Add Convert Arbitrary Number To Probability With Sigmoid as a Math TIL

README.md

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 ### Math
 
+- [Convert Arbitrary Number To Probability With Sigmoid](math/convert-arbitrary-number-to-probability-with-sigmoid.md)
 - [Generate Permutations Of All Valid 9-ball Racks](math/generate-permutations-of-all-valid-9-ball-racks.md)
 
 ### Mise

math/convert-arbitrary-number-to-probability-with-sigmoid.md

@@ -0,0 +1,45 @@
+# Convert Arbitrary Number To Probability With Sigmoid
+
+A sigmoid function is a useful function in statistics and machine learning for
+converting a number in the range of positive and negative real numbers into a
+value between 0 and 1. Sigmoid functions can be a bit more diverse than this,
+but this is a good basic definition.
+
+Wikipedia defines another characteristic of sigmoid functions:
+
+> A sigmoid function is any mathematical function whose graph has a
+> characteristic S-shaped or sigmoid curve.
+
+This S-shape is because it is asymptotic at the ends allowing it to cover all
+real numbers in either direction.
+
+A common sigmoid function and the one used by [PyTorch's `Sigmoid`](https://docs.pytorch.org/docs/2.13/generated/torch.nn.Sigmoid.html)
+is this exponential form -- `σ(x) = 1 / (1 + exp(-x))`.
+
+Here is what this looks like plotted on a graph:
+
+![sigmoid function plotted on a graph](https://cdn.visualmode.dev/images/3cadb482-matplot-sigmoid-function-graph.png)
+
+This function can be used any time we want to convert an arbitrary number into a
+probability. Large negative numbers will approach 0. Large positive numbers will
+approach 1. Numbers near 0 will settle somewhere in the middle.
+
+Here are a few examples run through PyTorch's `sigmoid` function:
+
+```python
+print("σ(-99) => ", torch.sigmoid(torch.tensor(-99.0)))
+print("σ(99) => ", torch.sigmoid(torch.tensor(99.0)))
+print("σ(0.123) => ", torch.sigmoid(torch.tensor(0.123)))
+print("σ(-2) => ", torch.sigmoid(torch.tensor(-2.0)))
+print("σ(1) => ", torch.sigmoid(torch.tensor(1.0)))
+```
+
+which prints out:
+
+```
+σ(-99) =>  tensor(0.)
+σ(99) =>  tensor(1.)
+σ(0.123) =>  tensor(0.5307)
+σ(-2) =>  tensor(0.1192)
+σ(1) =>  tensor(0.7311)
+```