Who'd Claim?

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.