Who'd Claim?

Predicting life insurance claims with neural networks

Neural networks can help predict life insurance claims.

These networks capture complex relationships in data that traditional actuarial models may miss. Let’s explore how a simple neural network works for this use case.

Neural Network Structure

A neural network consists of layers of interconnected nodes (neurons). The basic structure includes:

  • Input Layer: Takes input features (e.g., age, gender, health status).
  • Hidden Layers: Perform computations to detect patterns.
  • Output Layer: Produces the prediction (e.g., probability of a claim).

Algebraic Formulations

Input and Weights

Each neuron in a hidden layer receives input \( x_i \) (features) and has associated weights \( w_i \). The net input \( z \) to a neuron is given by:

\[ z = \sum_{i=1}^{n} w_i x_i + b \]

where \( b \) is the bias term.

Activation Function

The net input \( z \) is passed through an activation function \( f(z) \) to introduce non-linearity. Common activation functions include the sigmoid, ReLU, and tanh.

  • Sigmoid: \( f(z) = \frac{1}{1 + e^{-z}} \)
  • ReLU: \( f(z) = \max(0, z) \)
  • Tanh: \( f(z) = \tanh(z) \)

Output

For a binary classification problem (e.g., predicting whether a claim will be made or not), the output layer often uses the sigmoid activation function. The final output \( \hat{y} \) is:

\[ \hat{y} = \sigma \left( \sum_{j=1}^{m} w_j f(z_j) + b \right) \]

where \( \sigma \) is the sigmoid function, and \( m \) is the number of neurons in the previous layer.

Training the Network

The network is trained using a dataset with known outcomes. The objective is to minimize the loss function, typically the binary cross-entropy loss for classification problems:

\[ L(y, \hat{y}) = - \frac{1}{N} \sum_{i=1}^{N} \left[ y_i \log(\hat{y}_i) + (1 - y_i) \log(1 - \hat{y}_i) \right] \]

where \( y_i \) is the actual label, \( \hat{y}_i \) is the predicted probability, and \( N \) is the number of samples.

The optimization is done using algorithms like gradient descent, which updates the weights \( w_i \) and biases \( b \) iteratively:

\[ w_i := w_i - \eta \frac{\partial L}{\partial w_i} \] \[ b := b - \eta \frac{\partial L}{\partial b} \]

where \( \eta \) is the learning rate.

Numerical Example

Dataset

Assume we have a simplified dataset with the following features and labels:

Age Gender Health Score Claim (1/0)
45 0 0.8 1
34 1 0.5 0
50 0 0.6 1
28 1 0.9 0

Age and Health Score are numerical features, and Gender is a binary feature (0 for male, 1 for female).

Network Configuration

  • Input Layer: 3 neurons (Age, Gender, Health Score)
  • Hidden Layer: 2 neurons, ReLU activation
  • Output Layer: 1 neuron, Sigmoid activation

Input to Hidden Layer

Assume initial weights \( w_{11}, w_{12}, w_{21}, w_{22}, w_{31}, w_{32} \) and biases \( b_1, b_2 \):

\[ z_1 = w_{11} \cdot \text{Age} + w_{21} \cdot \text{Gender} + w_{31} \cdot \text{Health Score} + b_1 \] \[ z_2 = w_{12} \cdot \text{Age} + w_{22} \cdot \text{Gender} + w_{32} \cdot \text{Health Score} + b_2 \]

Using ReLU activation:

\[ a_1 = \max(0, z_1) \] \[ a_2 = \max(0, z_2) \]

Hidden to Output Layer

Assume weights \( w_{1o}, w_{2o} \) and bias \( b_o \):

\[ z_o = w_{1o} \cdot a_1 + w_{2o} \cdot a_2 + b_o \] Using sigmoid activation:

\[ \hat{y} = \frac{1}{1 + e^{-z_o}} \]

Example Calculation

For the first row in the dataset (Age=45, Gender=0, Health Score=0.8):

Assume the following weights and biases for simplicity: \[ w_{11} = 0.1, w_{21} = -0.2, w_{31} = 0.3 \] \[ w_{12} = 0.2, w_{22} = 0.1, w_{32} = -0.3 \] \[ b_1 = 0.1, b_2 = -0.1 \] \[ w_{1o} = 0.4, w_{2o} = -0.5, b_o = 0.2 \]

Calculate the hidden layer outputs: \[ z_1 = 0.1 \cdot 45 + (-0.2) \cdot 0 + 0.3 \cdot 0.8 + 0.1 = 4.14 \] \[ z_2 = 0.2 \cdot 45 + 0.1 \cdot 0 + (-0.3) \cdot 0.8 - 0.1 = 8.76 \]

\[ a_1 = \max(0, 4.14) = 4.14 \] \[ a_2 = \max(0, 8.76) = 8.76 \]

Calculate the output: \[ z_o = 0.4 \cdot 4.14 + (-0.5) \cdot 8.76 + 0.2 = -0.414 \] \[ \hat{y} = \frac{1}{1 + e^{0.414}} \approx 0.397 \]

The predicted probability of a claim for this input is approximately 0.397, or almost 40%.

Neural networks provide a new approach to predicting life insurance claims by capturing complex relationships in the data. Through proper training and tuning, these models can significantly enhance decision-making processes in the insurance industry.