'''simple docstring'''




from typing import Any

def 	UpperCamelCase_							(					_UpperCAmelCase  :      list			,	_UpperCAmelCase  :      list			,	_UpperCAmelCase  :      dict			,	_UpperCAmelCase  :      dict			,	_UpperCAmelCase  :      dict			,	)    ->    list:


       """simple docstring"""







       _validation(
           _UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	)
       # Creates data structures and fill initial step
       _UpperCAmelCase     :  dict							      =    {}
       _UpperCAmelCase     :  dict							      =    {}
       for state in states_space:
              _UpperCAmelCase     :  Union[str, Any]							      =    observations_space[0]
              _UpperCAmelCase     :  Tuple							      =    (
                  initial_probabilities[state] * emission_probabilities[state][observation]
              )
              _UpperCAmelCase     :  List[str]							      =    None

       # Fills the data structure with the probabilities of
       # different transitions and pointers to previous states
       for o in range(1			,	len(_UpperCAmelCase    )    ):
              _UpperCAmelCase     :  Optional[Any]							      =    observations_space[o]
              _UpperCAmelCase     :  int							      =    observations_space[o - 1]
              for state in states_space:
                     # Calculates the argmax for probability function
                     _UpperCAmelCase     :  str							      =    ""
                     _UpperCAmelCase     :  Tuple							      =    -1
                     for k_state in states_space:
                            _UpperCAmelCase     :  Any							      =    (
                                probabilities[(k_state, prior_observation)]
                                * transition_probabilities[k_state][state]
                                * emission_probabilities[state][observation]
                            )
                            if probability > max_probability:
                                   _UpperCAmelCase     :  Union[str, Any]							      =    probability
                                   _UpperCAmelCase     :  str							      =    k_state

            # Update probabilities and pointers dicts
                     _UpperCAmelCase     :  Optional[int]							      =    (
                         probabilities[(arg_max, prior_observation)]
                         * transition_probabilities[arg_max][state]
                         * emission_probabilities[state][observation]
                     )

                     _UpperCAmelCase     :  Tuple							      =    arg_max

    # The final observation
       _UpperCAmelCase     :  Optional[Any]							      =    observations_space[len(_UpperCAmelCase    ) - 1]

       # argmax for given final observation
       _UpperCAmelCase     :  List[str]							      =    ""
       _UpperCAmelCase     :  Any							      =    -1
       for k_state in states_space:
              _UpperCAmelCase     :  Optional[int]							      =    probabilities[(k_state, final_observation)]
              if probability > max_probability:
                     _UpperCAmelCase     :  int							      =    probability
                     _UpperCAmelCase     :  Dict							      =    k_state
       _UpperCAmelCase     :  Dict							      =    arg_max

       # Process pointers backwards
       _UpperCAmelCase     :  List[Any]							      =    last_state
       _UpperCAmelCase     :  str							      =    []
       for o in range(len(_UpperCAmelCase    ) - 1			,	-1			,	-1    ):
              result.append(_UpperCAmelCase    )
              _UpperCAmelCase     :  List[Any]							      =    pointers[previous, observations_space[o]]
       result.reverse()

       return result

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	)    ->    None:


       """simple docstring"""







       _validate_not_empty(
           _UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	)
       _validate_lists(_UpperCAmelCase			,	_UpperCAmelCase    )
       _validate_dicts(
           _UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase    )

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	)    ->    None:


       """simple docstring"""







       if not all(
           [
               observations_space,
               states_space,
               initial_probabilities,
               transition_probabilities,
               emission_probabilities,
           ]    ):
              raise ValueError("There's an empty parameter"    )

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any    )    ->    None:


       """simple docstring"""







       _validate_list(_UpperCAmelCase			,	"observations_space"    )
       _validate_list(_UpperCAmelCase			,	"states_space"    )

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      str    )    ->    None:


