我一直在研究数据融合方法，引起我注意的是使用卡尔曼滤波器的想法，它研究数据融合数据，研究高斯分布的均值和方差，并实现从弱传感器到更强/更准确传感器的预测和校正. 我查看了以下 GitHub 链接以进一步了解融合技术：
所以，在第一个链接中，我发现他们在谈论离散贝叶斯滤波器，但是，他们没有提到连续贝叶斯滤波器。所以，我想用卡尔曼滤波器的想法做同样的步骤，在 PyMC3 包的帮助下实现一个连续的贝叶斯滤波器。所涉及的步骤可以在第二个链接中找到，代码如下。
mean = mean0
var = var0
def predict(var, mean, varMove, meanMove):
new_var = var + varMove
new_mean= mean+ meanMove
return new_var, new_mean
def correct(var, mean, varSensor, meanSensor):
new_mean=(varSensor*mean + var*meanSensor) / (var+varSensor)
new_var = 1/(1/var +1/varSensor)
return new_var, new_mean
# Predict
var, mean = predict(var, mean, varMove, distances[m])
#print('mean: %.2f\tvar:%.2f' % (mean, var))
plt.plot(x,mlab.normpdf(x, mean, var), label='%i. step (Prediction)' % (m+1))
# Correct
var, mean = correct(var, mean, varSensor, positions[m])
print('After correction: mean= %.2f\tvar= %.2f' % (mean, var))
plt.plot(x,mlab.normpdf(x, mean, var), label='%i. step (Correction)' % (m+1))
因此，根据我使用的数据，我实现了以下链接，
从上面的链接和分布列表中，学生 T 分布符合我的数据。实现数据融合的思想是能够获得初始数据，然后使用来自第一个模型的参数并将它们实施到预测模型中，并从预测模型收集参数到校正模型。这是我的融合方法（我为冗长的代码道歉），
def S2_0_Bayesian_Interface(data):
########################################################################################################################
with pm.Model() as model_0:
# Prior Distributions for unknown model parameters:
nu_0 = pm.HalfNormal('nu_0', sigma=10)
sigma_0 = pm.HalfNormal('sigma_0', sigma=10)
mu_0 = pm.Normal('mu_0', mu=0, sigma=10)
# Observed data is from a Likelihood distributions (Likelihood (sampling distribution) of observations):
observed_data_0 = pm.StudentT('observed_data_0', nu=nu_0, mu=mu_0, sigma=sigma_0, observed=data)
# draw 5000 posterior samples
trace_0 = pm.sample(draws=5000, tune=3000, chains=2, cores=1)
# Obtaining Posterior Predictive Sampling:
post_pred_0 = pm.sample_posterior_predictive(trace_0, samples=3000)
print(post_pred_0['observed_data_0'].shape)
print('\nSummary: ')
print(pm.stats.summary(data=trace_0))
print(pm.stats.summary(data=post_pred_0))
########################################################################################################################
return trace_0, post_pred_0
def S2_P_Bayesian_Interface(nu, mu, sigma, data):
########################################################################################################################
with pm.Model() as model_P:
# Prior Distributions for unknown model parameters:
nu_P = pm.HalfNormal('nu_P', sigma=sigma)
sigma_P = pm.HalfNormal('sigma_P', sigma=sigma)
mu_P = pm.Normal('mu_P', mu=mu, sigma=sigma)
# Observed data is from a Likelihood distributions (Likelihood (sampling distribution) of observations):
observed_data_P = pm.StudentT('observed_data_P', nu=nu_P, mu=mu_P, sigma=sigma_P, observed=data)
# draw 5000 posterior samples
trace_P = pm.sample(draws=5000, tune=3000, chains=2, cores=1)
# Obtaining Posterior Predictive Sampling:
post_pred_P = pm.sample_posterior_predictive(trace_P, samples=3000)
print(post_pred_P['observed_data_P'].shape)
print('\nSummary: ')
print(pm.stats.summary(data=trace_P))
print(pm.stats.summary(data=post_pred_P))
########################################################################################################################
return trace_P, post_pred_P
def S1_C_Bayesian_Interface(nu, mu, sigma, data):
########################################################################################################################
