Interpreting Criminal Charge Prediction and Its Algorithmic Bias via Quantum-Inspired Complex Valued Networks

The real-life crime prediction data we use contains the criminal charge record in Newark, New Jersey, from $1997$ to $2017$, with $17,335$ suspect individuals. Each suspect is provided with her/his Personal ID, race, and a list of Bookings. Each booking has a list of sentences, where each sentence has the age of committing the crime and the individual criminal charge history, such as the NCIC crime code, NCIC category code, and the crime level. The NCIC category code includes the list of [”ASSL”,”TO”, ”DAD”, ”LARC”, ”FO”, ”PP”, ”BURG”, ”SV”, ”DP”, ”OP”]. The input features also include the number of bookings, age average, number of crimes at level $1$, $2$, $3$, and so on. We take the first 18 years as the training set and the last two years as the test set. A suspect with at least two bookings is one sample. The label for each sample is the crime level (or whether it is a specific crime level) during the last booking.

Our goal is to predict a person’s crime or crime level ($1$ for most severe and $3$ least severe) given her/his personal information and previous criminal records. We propose a multitude of tasks addressed by Question (Q)$1$/$2$/$3$/$4$:

Is a suspect going to commit a crime at level 1/2/3/any?

Essentially we train four different systems for each of the above four questions, viewed as a binary classification task. We observe from Table 1 that this data is highly imbalanced, with most of the instances having the “No crime” label. This resembles a real-life scenario, where a suspect stops committing a crime more often than continuing to commit crimes. It is technically challenging to handle imbalanced data. A fully connected feed-forward neural network does not work in our experiments since it may classify all samples as negative. For example, suppose there are only $2$ positive samples among $100$ samples. In that case, the precision is still $98\%$, although nothing is classified.

We will first describe our task and dataset in the following context, introduce our models, then demonstrate experimental results and interpret our models. Finally, we will discuss related work and conclude.

To solve the imbalanced data problem, we observe that the criminal charges of an individual follow temporal patterns. A suspect likely repeats a similar behavior, such as committing a crime with the same level or after an equal amount of time (i.e. somewhat periodically). This kind of temporary pattern can be learned with the whole dataset’s statistics and shared among different individuals. We introduce a Recurrent Neural Network (RNN) to capture the sequential patterns of individual criminal charge records along one’s lifetime. To reduce the vanishing gradient problem, we add LSTMs (Hochreiter & Schmidhuber, 1997) in the activation functions, which greatly reduced the data imbalance problem. As a data preparation step, we had a list of booking for each person. Each booking has an age that shows the person’s age when he/she gets the sentence. We find that the maximum number of bookings for a person of different ages (sorted ages from young ages to old ones) is 13. We decided to create sequential data with 12 parts, and we use the last booking as a label of the dataset.

The second challenge is that our models lack interpretability. We provide a certain degree of post-interpretability using the attention mechanism (Vaswani et al., 2017). In particular, the attention mechanism allows to focus on, or pay attention to, different parts of the input sequence to generate a prediction. The attention scores obtained while predicting will indicate the importance of the input features towards the decision and can provide post-hoc interpretability. To obtain the feature importance: First, we check the impact of all the input features on our model predictions by setting the window size to be the total number of input features. After calculating the embedding of each feature, we compute their relative weights. Then, we apply softmax overall features to obtain probabilities of the predictions. Each input is associated with a probability value, and the sum of all probabilities should be $1$.

Attention mechanisms only interpret the input feature weights, while the deep learning hidden layers and decision process are hard to explain. To increase the transparency of our prediction model, we introduce complex-valued networks based on the mathematical formulation of quantum physics. As a realization, we adopt the model of (Li et al., 2019) from the matching task to a binary classification task. The criminal history and other attributes are the feature inputs to the model, and the output is yes or no to the future crime level.

