Logistic Distribution

Cheat Sheet

PDF Logistic Distribution

$$f(x) = \frac{\exp(-x)}{(1 + \exp(-x))^2} = \frac{\exp(-z)}{\sigma(1 + \exp(-z))^2}$$

where $z = \frac{(x-\mu)}{\sigma}$.

• Support: $(-\infty, \infty)$

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CDF Logistic Distribution

$$F(x) = \frac{1}{1 + \exp(-x)} = \frac{1}{1 + \exp(-z)}$$

where $z = \frac{(x-\mu)}{\sigma}$.

• Support: $(-\infty, \infty) \rightarrow [0, 1]$

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Quantile Function Logistic Distribution

$$F^{-1}(x) = \log\left(\frac{p}{1-p}\right) = \mu + \sigma_{\log} \log \left( \frac{p}{1-p} \right)$$

where $p \sim \mathcal{U}([0,1])$.

Inverse sampling

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Log SoftMax

$$\text{LogSoftmax}(x_i) = \log \left( \frac{\exp(x_i)}{\sum_j \exp(x_i)} \right)$$

PyTorch Function - Functional.log_softmax

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Sigmoid

$$\text{Sigmoid}(x) = \frac{1}{1 +\exp(-x)}$$

PyTorch Function - Functional.sigmoid

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Log Sigmoid

$$\text{LogSigmoid}(x) = \log \left( \frac{1}{1 +\exp(-x)} \right)$$

PyTorch Function - Functional.logsigmoid

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Log Sum Exponential

$$\text{LogSumExp}(x)_i = \log \sum_j \exp(x_{ij})$$

PyTorch Function - torch.logsumexp

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SoftPlus

$$\text{SoftPlus}(x) = \frac{1}{\beta}\log \left(1 + \exp(\beta x) \right)$$

• PyTorch Function - Function.softplus

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