csv', skills = ["Plot imperfect radical",
                                          "Plot pi"],
                                multigs = True, forgets = True,
                                multilearn = True)

# We can use a different column to specify the different learn and 
# forget classes. In this case, we use student ID.
model.fit(data_path = 'ct.csv', skills = ["Plot imperfect radical",
                                          "Plot pi"],
                                multigs = True, forgets = True,
                                multilearn = 'Anon Student Id')

# View the trained parameters!
print(model.params())

Note that if we train on a dataset that has unfamiliar columns to pyBKT, you will be required to specify a mapping of column names in that dataset to expected pyBKT columns. This is referred to as the model defaults (i.e. it specifies the default column names to lookup in the dataset). An example usage is provided below for an unknown dataset which has column names "row", "skill_t", "answer", and "gs_classes".

# Load unfamiliar dataset.
df = pd.read_csv('mystery.csv')

# For other non-Assistments/CogTutor style datasets, we will need to specify the
# columns corresponding to each required column (i.e. the user ID, correct/incorrect).
# For that, we use a defaults dictionary.
# In this case, the order ID that pyBKT expects is specified by the column row in the
# dataset, the skill_name is specified by a column skill_t and the correctness is specified
# by the answer column in the dataset.
defaults = {'order_id': 'row', 'skill_name': 'skill_t', 'correct': 'answer'}

# This defaults dictionary contains columns specifying what columns correspond
# to the desired guess/slip classes, etc. In this case, our desired column for
# the guess/slip classes is a column named gs_classes.
defaults['multigs'] = 'gs_classes'

# Fit using the defaults (column mappings) specified in the dictionary.
model.fit(data = df, defaults = defaults)

# Predict/evaluate/etc.
training_acc = model.evaluate(data = df, metric = 'accuracy')

Model Prediction and Evaluation

Prediction and evaluation behave similarly to SciKit-Learn. pyBKT offers a variety of features for prediction and evaluation.

from pyBKT.models import Model

# Initialize the model with an optional seed
model = Model(seed = 42, num_fits = 1)

# Load the Cognitive Tutor data (not necessary, but shown
# for the purposes of the tutorial that pyBKT accepts
# DataFrames as well as file locations!).
ct_df = pd.read_csv('ct.csv', encoding = 'latin')

# Train a simple BKT model on all skills in the CT dataset
model.fit(data_path = 'ct.csv')

# Predict on all skills on the training data.
# This returns a Pandas DataFrame.
preds_df = model.predict(data_path = 'ct.csv')

# Evaluate the RMSE of the model on the training data.
# Note that the default evaluate metric is RMSE.
training_rmse = model.evaluate(data = ct_df)

# Evaluate the AUC of the model on the training data. The supported
# metrics are AUC, RMSE and accuracy (they should be lowercased in
# the argument!).
training_auc = model.evaluate(data_path = 'ct.csv', metric = 'auc')

# We can define a custom metric as well.
def mae(true_vals, pred_vals):
  """ Calculates the mean absolute error. """
  return np.mean(np.abs(true_vals - pred_vals))

training_mae = model.evaluate(data_path = 'ct.csv', metric = mae)

Crossvalidation

Crossvalidation is offered as a blackbox function similar to a combination of fit and evaluate that accepts a particular number of folds, a seed, and a metric (either one of the 3 provided that are 'rmse', 'auc' or 'accuracy' or a custom Python function taking 2 arguments). Similar arguments for the model types, data path/data, and skill names are accepted as with the fit function.

from pyBKT.models import Model

# Initialize the model with an optional seed
model = Model(seed = 42, num_fits = 1)

# Crossvalidate with 5 folds on all skills in the CT dataset.
crossvalidated_errors = model.crossvalidate(data_path = 'ct.csv', folds = 5)

# Crossvalidate on a particular set of skills with a given 
# seed, folds and metric.
def mae(true_vals, pred_vals):
  """ Calculates the mean absolute error. """
  return np.mean(np.abs(true_vals - pred_vals))

# Note that the skills argument accepts a REGEX pattern. In this case, this matches and 
# crossvalidates on all skills containing the word fraction.
crossvalidated_mae_errs = model.crossvalidate(data_path = 'ct.csv', skills = ".*fraction.*",
                                              folds = 10, metric = mae)

# Crossvalidate using multiple model variants.
crossvalidated_multigsf = model.crossvalidate(data_path = 'ct.csv', multigs = True, forgets = True)