       """simple docstring"""







       if not isinstance(_object			,	_UpperCAmelCase    ):
              _UpperCAmelCase     :  Optional[int]							      =    F"""{var_name} must be a list"""
              raise ValueError(_UpperCAmelCase    )
       else:
              for x in _object:
                     if not isinstance(_UpperCAmelCase			,	_UpperCAmelCase    ):
                            _UpperCAmelCase     :  Optional[int]							      =    F"""{var_name} must be a list of strings"""
                            raise ValueError(_UpperCAmelCase    )

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	)    ->    None:


       """simple docstring"""







       _validate_dict(_UpperCAmelCase			,	"initial_probabilities"			,	_UpperCAmelCase    )
       _validate_nested_dict(_UpperCAmelCase			,	"transition_probabilities"    )
       _validate_nested_dict(_UpperCAmelCase			,	"emission_probabilities"    )

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      str    )    ->    None:


       """simple docstring"""







       _validate_dict(_object			,	_UpperCAmelCase			,	_UpperCAmelCase    )
       for x in _object.values():
              _validate_dict(_UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase    )

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      str			,	_UpperCAmelCase  :      type			,	_UpperCAmelCase  :      bool = False    )    ->    None:


       """simple docstring"""







       if not isinstance(_object			,	_UpperCAmelCase    ):
              _UpperCAmelCase     :  Any							      =    F"""{var_name} must be a dict"""
              raise ValueError(_UpperCAmelCase    )
       if not all(isinstance(_UpperCAmelCase			,	_UpperCAmelCase    ) for x in _object    ):
              _UpperCAmelCase     :  Tuple							      =    F"""{var_name} all keys must be strings"""
              raise ValueError(_UpperCAmelCase    )
       if not all(isinstance(_UpperCAmelCase			,	_UpperCAmelCase    ) for x in _object.values()    ):
              _UpperCAmelCase     :  List[str]							      =    "nested dictionary " if nested else ""
              _UpperCAmelCase     :  List[str]							      =    F"""{var_name} {nested_text}all values must be {value_type.__name__}"""
              raise ValueError(_UpperCAmelCase    )


if __name__ == "__main__":
  from doctest import testmod

  testmod()


'''simple docstring'''




from typing import Any

def 	UpperCamelCase_							(					_UpperCAmelCase  :      list			,	_UpperCAmelCase  :      list			,	_UpperCAmelCase  :      dict			,	_UpperCAmelCase  :      dict			,	_UpperCAmelCase  :      dict			,	)    ->    list:


       """simple docstring"""







       _validation(
           _UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	)
       # Creates data structures and fill initial step
       _UpperCAmelCase     :  dict							      =    {}
       _UpperCAmelCase     :  dict							      =    {}
       for state in states_space:
              _UpperCAmelCase     :  Union[str, Any]							      =    observations_space[0]
              _UpperCAmelCase     :  Tuple							      =    (
                  initial_probabilities[state] * emission_probabilities[state][observation]
              )
              _UpperCAmelCase     :  List[str]							      =    None

       # Fills the data structure with the probabilities of
       # different transitions and pointers to previous states
       for o in range(1			,	len(_UpperCAmelCase    )    ):
              _UpperCAmelCase     :  Optional[Any]							      =    observations_space[o]
              _UpperCAmelCase     :  int							      =    observations_space[o - 1]
              for state in states_space:
                     # Calculates the argmax for probability function
                     _UpperCAmelCase     :  str							      =    ""
                     _UpperCAmelCase     :  Tuple							      =    -1
                     for k_state in states_space:
                            _UpperCAmelCase     :  Any							      =    (
                                probabilities[(k_state, prior_observation)]
                                * transition_probabilities[k_state][state]
                                * emission_probabilities[state][observation]
                            )
                            if probability > max_probability:
                                   _UpperCAmelCase     :  Union[str, Any]							      =    probability
                                   _UpperCAmelCase     :  str							      =    k_state

            # Update probabilities and pointers dicts
                     _UpperCAmelCase     :  Optional[int]							      =    (
                         probabilities[(arg_max, prior_observation)]
                         * transition_probabilities[arg_max][state]
                         * emission_probabilities[state][observation]
                     )

                     _UpperCAmelCase     :  Tuple							      =    arg_max

    # The final observation
       _UpperCAmelCase     :  Optional[Any]							      =    observations_space[len(_UpperCAmelCase    ) - 1]

