
from typing import Any

def  __lowerCamelCase					( _lowercase						,     _lowercase						,     _lowercase						,     _lowercase						,     _lowercase						,     )       ->      List[Any]:
     _validation(
         lowerCAmelCase__						,     lowerCAmelCase__						,     lowerCAmelCase__						,     lowerCAmelCase__						,     lowerCAmelCase__						,     )
     # Creates data structures and fill initial step
     UpperCamelCase						    =					{}
     UpperCamelCase						    =					{}
     for state in states_space:
          UpperCamelCase						    =					observations_space[0]
          UpperCamelCase						    =					(
              initial_probabilities[state] * emission_probabilities[state][observation]
          )
          UpperCamelCase						    =					None

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

                    # Update probabilities and pointers dicts
               UpperCamelCase						    =					(
                   probabilities[(arg_max, prior_observation)]
                   * transition_probabilities[arg_max][state]
                   * emission_probabilities[state][observation]
               )

               UpperCamelCase						    =					arg_max

    # The final observation
     UpperCamelCase						    =					observations_space[len(lowerCAmelCase__			) - 1]

     # argmax for given final observation
     UpperCamelCase						    =					''
     UpperCamelCase						    =					-1
     for k_state in states_space:
          UpperCamelCase						    =					probabilities[(k_state, final_observation)]
          if probability > max_probability:
               UpperCamelCase						    =					probability
               UpperCamelCase						    =					k_state
     UpperCamelCase						    =					arg_max

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

     return result

def  __lowerCamelCase					( _lowercase						,     _lowercase						,     _lowercase						,     _lowercase						,     _lowercase						,     )       ->      Tuple:
     _validate_not_empty(
         lowerCAmelCase__						,     lowerCAmelCase__						,     lowerCAmelCase__						,     lowerCAmelCase__						,     lowerCAmelCase__						,     )
     _validate_lists(lowerCAmelCase__						,     lowerCAmelCase__			)
     _validate_dicts(
         lowerCAmelCase__						,     lowerCAmelCase__						,     lowerCAmelCase__			)

def  __lowerCamelCase					( _lowercase						,     _lowercase						,     _lowercase						,     _lowercase						,     _lowercase						,     )       ->      Optional[Any]:
     if not all(
         [
             observations_space,
             states_space,
             initial_probabilities,
             transition_probabilities,
             emission_probabilities,
         ]			):
          raise ValueError('There\'s an empty parameter'			)

def  __lowerCamelCase					( _lowercase						,     _lowercase			)       ->      Tuple:
     _validate_list(lowerCAmelCase__						,     'observations_space'			)
     _validate_list(lowerCAmelCase__						,     'states_space'			)

def  __lowerCamelCase					( _lowercase						,     _lowercase			)       ->      Tuple:
     if not isinstance(_object						,     lowerCAmelCase__			):
          UpperCamelCase						    =					F'{var_name} must be a list'
          raise ValueError(lowerCAmelCase__			)
     else:
          for x in _object:
               if not isinstance(lowerCAmelCase__						,     lowerCAmelCase__			):
                    UpperCamelCase						    =					F'{var_name} must be a list of strings'
                    raise ValueError(lowerCAmelCase__			)

def  __lowerCamelCase					( _lowercase						,     _lowercase						,     _lowercase						,     )       ->      List[Any]:
     _validate_dict(lowerCAmelCase__						,     'initial_probabilities'						,     lowerCAmelCase__			)
     _validate_nested_dict(lowerCAmelCase__						,     'transition_probabilities'			)
     _validate_nested_dict(lowerCAmelCase__						,     'emission_probabilities'			)

def  __lowerCamelCase					( _lowercase						,     _lowercase			)       ->      Dict:
     _validate_dict(_object						,     lowerCAmelCase__						,     lowerCAmelCase__			)
     for x in _object.values():
          _validate_dict(lowerCAmelCase__						,     lowerCAmelCase__						,     lowerCAmelCase__						,     lowerCAmelCase__			)

def  __lowerCamelCase					( _lowercase						,     _lowercase						,     _lowercase						,     _lowercase = False			)       ->      Optional[int]:
     if not isinstance(_object						,     lowerCAmelCase__			):
          UpperCamelCase						    =					F'{var_name} must be a dict'
          raise ValueError(lowerCAmelCase__			)
     if not all(isinstance(lowerCAmelCase__						,     lowerCAmelCase__			) for x in _object			):
          UpperCamelCase						    =					F'{var_name} all keys must be strings'
          raise ValueError(lowerCAmelCase__			)
     if not all(isinstance(lowerCAmelCase__						,     lowerCAmelCase__			) for x in _object.values()			):
          UpperCamelCase						    =					'nested dictionary ' if nested else ''
          UpperCamelCase						    =					F'{var_name} {nested_text}all values must be {value_type.__name__}'
          raise ValueError(lowerCAmelCase__			)


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

 testmod()






from collections.abc import Callable




def  a   (lowerCAmelCase__		,				lowerCAmelCase__		,				lowerCAmelCase__   ):
   __a					=  a
   __a					=  b
   if function(lowerCAmelCase__   ) == 0:  # one of the a or b is a root for the function
      return a
   elif function(lowerCAmelCase__   ) == 0:
      return b
   elif (
       function(lowerCAmelCase__   ) * function(lowerCAmelCase__   ) > 0
   ):  # if none of these are root and they are both positive or negative,
      # then this algorithm can't find the root
      raise ValueError("""could not find root in given interval."""   )
   else:
      __a					=  start + (end - start) / 2.0
      while abs(start - mid   ) > 10**-7:  # until precisely equals to 10^-7
         if function(lowerCAmelCase__   ) == 0:
            return mid
         elif function(lowerCAmelCase__   ) * function(lowerCAmelCase__   ) < 0:
            __a					=  mid
         else:
            __a					=  mid
         __a					=  start + (end - start) / 2.0
      return mid




def  a   (lowerCAmelCase__   ):
   return x**3 - 2 * x - 5


if __name__ == "__main__":
  print(bisection(f, 1, 1_0_0_0))

  import doctest

  doctest.testmod()



