def conditional_probability_correct(abilities, ex_parameters, exercise_ind):
"""Predict the probabilities of getting questions correct for a set of
exercise indices, given model parameters in couplings and the
abilities vector for the user.
Args:
abilities: An ndarray with shape = (a, 1), where a = the
number of abilities in the model (not including the bias)
couplings: An ndarray with shape (n, a + 1), where n = the
number of exercises known by the model.
execises_ind: An ndarray of exercise indices in the coupling matrix.
The argument specifies which exercises the caller would like
conditional probabilities for.
Should be 1-d with shape = (# of exercises queried for)
Returns:
An ndarray of floats with shape = (exercise_ind.size)
"""
# Pad the abilities vector with a 1 to act as a bias.
# The shape of abilities will become (a+1, 1).
abilities = np.append(abilities.copy(), np.ones((1, 1)), axis=0)
difficulties = ex_parameters.W_correct[exercise_ind, :]
Z = sigmoid(np.dot(difficulties, abilities))
Z = np.reshape(Z, Z.size) # flatten to 1-d ndarray
return Z
def conditional_probability_correct(abilities, ex_parameters, exercise_ind):
"""Predict the probabilities of getting questions correct for a set of
exercise indices, given model parameters in couplings and the
abilities vector for the user.
Args:
abilities: An ndarray with shape = (a, 1), where a = the
number of abilities in the model (not including the bias)
couplings: An ndarray with shape (n, a + 1), where n = the
number of exercises known by the model.
execises_ind: An ndarray of exercise indices in the coupling matrix.
The argument specifies which exercises the caller would like
conditional probabilities for.
Should be 1-d with shape = (# of exercises queried for)
Returns:
An ndarray of floats with shape = (exercise_ind.size)
"""
# Pad the abilities vector with a 1 to act as a bias.
# The shape of abilities will become (a+1, 1).
abilities = np.append(abilities.copy(), np.ones((1, 1)), axis=0)
difficulties = ex_parameters.W_correct[exercise_ind, :]
Z = sigmoid(np.dot(difficulties, abilities))
Z = np.reshape(Z, Z.size) # flatten to 1-d ndarray
return Z
def L_dL_singleuser(arg):
"""Calculate log likelihood and gradient wrt couplings of mIRT model
for single user """
theta, state, options = arg
abilities = state['abilities'].copy()
correct = state['correct']
exercises_ind = state['exercises_ind']
dL = Parameters(theta.num_abilities, theta.num_exercises)
# pad the abilities vector with a 1 to act as a bias
abilities = np.append(abilities.copy(),
np.ones((1, abilities.shape[1])),
axis=0)
# the abilities to exercise coupling parameters for this exercise
W_correct = theta.W_correct[exercises_ind, :]
# calculate the probability of getting a question in this exercise correct
Y = np.dot(W_correct, abilities)
Z = sigmoid(Y) # predicted correctness value
Zt = correct.reshape(Z.shape) # true correctness value
pdata = Zt * Z + (1. - Zt) * (1. - Z) # = 2*Zt*Z - Z + const
dLdY = ((2. * Zt - 1.) * Z * (1. - Z)) / pdata
L = -np.sum(np.log(pdata))
dL.W_correct = -np.dot(dLdY, abilities.T)
if not options.correct_only:
# calculate the probability of taking time response_time to answer
log_time_taken = state['log_time_taken']
# the abilities to time coupling parameters for this exercise
W_time = theta.W_time[exercises_ind, :]
sigma = theta.sigma_time[exercises_ind].reshape((-1, 1))
Y = np.dot(W_time, abilities)
err = (Y - log_time_taken.reshape((-1, 1)))
L += np.sum(err**2 / sigma**2) / 2.
dLdY = err / sigma**2
dL.W_time = np.dot(dLdY, abilities.T)
dL.sigma_time = (-err**2 / sigma**3).ravel()
# normalization for the Gaussian
L += np.sum(0.5 * np.log(sigma**2))
dL.sigma_time += 1. / sigma.ravel()
return L, dL, exercises_ind
def L_dL_singleuser(arg):
"""Calculate log likelihood and gradient wrt couplings of mIRT model
for single user """
theta, state, options = arg
abilities = state['abilities'].copy()
correct = state['correct']
exercises_ind = state['exercises_ind']
dL = Parameters(theta.num_abilities, theta.num_exercises)
# pad the abilities vector with a 1 to act as a bias
abilities = np.append(abilities.copy(),
np.ones((1, abilities.shape[1])),
axis=0)
# the abilities to exercise coupling parameters for this exercise
W_correct = theta.W_correct[exercises_ind, :]
# calculate the probability of getting a question in this exercise correct
Y = np.dot(W_correct, abilities)
Z = sigmoid(Y) # predicted correctness value
Zt = correct.reshape(Z.shape) # true correctness value
pdata = Zt * Z + (1. - Zt) * (1. - Z) # = 2*Zt*Z - Z + const
dLdY = ((2. * Zt - 1.) * Z * (1. - Z)) / pdata
L = -np.sum(np.log(pdata))
dL.W_correct = -np.dot(dLdY, abilities.T)
if not options.correct_only:
# calculate the probability of taking time response_time to answer
log_time_taken = state['log_time_taken']
# the abilities to time coupling parameters for this exercise
W_time = theta.W_time[exercises_ind, :]
sigma = theta.sigma_time[exercises_ind].reshape((-1, 1))
Y = np.dot(W_time, abilities)
err = (Y - log_time_taken.reshape((-1, 1)))
L += np.sum(err ** 2 / sigma ** 2) / 2.
dLdY = err / sigma ** 2
dL.W_time = np.dot(dLdY, abilities.T)
dL.sigma_time = (-err ** 2 / sigma ** 3).ravel()
# normalization for the Gaussian
L += np.sum(0.5 * np.log(sigma ** 2))
dL.sigma_time += 1. / sigma.ravel()
return L, dL, exercises_ind