       # argmax for given final observation
       _UpperCAmelCase     :  List[str]							      =    ""
       _UpperCAmelCase     :  Any							      =    -1
       for k_state in states_space:
              _UpperCAmelCase     :  Optional[int]							      =    probabilities[(k_state, final_observation)]
              if probability > max_probability:
                     _UpperCAmelCase     :  int							      =    probability
                     _UpperCAmelCase     :  Dict							      =    k_state
       _UpperCAmelCase     :  Dict							      =    arg_max

       # Process pointers backwards
       _UpperCAmelCase     :  List[Any]							      =    last_state
       _UpperCAmelCase     :  str							      =    []
       for o in range(len(_UpperCAmelCase    ) - 1			,	-1			,	-1    ):
              result.append(_UpperCAmelCase    )
              _UpperCAmelCase     :  List[Any]							      =    pointers[previous, observations_space[o]]
       result.reverse()

       return result

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	)    ->    None:


       """simple docstring"""







       _validate_not_empty(
           _UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	)
       _validate_lists(_UpperCAmelCase			,	_UpperCAmelCase    )
       _validate_dicts(
           _UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase    )

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	)    ->    None:


       """simple docstring"""







       if not all(
           [
               observations_space,
               states_space,
               initial_probabilities,
               transition_probabilities,
               emission_probabilities,
           ]    ):
              raise ValueError("There's an empty parameter"    )

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any    )    ->    None:


       """simple docstring"""







       _validate_list(_UpperCAmelCase			,	"observations_space"    )
       _validate_list(_UpperCAmelCase			,	"states_space"    )

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      str    )    ->    None:


       """simple docstring"""







       if not isinstance(_object			,	_UpperCAmelCase    ):
              _UpperCAmelCase     :  Optional[int]							      =    F"""{var_name} must be a list"""
              raise ValueError(_UpperCAmelCase    )
       else:
              for x in _object:
                     if not isinstance(_UpperCAmelCase			,	_UpperCAmelCase    ):
                            _UpperCAmelCase     :  Optional[int]							      =    F"""{var_name} must be a list of strings"""
                            raise ValueError(_UpperCAmelCase    )

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      Any			,	)    ->    None:


       """simple docstring"""







       _validate_dict(_UpperCAmelCase			,	"initial_probabilities"			,	_UpperCAmelCase    )
       _validate_nested_dict(_UpperCAmelCase			,	"transition_probabilities"    )
       _validate_nested_dict(_UpperCAmelCase			,	"emission_probabilities"    )

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      str    )    ->    None:


       """simple docstring"""







       _validate_dict(_object			,	_UpperCAmelCase			,	_UpperCAmelCase    )
       for x in _object.values():
              _validate_dict(_UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase			,	_UpperCAmelCase    )

def 	UpperCamelCase_							(					_UpperCAmelCase  :      Any			,	_UpperCAmelCase  :      str			,	_UpperCAmelCase  :      type			,	_UpperCAmelCase  :      bool = False    )    ->    None:


       """simple docstring"""







       if not isinstance(_object			,	_UpperCAmelCase    ):
              _UpperCAmelCase     :  Any							      =    F"""{var_name} must be a dict"""
              raise ValueError(_UpperCAmelCase    )
       if not all(isinstance(_UpperCAmelCase			,	_UpperCAmelCase    ) for x in _object    ):
              _UpperCAmelCase     :  Tuple							      =    F"""{var_name} all keys must be strings"""
              raise ValueError(_UpperCAmelCase    )
       if not all(isinstance(_UpperCAmelCase			,	_UpperCAmelCase    ) for x in _object.values()    ):
              _UpperCAmelCase     :  List[str]							      =    "nested dictionary " if nested else ""
              _UpperCAmelCase     :  List[str]							      =    F"""{var_name} {nested_text}all values must be {value_type.__name__}"""
              raise ValueError(_UpperCAmelCase    )


if __name__ == "__main__":
  from doctest import testmod

  testmod()