Roster

The model has been extended into the Roster to accomodate and simulate the learning environment for a cohort of students learning any combination of individual skills. The Roster feature has the efficient ability to track individuals' progress through the mastery and correctness probabilities outputted by BKT by storing only the current latent and observable state of the student. The following shows an example of Roster being used in practise:

from pyBKT.models import *
import numpy as np

# Create a backend pyBKT model and fit it on the CT data
model = Model()
model.fit(data_path = 'ct.csv')

# Create a Roster with two students, Jeff and Bob, who are participating in the roster
# for one skill (Calculate Unit Rate) using the pyBKT model above.
roster = Roster(students = ['Jeff', 'Bob'], skills = 'Calculate unit rate', model = model)

# Initial mastery state (prior) for Jeff, should be unmastered with low probability of mastery
# get_state_type returns whether a student has mastered the skill or not
# get_mastery_prob returns the probability a student has mastered the skill
print("Jeff's mastery (t = 0):", roster.get_state_type('Calculate unit rate', 'Jeff'))
print("Jeff's probability of mastery (t = 0):", roster.get_mastery_prob('Calculate unit rate', 'Jeff'))

# We can update Jeff's state by adding one or more responses to a particular skill. In this case,
# we observed a correct response for the one skill in the roster.
jeff_new_state = roster.update_state('Calculate unit rate', 'Jeff', 1)

# Check the updated mastery state and probability.
print("Jeff's mastery (t = 1):", roster.get_state_type('Calculate unit rate', 'Jeff'))
print("Jeff's probability of mastery (t = 1):", roster.get_mastery_prob('Calculate unit rate', 'Jeff'))

# We can update his state with multiple correct responses (ten of them).
roster.update_state('Calculate unit rate', 'Jeff', np.ones(10))

# After 10 consecutive correct responses, he should have mastered the skill.
print("Jeff's mastery (t = 11):", roster.get_state_type('Calculate unit rate', 'Jeff'))
print("Jeff's probability of mastery (t = 11):", roster.get_mastery_prob('Calculate unit rate', 'Jeff'))

# Programmatically check whether he has mastered the skill
if roster.get_state_type('Calculate unit rate', 'Jeff') == StateType.MASTERED:
    print("Jeff has mastered the skill!")
    
# We can update Bob's state with two correct responses.
roster.update_state('Calculate unit rate', 'Bob', np.ones(2))

# He should remain unmastered.
print("Bob's mastery (t = 2):", roster.get_state_type('Calculate unit rate', 'Bob'))
print("Bob's probability of mastery (t = 2):", roster.get_mastery_prob('Calculate unit rate', 'Bob'))

# We can print aggregate statistics for mastery and correctness.
print("Both students' probabilites of correctness:", roster.get_correct_probs('Calculate unit rate'))
print("Both students' probabilites of mastery:", roster.get_mastery_probs('Calculate unit rate'))

# Add a new student, Sarah.
roster.add_student('Calculate unit rate', 'Sarah')

# Update Sarah's state with a sequence of correct and incorrect responses.
sarah_new_state = roster.update_state('Calculate unit rate', 'Sarah', np.array([1, 0, 1, 0, 1, 1, 1]))

# Print Sarah's correctness and mastery probability.
print("Sarah's correctness probability:", sarah_new_state.get_correct_prob()
print("Sarah's mastery probability:", sarah_new_state.get_mastery_prob())

# Delete Bob from the roster.
roster.remove_student('Calculate unit rate', 'Bob')

# Reset student's state (i.e. latent and observable).
roster.reset_state('Calculate unit rate', 'Jeff')

# Jeff should be back to the initial prior as the mastery probability and should be unmastered.
print("Jeff's mastery (t' = 0):", roster.get_state_type('Calculate unit rate', 'Jeff'))
print("Jeff's probability of mastery (t' = 0):", roster.get_mastery_prob('Calculate unit rate', 'Jeff'))

Parameter Fixing

Another advanced feature supported by pyBKT is parameter fixing, where we can fix one or more parameters and train the model conditioned on those fixed parameters. This can be useful if you already know the ground truth value of some parameters beforehand, or to avoid degenerate model creation by fixing parameters at reasonable values. To specify which parameters and values we want fixed for any skill, we can pass in a dictionary to model.coef_, and then specify fixed=True in the model.fit call:

from pyBKT.models import *
import numpy as np
model = Model()

# Fixes the prior rate and learn rate to 0.1 for the Plot imperfect radical skill, and trains the model given those fixed parameters.
model.coef_ = {'Plot imperfect radical': {'prior': 0.1, 'learns': np.array([0.1])}}
model.fit(data_path = 'ct.csv', skills='Plot imperfect radical', fixed=True)
model.params()

Within the model.coef_ dictionary, the 'prior' parameter takes a scalar, while 'learns', 'forgets', 'guesses', and 'slips' takes an np.array, in order to provide support for parameter fixing in model extensions with multiple learn or guess classes. An example of such is shown below.

