模型估计与似然

Python 线性建模入门

Jason Vestuto

Data Scientist

估计

直方图：标准化计数对距离分箱的灰色柱；模型为红色高斯曲线，紧贴灰柱顶部

估计

# Define gaussian model function
def gaussian_model(x, mu, sigma):
    coeff_part = 1/(np.sqrt(2 * np.pi * sigma**2))
    exp_part = np.exp( - (x - mu)**2 / (2 * sigma**2) )
    return coeff_part*exp_part

# Compute sample statistics
mean = np.mean(sample)
stdev = np.std(sample)

# Model the population using sample statistics
population_model = gaussian(sample, mu=mean, sigma=stdev)

似然 vs 概率

条件概率：$P( \text{结果A} | \text{给定B})$
概率：$P( \text{数据} | \text{模型} )$
似然：$L( \text{模型} | \text{数据} )$

计算似然

概率对距离的曲线为红色高斯钟形；左侧近边有一点，水平和垂直线段将该点连到各坐标轴

计算似然

概率对距离的曲线为红色高斯钟形；从左侧到中心有6个点，水平和垂直线段将每个点连到两轴

由概率到似然

# Guess parameters
mu_guess = np.mean(sample_distances)
sigma_guess = np.std(sample_distances)

# For each sample point, compute a probability
probabilities = np.zeros(len(sample_distances))
for n, distance in enumerate(sample_distances):
    probabilities[n] = gaussian_model(distance, mu=mu_guess, sigma=sigma_guess)

likelihood = np.product(probs)
loglikelihood = np.sum(np.log(probs))

最大似然估计

# Create an array of mu guesses
low_guess = sample_mean - 2*sample_stdev
high_guess = sample_mean + 2*sample_stdev
mu_guesses = np.linspace(low_guess, high_guess, 101)

# Compute the loglikelihood for each guess
loglikelihoods = np.zeros(len(mu_guesses))
for n, mu_guess in enumerate(mu_guesses):
    loglikelihoods[n] = compute_loglikelihood(sample_distances, mu=mu_guess, sigma=sample_stdev)

# Find the best guess
max_loglikelihood = np.max(loglikelihoods)
best_mu = mu_guesses[loglikelihoods == max_loglikelihood]

最大似然估计

向下开的抛物线图，红点位于最大值处，坐标轴为对数似然与 mu 值

Passons à la pratique !

Python 线性建模入门