问对期望最大化技术的直观解释是什么？

EN

import numpy as np
from scipy import stats

np.random.seed(110) # for reproducible results

# set parameters
red_mean = 3
red_std = 0.8

blue_mean = 7
blue_std = 2

# draw 20 samples from normal distributions with red/blue parameters
red = np.random.normal(red_mean, red_std, size=20)
blue = np.random.normal(blue_mean, blue_std, size=20)

both_colours = np.sort(np.concatenate((red, blue))) # for later use...

>>> np.mean(red)
2.802
>>> np.std(red)
0.871
>>> np.mean(blue)
6.932
>>> np.std(blue)
2.195

# estimates for the mean
red_mean_guess = 1.1
blue_mean_guess = 9

# estimates for the standard deviation
red_std_guess = 2
blue_std_guess = 1.7

likelihood_of_red = stats.norm(red_mean_guess, red_std_guess).pdf(both_colours)
likelihood_of_blue = stats.norm(blue_mean_guess, blue_std_guess).pdf(both_colours)

likelihood_total = likelihood_of_red + likelihood_of_blue

red_weight = likelihood_of_red / likelihood_total
blue_weight = likelihood_of_blue / likelihood_total

def estimate_mean(data, weight):
    """
    For each data point, multiply the point by the probability it
    was drawn from the colour's distribution (its "weight").

    Divide by the total weight: essentially, we're finding where 
    the weight is centred among our data points.
    """
    return np.sum(data * weight) / np.sum(weight)

def estimate_std(data, weight, mean):
    """
    For each data point, multiply the point's squared difference
    from a mean value by the probability it was drawn from
    that distribution (its "weight").

    Divide by the total weight: essentially, we're finding where 
    the weight is centred among the values for the difference of
    each data point from the mean.

    This is the estimate of the variance, take the positive square
    root to find the standard deviation.
    """
    variance = np.sum(weight * (data - mean)**2) / np.sum(weight)
    return np.sqrt(variance)

# new estimates for standard deviation
blue_std_guess = estimate_std(both_colours, blue_weight, blue_mean_guess)
red_std_guess = estimate_std(both_colours, red_weight, red_mean_guess)

# new estimates for mean
red_mean_guess = estimate_mean(both_colours, red_weight)
blue_mean_guess = estimate_mean(both_colours, blue_weight)

          | EM guess | Actual |  Delta
----------+----------+--------+-------
Red mean  |    2.910 |  2.802 |  0.108
Red std   |    0.854 |  0.871 | -0.017
Blue mean |    6.838 |  6.932 | -0.094
Blue std  |    2.227 |  2.195 |  0.032

import numpy as np
import math

#### E-M Coin Toss Example as given in the EM tutorial paper by Do and Batzoglou* #### 

def get_mn_log_likelihood(obs,probs):
    """ Return the (log)likelihood of obs, given the probs"""
    # Multinomial Distribution Log PMF
    # ln (pdf)      =             multinomial coeff            *   product of probabilities
    # ln[f(x|n, p)] = [ln(n!) - (ln(x1!)+ln(x2!)+...+ln(xk!))] + [x1*ln(p1)+x2*ln(p2)+...+xk*ln(pk)]     

    multinomial_coeff_denom= 0
    prod_probs = 0
    for x in range(0,len(obs)): # loop through state counts in each observation
        multinomial_coeff_denom = multinomial_coeff_denom + math.log(math.factorial(obs[x]))
        prod_probs = prod_probs + obs[x]*math.log(probs[x])

    multinomial_coeff = math.log(math.factorial(sum(obs))) -  multinomial_coeff_denom
    likelihood = multinomial_coeff + prod_probs
    return likelihood

# 1st:  Coin B, {HTTTHHTHTH}, 5H,5T
# 2nd:  Coin A, {HHHHTHHHHH}, 9H,1T
# 3rd:  Coin A, {HTHHHHHTHH}, 8H,2T
# 4th:  Coin B, {HTHTTTHHTT}, 4H,6T
# 5th:  Coin A, {THHHTHHHTH}, 7H,3T
# so, from MLE: pA(heads) = 0.80 and pB(heads)=0.45

# represent the experiments
head_counts = np.array([5,9,8,4,7])
tail_counts = 10-head_counts
experiments = zip(head_counts,tail_counts)

# initialise the pA(heads) and pB(heads)
pA_heads = np.zeros(100); pA_heads[0] = 0.60
pB_heads = np.zeros(100); pB_heads[0] = 0.50

# E-M begins!
delta = 0.001  
j = 0 # iteration counter
improvement = float('inf')
while (improvement>delta):
    expectation_A = np.zeros((5,2), dtype=float) 
    expectation_B = np.zeros((5,2), dtype=float)
    for i in range(0,len(experiments)):
        e = experiments[i] # i'th experiment
        ll_A = get_mn_log_likelihood(e,np.array([pA_heads[j],1-pA_heads[j]])) # loglikelihood of e given coin A
        ll_B = get_mn_log_likelihood(e,np.array([pB_heads[j],1-pB_heads[j]])) # loglikelihood of e given coin B

        weightA = math.exp(ll_A) / ( math.exp(ll_A) + math.exp(ll_B) ) # corresponding weight of A proportional to likelihood of A 
        weightB = math.exp(ll_B) / ( math.exp(ll_A) + math.exp(ll_B) ) # corresponding weight of B proportional to likelihood of B                            

        expectation_A[i] = np.dot(weightA, e) 
        expectation_B[i] = np.dot(weightB, e)

    pA_heads[j+1] = sum(expectation_A)[0] / sum(sum(expectation_A)); 
    pB_heads[j+1] = sum(expectation_B)[0] / sum(sum(expectation_B)); 

    improvement = max( abs(np.array([pA_heads[j+1],pB_heads[j+1]]) - np.array([pA_heads[j],pB_heads[j]]) ))
    j = j+1