Specifically, we first convert each feature value to a complex embedding of size $d$. An embedding layer with a ReLU activation (Nair & Hinton, 2010) takes as input the features one-by-one and converts them into two $d$-dimensional vectors, one for representing amplitude and the other for phase. Each number in the phase embedding is normalized to the range $[-\pi,\pi]$. Given the complex embedding, we compute the relative weight of the $i$-th feature as the $\ell_{2}$ norm of its amplitude. We then normalize the amplitude by dividing by the $\ell_{2}$ norm. Using the normalized amplitude $A_{i}$ and the phase $\varphi_{i}$, we compute the real ($R$) and imaginary ($G$) parts of $i$-th feature as $R_{i}=A_{i}\cos(\varphi_{i})$ and $G_{i}=A_{i}\sin(\varphi_{i})$. As in Figure 1, we compute the mixture ($M$) for each feature in the n-gram as complex matrix multiplication, and then each mixture is multiplied by a softmax probability of the feature’s relative weight in the n-gram. After that, all the mixtures in the n-gram are summed up to obtain the local mixture density matrix. The local mixtures are projected using $K$ measurement matrices as in (Li et al., 2019). The resulting vectors are max-pooled along each dimension, followed by a linear layer with softmax activation.

Our model (Winata et al., 2018) is based on long short-term memory (LSTM) network, and Bi-LSTM with attentions (Hochreiter & Schmidhuber, 1997) with two hidden layers, where each hidden layer has ten nodes with or without attention layers. The inputs are the features, and the outputs are fed to a linear layer mapping to the target classes. The dropout parameter is set to 0.1. The parameters were updated using the Adam optimizer (Kingma & Ba, 2014) with a learning rate of 0.001 and random initialization.

Table 2 shows the accuracy scores on the test data using different models on four research questions in Section 2. We introduce sequential models, including LSTM, Bi-LSTM with and without attention. We achieve both high precision and high recall overcoming the data imbalance problem by considering the criminal charge records along the previous eighteen years and learning the common patterns across samples.

Our attention model provides some interpretations for its predictions, suggesting that the prediction relies on historical criminal records rather than personal data, such as race and age. Feature weights are learned as a means to measure the “causality” of model decisions in our algorithm. We visualize softmax probabilities for randomly selected test samples. See Figure 2. We see that surprisingly, although there are many input features, only a few of them have the highest impact. Importantly, Figure 2 shows that personal information such as race and age contribute very little towards the prediction. Instead, an individual’s criminal history features greatly impact the prediction, such as “time since last crime” and “variance of the time gap between successive crimes”. This indicates the personal information features do not explicitly contribute to our algorithmic decisions. Note that our work only focuses on the feature importance explicitly used in our model. Discussions on any bias made by humans in the previous charge decisions (Brantingham et al., 2018; Arnold et al., 2018) are out of the scope of this paper.

The correlation is computed as the probability (relative frequency) of the lowest level (most severe one) of the crimes committed in the individual history given a certain crime level $L$ in the test set. The algorithmic “causality” of level an on predicting level b is computed as follows: For each test sample, we sum up the weights of all features of level a, such as the number of level a, yearly level a, on the model to predict whether a suspect is going to commit a crime at level b. Then we normalize it with all level-related weights and average the relative weights over all test samples. This average weight shows how much level a influences the decision on predicting level b.

The correlation and causality are different. The former computes the probability of a future crime level given a previous crime level in our data. The latter measures how much the features of a prior crime level contribute to the decision on whether a person will commit a certain level of crime in the future. Thus, the correlation is data-dependent, and the causality is model-dependent. In our study on the transitions of a suspect from one crime level to another, we find that crime level features are essential to predict a new crime level. Table 3 shows whether a lower level crime record indicates a higher level of crime or vice versa. Both our algorithmic causality (from our model of Bi-LSTM with attention mechanisms) and correlation results indicate that a suspect is likely to commit a crime with the same or similar level again than a much higher or lower level.