Derive an expression for the mutual information I(X; Y)
时间: 2024-05-19 22:15:34 浏览: 12
Sure, the mutual information I(X; Y) can be expressed as the sum of the joint entropy H(X,Y) and the sum of the individual entropies H(X) and H(Y) with opposite signs, as follows:
I(X; Y) = H(X) + H(Y) - H(X,Y)
Is there anything else I can help you with?
相关问题
Derive the stochastic gradient descent algorithm
The stochastic gradient descent (SGD) algorithm is a popular optimization algorithm used in machine learning. It is an iterative algorithm that updates the model parameters in small steps based on the gradient of the loss function with respect to the parameters. The algorithm works as follows:
1. Initialize the model parameters randomly.
2. Set the learning rate, which determines the step size of the updates.
3. For each training example:
- Compute the gradient of the loss function with respect to the parameters using the current example.
- Update the model parameters by subtracting the gradient multiplied by the learning rate.
The key difference between SGD and regular gradient descent is that in SGD, the gradient is computed and the parameters are updated for each training example, rather than for the entire training set. This makes the algorithm faster and more scalable for large datasets.
The stochastic aspect of the algorithm comes from the fact that the training examples are sampled randomly from the training set, rather than being processed in a fixed order. This randomness can help the algorithm escape from local minima and find better solutions.
Here is the pseudocode for the SGD algorithm:
```
Input: Training set (X, Y), learning rate α, number of iterations T
Output: Model parameters θ
Initialize θ randomly
for t = 1 to T do
Sample a training example (x, y) from (X, Y) randomly
Compute the gradient ∇θ L(θ; x, y) using the current example
Update the parameters: θ ← θ - α * ∇θ L(θ; x, y)
end for
return θ
```
Simplify the expression (2i − 2)(1 − 3i)/(2i + 1). Derive your answer step-by-step.
答案: 步骤1:将分子中的2i和1-3i相乘:
2i × (1 - 3i) = 2i - 6i² 步骤2:将分子和分母相除:
(2i - 6i²) / (2i + 1) 步骤3:使用分数的乘法变换法则:
(2i - 6i²) × (2i - 1) / (2i + 1) × (2i - 1) 步骤4:将分子和分母中的因数相减:
(-12i³ + 4i² - 2i + 6i²) / (4i² - 2i - 1) 步骤5:将分子和分母中的因数相加:
(-12i³ + 6i² - 2i - 1) / (4i² - 2i - 1)
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