


from typing import Callable, Dict, Optional, Tuple

import torch
from torch import nn
from torch.distributions import (
    AffineTransform,
    Distribution,
    Independent,
    NegativeBinomial,
    Normal,
    StudentT,
    TransformedDistribution,
)






class   __lowercase				(  __lowerCamelCase  ):





      def __init__(		self						:					Optional[Any]					,       __lowerCamelCase						:					Optional[int]					,       __lowerCamelCase						:					int=None					,       __lowerCamelCase						:					List[str]=None					,       __lowerCamelCase						:					Any=0       )    ->   int:






         """simple docstring"""







         UpperCAmelCase							=     1.0 if scale is None else scale
         UpperCAmelCase							=     0.0 if loc is None else loc

         super().__init__(SCREAMING_SNAKE_CASE_					,       [AffineTransform(loc=self.loc					,       scale=self.scale					,       event_dim=SCREAMING_SNAKE_CASE_       )]       )





      @property
      def        _lowercase				(		self						:					Dict       )    ->   List[Any]:






         """simple docstring"""







         return self.base_dist.mean * self.scale + self.loc





      @property
      def        _lowercase				(		self						:					Dict       )    ->   Tuple:






         """simple docstring"""







         return self.base_dist.variance * self.scale**2





      @property
      def        _lowercase				(		self						:					int       )    ->   List[str]:






         """simple docstring"""







         return self.variance.sqrt()






class   __lowercase				(  nn.Module  ):





      def __init__(		self						:					Optional[Any]					,       __lowerCamelCase						:					str					,       __lowerCamelCase						:					str					,       __lowerCamelCase						:					Tuple					,       **__lowerCamelCase						:					Dict       )    ->   str:






         """simple docstring"""







         super().__init__(**SCREAMING_SNAKE_CASE_       )
         UpperCAmelCase							=     args_dim
         UpperCAmelCase							=     nn.ModuleList([nn.Linear(SCREAMING_SNAKE_CASE_					,       SCREAMING_SNAKE_CASE_       ) for dim in args_dim.values()]       )
         UpperCAmelCase							=     domain_map





      def        _lowercase				(		self						:					Dict					,       __lowerCamelCase						:					int       )    ->   Union[str, Any]:






         """simple docstring"""







         UpperCAmelCase							=     [proj(SCREAMING_SNAKE_CASE_       ) for proj in self.proj]

         return self.domain_map(*SCREAMING_SNAKE_CASE_       )






class   __lowercase				(  nn.Module  ):





      def __init__(		self						:					List[Any]					,       __lowerCamelCase						:					int       )    ->   Union[str, Any]:






         """simple docstring"""







         super().__init__()
         UpperCAmelCase							=     function





      def        _lowercase				(		self						:					int					,       __lowerCamelCase						:					Union[str, Any]					,       *__lowerCamelCase						:					List[str]       )    ->   str:






         """simple docstring"""







         return self.function(SCREAMING_SNAKE_CASE_					,       *SCREAMING_SNAKE_CASE_       )






class   __lowercase				:
      UpperCamelCase								=       42
      UpperCamelCase								=       42
      UpperCamelCase								=       42





      def __init__(		self						:					List[Any]					,       __lowerCamelCase						:					Any = 1       )    ->   Any:






         """simple docstring"""







         UpperCAmelCase							=     dim
         UpperCAmelCase							=     {k: dim * self.args_dim[k] for k in self.args_dim}





      def        _lowercase				(		self						:					List[str]					,       __lowerCamelCase						:					Union[str, Any]       )    ->   Any:






         """simple docstring"""







         if self.dim == 1:
            return self.distribution_class(*SCREAMING_SNAKE_CASE_       )
         else:
            return Independent(self.distribution_class(*SCREAMING_SNAKE_CASE_       )					,       1       )





      def        _lowercase				(		self						:					Union[str, Any]					,       __lowerCamelCase						:					Optional[int]					,       __lowerCamelCase						:					Optional[Any] = None					,       __lowerCamelCase						:					List[Any] = None					,       )    ->   List[Any]:






         """simple docstring"""







         UpperCAmelCase							=     self._base_distribution(SCREAMING_SNAKE_CASE_       )
         if loc is None and scale is None:
            return distr
         else:
            return AffineTransformed(SCREAMING_SNAKE_CASE_					,       loc=SCREAMING_SNAKE_CASE_					,       scale=SCREAMING_SNAKE_CASE_					,       event_dim=self.event_dim       )





      @property
      def        _lowercase				(		self						:					Dict       )    ->   int:






         """simple docstring"""







         return () if self.dim == 1 else (self.dim,)





      @property
      def        _lowercase				(		self						:					str       )    ->   List[Any]:






         """simple docstring"""







         return len(self.event_shape       )





      @property
      def        _lowercase				(		self						:					Optional[int]       )    ->   List[Any]:






         """simple docstring"""







         return 0.0





      def        _lowercase				(		self						:					List[str]					,       __lowerCamelCase						:					List[str]       )    ->   Union[str, Any]:






         """simple docstring"""







         return ParameterProjection(
             in_features=SCREAMING_SNAKE_CASE_					,       args_dim=self.args_dim					,       domain_map=LambdaLayer(self.domain_map       )					,       )





      def        _lowercase				(		self						:					str					,       *__lowerCamelCase						:					Dict       )    ->   Tuple:






