[typing] Add type hints to __init__ methods in torch.distributions. (#144197)

Fixes #144196
Extends #144106 and #144110

## Open Problems:

- [ ] Annotating with `numbers.Number` is a bad idea, should consider using `float`, `SupportsFloat` or some `Procotol`. https://github.com/pytorch/pytorch/pull/144197#discussion_r1903324769

# Notes

- `beta.py`: needed to add `type: ignore` since `broadcast_all` is untyped.
- `categorical.py`: converted `else` branches of mutually exclusive arguments to `if` branch[^2].
- ~~`dirichlet.py`: replaced `axis` with `dim` arguments.~~ #144402
- `gemoetric.py`: converted `else` branches of mutually exclusive arguments to `if` branch[^2].
- ~~`independent.py`: fixed bug in `Independent.__init__` where `tuple[int, ...]` could be passed to `Distribution.__init__` instead of `torch.Size`.~~ **EDIT:** turns out the bug is related to typing of `torch.Size`. #144218
- `independent.py`: made `Independent` a generic class of its base distribution.
- `multivariate_normal.py`: converted `else` branches of mutually exclusive arguments to `if` branch[^2].
- `relaxed_bernoulli.py`: added class-level type hint for `base_dist`.
- `relaxed_categorical.py`: added class-level type hint for `base_dist`.
- ~~`transforms.py`: Added missing argument to docstring of `ReshapeTransform`~~ #144401
- ~~`transforms.py`: Fixed bug in `AffineTransform.sign` (could return `Tensor` instead of `int`).~~ #144400
- `transforms.py`: Added `type: ignore` comments to `AffineTransform.log_abs_det_jacobian`[^1]; replaced `torch.abs(scale)` with `scale.abs()`.
- `transforms.py`: Added `type: ignore` comments to `AffineTransform.__eq__`[^1].
- `transforms.py`: Fixed type hint on `CumulativeDistributionTransform.domain`. Note that this is still an LSP violation, because `Transform.domain` is defined as `Constraint`, but `Distribution.domain` is defined as `Optional[Constraint]`.
- skipped: `constraints.py`, `constraints_registry.py`, `kl.py`, `utils.py`, `exp_family.py`, `__init__.py`.

## Remark

`TransformedDistribution`: `__init__` uses the check `if reinterpreted_batch_ndims > 0:`, which can lead to the creation of `Independent` distributions with only 1 component. This results in awkward code like `base_dist.base_dist` in `LogisticNormal`.

```python
import torch
from torch.distributions import *
b1 = Normal(torch.tensor([0.0]), torch.tensor([1.0]))
b2 = MultivariateNormal(torch.tensor([0.0]), torch.eye(1))
t = StickBreakingTransform()
d1 = TransformedDistribution(b1, t)
d2 = TransformedDistribution(b2, t)
print(d1.base_dist)  # Independent with 1 dimension
print(d2.base_dist)  # MultivariateNormal
```

One could consider changing this to `if reinterpreted_batch_ndims > 1:`.

[^1]: Usage of `isinstance(value, numbers.Real)` leads to problems with static typing, as the `numbers` module is not supported by `mypy` (see <https://github.com/python/mypy/issues/3186>). This results in us having to add type-ignore comments in several places
[^2]: Otherwise, we would have to add a bunch of `type: ignore` comments to make `mypy` happy, as it isn't able to perform the type narrowing. Ideally, such code should be replaced with structural pattern matching once support for Python 3.9 is dropped.

Pull Request resolved: https://github.com/pytorch/pytorch/pull/144197
Approved by: https://github.com/malfet

Co-authored-by: Aaron Gokaslan <aaronGokaslan@gmail.com>

torch/distributions/utils.py

@@ -7,7 +7,7 @@ import torch
 import torch.nn.functional as F
 from torch import Tensor
 from torch.overrides import is_tensor_like
-from torch.types import _Number
+from torch.types import _Number, Number
 
 
 euler_constant = 0.57721566490153286060  # Euler Mascheroni Constant
@@ -23,7 +23,9 @@ __all__ = [
 ]
 
 
-def broadcast_all(*values):
+# FIXME: Use (*values: *Ts) -> tuple[Tensor for T in Ts] if Mapping-Type is ever added.
+#   See https://github.com/python/typing/issues/1216#issuecomment-2126153831
+def broadcast_all(*values: Union[Tensor, Number]) -> tuple[Tensor, ...]:
     r"""
     Given a list of values (possibly containing numbers), returns a list where each
     value is broadcasted based on the following rules: