Can somebody help explain a line of code in an example of Theano tutorial?

In the logistic regression example provided in the Theano tutorial, there is one line of code in the negative_log_likelihood function as below:

def negative_log_likelihood(self, y):
    """Return the mean of the negative log-likelihood of the prediction
    of this model under a given target distribution.

    .. math::

        \frac{1}{|\mathcal{D}|} \mathcal{L} (\theta=\{W,b\}, \mathcal{D}) =
        \frac{1}{|\mathcal{D}|} \sum_{i=0}^{|\mathcal{D}|} \log(P(Y=y^{(i)}|x^{(i)}, W,b)) \\
            \ell (\theta=\{W,b\}, \mathcal{D})

    :type y: theano.tensor.TensorType
    :param y: corresponds to a vector that gives for each example the
              correct label

    Note: we use the mean instead of the sum so that
          the learning rate is less dependent on the batch size
    """
    # y.shape[0] is (symbolically) the number of rows in y, i.e.,
    # number of examples (call it n) in the minibatch
    # T.arange(y.shape[0]) is a symbolic vector which will contain
    # [0,1,2,... n-1] T.log(self.p_y_given_x) is a matrix of
    # Log-Probabilities (call it LP) with one row per example and
    # one column per class LP[T.arange(y.shape[0]),y] is a vector
    # v containing [LP[0,y[0]], LP[1,y[1]], LP[2,y[2]], ...,
    # LP[n-1,y[n-1]]] and T.mean(LP[T.arange(y.shape[0]),y]) is
    # the mean (across minibatch examples) of the elements in v,
    # i.e., the mean log-likelihood across the minibatch.
    return -T.mean(T.log(self.p_y_given_x)[T.arange(y.shape[0]), y])

Can someone help explain what exactly the use of square bracket in the last line of the above code? How is [T.arange(y.shape[0]), y] gonna be interpreted?

Thanks!

Solution

You have most of the information you need in the comments of the function.

T.log(self.p_y_give_x) returns a numpy matrix.

So the [T.arange(y.shape[0]), y] is a slice of the matrix. Here we are using numpy advanced slicing. See: http://docs.scipy.org/doc/numpy/reference/arrays.indexing.html