         """simple docstring"""







         raise NotImplementedError()





      @staticmethod
      def        _lowercase				(		__lowerCamelCase						:					Optional[int]       )    ->   Optional[Any]:






         """simple docstring"""







         return (x + torch.sqrt(torch.square(SCREAMING_SNAKE_CASE_       ) + 4.0       )) / 2.0






class   __lowercase				(  __lowerCamelCase  ):
      UpperCamelCase								=       {'''df''': 1, '''loc''': 1, '''scale''': 1}
      UpperCamelCase								=       StudentT





      @classmethod
      def        _lowercase				(		cls						:					List[str]					,       __lowerCamelCase						:					Optional[Any]					,       __lowerCamelCase						:					Tuple					,       __lowerCamelCase						:					Dict       )    ->   Union[str, Any]:






         """simple docstring"""







         UpperCAmelCase							=     cls.squareplus(SCREAMING_SNAKE_CASE_       ).clamp_min(torch.finfo(scale.dtype       ).eps       )
         UpperCAmelCase							=     2.0 + cls.squareplus(SCREAMING_SNAKE_CASE_       )
         return df.squeeze(-1       ), loc.squeeze(-1       ), scale.squeeze(-1       )






class   __lowercase				(  __lowerCamelCase  ):
      UpperCamelCase								=       {'''loc''': 1, '''scale''': 1}
      UpperCamelCase								=       Normal





      @classmethod
      def        _lowercase				(		cls						:					Optional[Any]					,       __lowerCamelCase						:					Optional[Any]					,       __lowerCamelCase						:					Tuple       )    ->   Optional[int]:






         """simple docstring"""







         UpperCAmelCase							=     cls.squareplus(SCREAMING_SNAKE_CASE_       ).clamp_min(torch.finfo(scale.dtype       ).eps       )
         return loc.squeeze(-1       ), scale.squeeze(-1       )






class   __lowercase				(  __lowerCamelCase  ):
      UpperCamelCase								=       {'''total_count''': 1, '''logits''': 1}
      UpperCamelCase								=       NegativeBinomial





      @classmethod
      def        _lowercase				(		cls						:					Optional[int]					,       __lowerCamelCase						:					Optional[int]					,       __lowerCamelCase						:					int       )    ->   Optional[Any]:






         """simple docstring"""







         UpperCAmelCase							=     cls.squareplus(SCREAMING_SNAKE_CASE_       )
         return total_count.squeeze(-1       ), logits.squeeze(-1       )





      def        _lowercase				(		self						:					Union[str, Any]					,       __lowerCamelCase						:					Union[str, Any]       )    ->   Tuple:






         """simple docstring"""







         UpperCAmelCase				,  UpperCAmelCase							=     distr_args
         if self.dim == 1:
            return self.distribution_class(total_count=SCREAMING_SNAKE_CASE_					,       logits=SCREAMING_SNAKE_CASE_       )
         else:
            return Independent(self.distribution_class(total_count=SCREAMING_SNAKE_CASE_					,       logits=SCREAMING_SNAKE_CASE_       )					,       1       )





      def        _lowercase				(		self						:					Union[str, Any]					,       __lowerCamelCase						:					Dict					,       __lowerCamelCase						:					int = None					,       __lowerCamelCase						:					Dict = None       )    ->   Union[str, Any]:






         """simple docstring"""







         UpperCAmelCase				,  UpperCAmelCase							=     distr_args

         if scale is not None:
            # See scaling property of Gamma.
            logits += scale.log()

         return self._base_distribution((total_count, logits)       )








__a     =			[
    (1000, """M"""),
    (900, """CM"""),
    (500, """D"""),
    (400, """CD"""),
    (100, """C"""),
    (90, """XC"""),
    (50, """L"""),
    (40, """XL"""),
    (10, """X"""),
    (9, """IX"""),
    (5, """V"""),
    (4, """IV"""),
    (1, """I"""),
]
def   _UpperCamelCase				(						lowerCAmelCase_   )						->int:
   UpperCAmelCase							=     {"""I""": 1, """V""": 5, """X""": 1_0, """L""": 5_0, """C""": 1_0_0, """D""": 5_0_0, """M""": 1_0_0_0}
   UpperCAmelCase							=     0
   UpperCAmelCase							=     0
   while place < len(lowerCAmelCase_   ):
      if (place + 1 < len(lowerCAmelCase_   )) and (vals[roman[place]] < vals[roman[place + 1]]):
         total += vals[roman[place + 1]] - vals[roman[place]]
         place += 2
      else:
         total += vals[roman[place]]
         place += 1
   return total
def   _UpperCamelCase				(						lowerCAmelCase_   )						->str:
   UpperCAmelCase							=     []
   for arabic, roman in ROMAN:
      ((UpperCAmelCase)				,  (UpperCAmelCase))							=     divmod(lowerCAmelCase_  ,						lowerCAmelCase_   )
      result.append(roman * factor   )
      if number == 0:
         break
   return "".join(lowerCAmelCase_   )


if __name__ == "__main__":
 import doctest

 doctest.testmod()



