Example of calculating the probability of a word, from character probabilities

word_probability.py

# A python example of calculating the probability of the word 'hine'
# See https://discourse.mozilla.org/t/quick-heads-up-on-some-metadata-confidence-estimate-work-were-doing/40618/2

# The probability that a word is incorrect before the nth letter:               
def p_not_word(n, probabilities):                                               
    if n == 0:                                                                  
        return 0                                                                
    else:                                                                       
        return probabilities[n - 1] * p_not_word(n - 1, probabilities)  + \     
            (1 - probabilities[n - 1])                                          
                                                                                
# The probability of a word being correctly emitted, given the character        
# probabilities:                                                                
def p_word(probabilities):                                                      
    return 1 - p_not_word(len(probabilities), probabilities)                                                                                                    
                                                                                
# Character probabilities for the word 'hine' (including the following space):  
#       h         i         n         e         blank
hine = [0.994121, 0.998789, 0.999938, 0.998212, 0.841081]                       
                                                                                
# The probability of the word 'hine', assuming independent draws with those     
# probabilities, following the probability tree:                        
p_hine = 1 - (                                                                  
    (1 - 0.994121) +                                           # not h                 
    0.994121 * (1 - 0.998789) +                                # h * not i                 
    0.994121 * 0.998789 * (1 - 0.999938) +                     # h * i * not n                 
    0.994121 * 0.998789 * 0.999938 * (1 - 0.998212) +          # h * i * n * not e                 
    0.994121 * 0.998789 * 0.999938 * 0.998212 * (1 - 0.841081) # h * i * n * e * not space                 
    )                                                                           

# Check the manual and the recursive calculation are the same
assert round(p_hine, 5) == round(p_word(hine), 5), \                            
    'Error in the calculation of word probabilities'                            

# And here is the probability of the word 'hine' (0.83357883745)
print(p_word(hine))