def compute_mean_mles(train_data, train_labels):
    '''
    Compute the mean estimate for each digit class

    Should return a numpy array of size (10,64)
    The ith row will correspond to the mean estimate for digit class i
    '''
    means = np.zeros((10, 64))
    # Compute means
    return means

def compute_sigma_mles(train_data, train_labels):
    '''
    Compute the covariance estimate for each digit class

    Should return a three dimensional numpy array of shape (10, 64, 64)
    consisting of a covariance matrix for each digit class
    '''
    covariances = np.zeros((10, 64, 64))
    mus = compute_mean_mles(train_data,train_labels)
    identity_add = np.identity(64)*0.01
    for i in range(10):
        c = train_data[train_labels == i,:] - mus[i]
        covariances[i,:,:] = (np.dot(c.T,c) / (train_labels == i).sum()) + identity_add
    return covariances

def generative_likelihood(digits, means, covariances):
    '''
    Compute the generative log-likelihood:
        log p(x|y,mu,Sigma)

    Should return an n x 10 numpy array
    '''
    likelihoods = np.zeros((digits.shape,10))
    for d in range(digits.shape[0]):
        for i in range(10):
            x = digits[d,:]
            mu = means[i,:]
            cov = covariances[i]
            d = 64
            det_cv = np.log(np.linalg.det(cov))
            last_term = d*np.log(2*np.pi)
            likelihoods[d,i] = 0.5*(-det_cv - np.linalg.multi_dot([(x-mu).T,np.linalg.inv(cov),(x-mu)]) - last_term)
    return likelihoods