




from typing import Any




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  ):
				_validation(
				    lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  )
				# Creates data structures and fill initial step
				__magic_name__						:						dict										={}
				__magic_name__						:						dict										={}
				for state in states_space:
								__magic_name__						:						Optional[int]										=observations_space[0]
								__magic_name__						:						List[Any]										=(
								    initial_probabilities[state] * emission_probabilities[state][observation]
								)
								__magic_name__						:						Any										=None

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

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

												__magic_name__						:						List[str]										=arg_max

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

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

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

				return result




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  ):
				_validate_not_empty(
				    lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  )
				_validate_lists(lowerCamelCase						,  lowerCamelCase )
				_validate_dicts(
				    lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase )




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




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase ):
				_validate_list(lowerCamelCase						,  """observations_space""" )
				_validate_list(lowerCamelCase						,  """states_space""" )




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase ):
				if not isinstance(_object						,  lowerCamelCase ):
								__magic_name__						:						Any										=F"{var_name} must be a list"
								raise ValueError(lowerCamelCase )
				else:
								for x in _object:
												if not isinstance(lowerCamelCase						,  lowerCamelCase ):
																__magic_name__						:						List[str]										=F"{var_name} must be a list of strings"
																raise ValueError(lowerCamelCase )




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  ):
				_validate_dict(lowerCamelCase						,  """initial_probabilities"""						,  lowerCamelCase )
				_validate_nested_dict(lowerCamelCase						,  """transition_probabilities""" )
				_validate_nested_dict(lowerCamelCase						,  """emission_probabilities""" )




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase ):
				_validate_dict(_object						,  lowerCamelCase						,  lowerCamelCase )
				for x in _object.values():
								_validate_dict(lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase )




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase = False ):
				if not isinstance(_object						,  lowerCamelCase ):
								__magic_name__						:						int										=F"{var_name} must be a dict"
								raise ValueError(lowerCamelCase )
				if not all(isinstance(lowerCamelCase						,  lowerCamelCase ) for x in _object ):
								__magic_name__						:						Tuple										=F"{var_name} all keys must be strings"
								raise ValueError(lowerCamelCase )
				if not all(isinstance(lowerCamelCase						,  lowerCamelCase ) for x in _object.values() ):
								__magic_name__						:						Tuple										="""nested dictionary """ if nested else """"""
								__magic_name__						:						Any										=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
from math import pi, sqrt
from random import uniform
from statistics import mean




def 		lowerCAmelCase_ (						lowerCamelCase ):

				# A local function to see if a dot lands in the circle.
				def is_in_circle(lowerCamelCase						,  lowerCamelCase ) -> bool:
								__magic_name__						:						Dict										=sqrt((x**2) + (y**2) )
								# Our circle has a radius of 1, so a distance
								# greater than 1 would land outside the circle.
								return distance_from_centre <= 1

				# The proportion of guesses that landed in the circle
				__magic_name__						:						Union[str, Any]										=mean(
				    int(is_in_circle(uniform(-1.0						,  1.0 )						,  uniform(-1.0						,  1.0 ) ) )
				    for _ in range(lowerCamelCase ) )
				# The ratio of the area for circle to square is pi/4.
				__magic_name__						:						List[Any]										=proportion * 4
				print(F"The estimated value of pi is {pi_estimate}" )
				print(F"The numpy value of pi is {pi}" )
				print(F"The total error is {abs(pi - pi_estimate )}" )




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase = 0.0						,  lowerCamelCase = 1.0						,  ):
				return mean(
				    function_to_integrate(uniform(lowerCamelCase						,  lowerCamelCase ) ) for _ in range(lowerCamelCase ) ) * (max_value - min_value)




def 		lowerCAmelCase_ (						lowerCamelCase						,  lowerCamelCase = 0.0						,  lowerCamelCase = 1.0 ):

				def identity_function(lowerCamelCase ) -> float:
								return x

				__magic_name__						:						Optional[int]										=area_under_curve_estimator(
				    lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase						,  lowerCamelCase )
				__magic_name__						:						str										=(max_value * max_value - min_value * min_value) / 2

				print("""******************""" )
				print(F"Estimating area under y=x where x varies from {min_value} to {max_value}" )
				print(F"Estimated value is {estimated_value}" )
				print(F"Expected value is {expected_value}" )
				print(F"Total error is {abs(estimated_value - expected_value )}" )
				print("""******************""" )




def 		lowerCAmelCase_ (						lowerCamelCase ):

				def function_to_integrate(lowerCamelCase ) -> float:
								return sqrt(4.0 - x * x )

				__magic_name__						:						Dict										=area_under_curve_estimator(
				    lowerCamelCase						,  lowerCamelCase						,  0.0						,  2.0 )

				print("""******************""" )
				print("""Estimating pi using area_under_curve_estimator""" )
				print(F"Estimated value is {estimated_value}" )
				print(F"Expected value is {pi}" )
				print(F"Total error is {abs(estimated_value - pi )}" )
				print("""******************""" )


if __name__ == "__main__":
							import doctest

							doctest.testmod()




