[Term Entry] PyTorch Tensor Operations: .igamma()

content/pytorch/concepts/tensor-operations/terms/igamma/igamma.md

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+---
+Title: '.igamma()'
+Description: 'Computes the lower incomplete gamma function for tensor inputs.'
+Subjects:
+  - 'Computer Science'
+  - 'Data Science'
+  - 'Machine Learning'
+Tags:
+  - 'AI'
+  - 'Deep Learning'
+  - 'Methods'
+  - 'PyTorch'
+  - 'Tensor'
+CatalogContent:
+  - 'intro-to-py-torch-and-neural-networks'
+  - 'paths/data-science'
+---
+
+The **`torch.igamma()`** function in PyTorch computes the lower regularized incomplete gamma function, a special mathematical function often used in probability, statistics, and machine learning. `torch.igamma()` is an alias for `torch.special.gammainc()`. This means both functions compute the regularized lower incomplete gamma function and can be used interchangeably.
+
+## Syntax
+
+```pseudo
+torch.igamma(input, other, *, out=None)
+```
+
+Or, alternatively:
+
+```pseudo
+torch.special.gammainc(input, other, *, out=None)
+```
+
+**Parameters:**
+
+- `input` (Tensor): The shape parameter `a` of the Gamma function.
+- `other` (Tensor): The upper limit `x` of the integral.
+- `out` (Tensor, optional): The output tensor to store results.
+
+**Return value:**
+
+Returns a tensor containing the lower regularized incomplete gamma function values for each corresponding pair of elements in `input` and `other`.
+
+## Example 1: Basic Element-Wise Computation
+
+In this example, `torch.igamma()` computes the lower regularized incomplete gamma function for corresponding elements of two 1D tensors:
+
+```py
+import torch
+
+a = torch.tensor([2.0, 3.0, 4.0])
+x = torch.tensor([1.0, 2.0, 3.0])
+
+result = torch.igamma(a, x)
+print(result)
+```
+
+This example produces the following output:
+
+```shell
+tensor([0.2642, 0.3233, 0.3528])
+```
+
+## Example 2: Gamma Distribution CDF
+
+In this example, `torch.igamma()` calculates the cumulative distribution function (CDF) of a Gamma distribution with shape ${a\}$ and rate 1:
+
+```py
+import torch
+
+a = torch.tensor([2.0])
+x = torch.linspace(0, 5, 6)
+gamma_cdf = torch.igamma(a, x)
+
+print(gamma_cdf)
+```
+
+The output of this code is:
+
+```shell
+tensor([0.0000, 0.2642, 0.5940, 0.8009, 0.9084, 0.9596])
+```
+
+The `.igamma()` function is useful for:
+
+- Computing CDFs of Gamma, Chi-square, or Exponential distributions.
+- Performing Bayesian statistical modeling (priors and posteriors).
+- Implementing neural network activations and loss functions involving special functions.