# The Plot pi skill has 10 different guess/slip classes. This is how you would fix those slip classes to 0, 0.1, ..., 0.9 and train the model conditioned on those slip values.
model.coef_ = {'Plot pi': {'slips': np.array([0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9])}}
model.fit(data_path = 'ct.csv', skills='Plot pi', multigs=True, fixed=True)
model.params()

Extended Features

Extended features include model parameter initialization by setting model.coef_, providing a configuration dictionary, setting model default columns, and more. For more information about these features, take a look at the Colab notebook provided at the top of the README.

Internal Data Format

pyBKT models student mastery of a skills as they progress through series of learning resources and checks for understanding. Mastery is modelled as a latent variable has two states - "knowing" and "not knowing". At each checkpoint, students may be given a learning resource (i.e. watch a video) and/or question(s) to check for understanding. The model finds the probability of learning, forgetting, slipping and guessing that maximizes the likelihood of observed student responses to questions.

To run the pyBKT model, define the following variables:

• num_subparts: The number of unique questions used to check understanding. Each subpart has a unique set of emission probabilities.
• num_resources: The number of unique learning resources available to students.
• num_fit_initialization: The number of iterations in the EM step.

Next, create an input object Data, containing the following attributes:

• data: a matrix containing sequential checkpoints for all students, with their responses. Each row represents a different subpart, and each column a checkpoint for a student. There are three potential values: {0 = no response or no question asked, 1 = wrong response, 2 = correct response}. If at a checkpoint, a resource was given but no question asked, the associated column would have 0 values in all rows. For example, to set up data containing 5 subparts given to two students over 2-3 checkpoints, the matrix would look as follows:

    | 0  0  0  0  2 |
    | 0  1  0  0  0 |
    | 0  0  0  0  0 |
    | 0  0  0  0  0 |
    | 0  0  2  0  0 |   
  

  In the above example, the first student starts out with just a learning resource, and no checks for understanding. In subsequent checkpoints, this student also responds to subpart 2 and 5, and gets the first wrong and the second correct.

• starts: defines each student's starting column on the data matrix. For the above matrix, starts would be defined as:

    | 1  4 |
  

• lengths: defines the number of check point for each student. For the above matrix, lengths would be defined as:

    | 3  2 |
  

• resources: defines the sequential id of the resources at each checkpoint. Each position in the vector corresponds to the column in the data matrix. For the above matrix, the learning resources at each checkpoint would be structured as:

    | 1  2  1  1  3 |
  

• stateseqs: this attribute is the true knowledge state for above data and should be left undefined before running the pyBKT model.

The output of the model can will be stored in a fitmodel object, containing the following probabilities as attributes:

• As: the transition probability between the "knowing" and "not knowing" state. Includes both the learns and forgets probabilities, and their inverse. As creates a separate transition probability for each resource.
• learns: the probability of transitioning to the "knowing" state given "not known".
• forgets: the probability of transitioning to the "not knowing" state given "known".
• prior: the prior probability of "knowing".

The fitmodel also includes the following emission probabilities:

• guesses: the probability of guessing correctly, given "not knowing" state.
• slips: the probability of picking incorrect answer, given "knowing" state.

Citation

To credit this library, please cite our paper published in the Educaitonal Data Mining Conference:

Badrinath, A., Wang, F., Pardos, Z.A. (2021) pyBKT: An Accessible Python Library of Bayesian Knowledge Tracing Models. In S. Hsiao, & S. Sahebi (Eds.) Proceedings of the 14th International Conference on Educational Data Mining (EDM). Pages 468-474.

@inproceedings{badrinath2021pybkt,
  title={pyBKT: An Accessible Python Library of Bayesian Knowledge Tracing Models},
  author={Badrinath, Anirudhan and Wang, Frederic and Pardos, Zachary},
  booktitle={Proceedings of the 14th International Conference on Educational Data Mining},
  pages={468--474},
  year={2021}
}

TODOs

• Exporting compiled binaries on PyPi
• Additional model variants