with pm.Model() as model_C:
# Prior Distributions for unknown model parameters:
nu_C = pm.HalfNormal('nu_C', sigma=sigma)
sigma_C = pm.HalfNormal('sigma_C', sigma=sigma)
mu_C = pm.Normal('mu_C', mu=mu, sigma=sigma)
# Observed data is from a Likelihood distributions (Likelihood (sampling distribution) of observations):
observed_data_C = pm.StudentT('observed_data_C', nu=nu_C, mu=mu_C, sigma=sigma_C, observed=data)
# draw 5000 posterior samples
trace_C = pm.sample(draws=5000, tune=3000, chains=2, cores=1)
# Obtaining Posterior Predictive Sampling:
post_pred_C = pm.sample_posterior_predictive(trace_C, samples=3000)
print(post_pred_C['observed_data_C'].shape)
print('\nSummary: ')
print(pm.stats.summary(data=trace_C))
print(pm.stats.summary(data=post_pred_C))
########################################################################################################################
return trace_C, post_pred_C
for m in range(len(S2_data_list)):
print('\n')
print('Acessing list number: ', m)
# print(S2_data_list)
# print(len(S2_data_list))
# Initializing the PDF Distribution:
print('#######################################################################################################')
print('Designing the Bayesian PDF for initial Sensor 2:')
trace_S2_0, post_pred_S2_0 = S2_0_Bayesian_Interface(data=S2_data_list[0])
best_dist = getattr(st, 't')
param_S2_0 = [np.mean(trace_S2_0['nu_0']), np.mean(trace_S2_0['mu_0']), np.mean(trace_S2_0['sigma_0'])]
print('Parameters values = ', param_S2_0)
pdf_S2_0, B_x_axis_pdf_S2_0, y_axis_pdf_S2_0 = make_pdf(data=S2_data_list[0], dist=best_dist, params=param_S2_0, size=len(trace_S2_0))
print('#######################################################################################################')
plt.plot(B_x_axis_pdf_S2_0, y_axis_pdf_S2_0, label='%i. step (Initial[S2])' % (m + 1))
# Predict:
print('#######################################################################################################')
print('Designing the Bayesian PDF for predictive Sensor 2:')
trace_S2_P, post_pred_S2_P = S2_P_Bayesian_Interface(nu=np.mean(trace_S2_0['nu_0']), mu=np.mean(trace_S2_0['mu_0']), sigma=np.mean(trace_S2_0['sigma_0']), data=S2_data_list[m])
best_dist = getattr(st, 't')
param_S2_P = [np.mean(trace_S2_P['nu_P']), np.mean(trace_S2_P['mu_P']), np.mean(trace_S2_P['sigma_P'])]
print('Parameters values = ', param_S2_P)
pdf_S2_P, B_x_axis_pdf_S2_P, y_axis_pdf_S2_P = make_pdf(data=S2_data_list[m], dist=best_dist, params=param_S2_P, size=len(trace_S2_P))
print('#######################################################################################################')
plt.plot(B_x_axis_pdf_S2_P, y_axis_pdf_S2_P, label='%i. step (Prediction[S2])' % (m + 1))
# Correct:
print('#######################################################################################################')
print('Designing the Bayesian PDF for correction Sensor 1:')
trace_S1_C, post_pred_S1_C = S1_C_Bayesian_Interface(nu=np.mean(trace_S2_P['nu_P']), mu=np.mean(trace_S2_P['mu_P']), sigma=np.mean(trace_S2_P['sigma_P']), data=S1_data_list[m])
best_dist = getattr(st, 't')
param_S1_C = [np.mean(trace_S1_C['nu_C']), np.mean(trace_S1_C['mu_C']), np.mean(trace_S1_C['sigma_C'])]
print('Parameters values = ', param_S1_C)
pdf_S1_C, B_x_axis_pdf_S1_C, y_axis_pdf_S1_C = make_pdf(data=S1_data_list[m], dist=best_dist, params=param_S1_C, size=len(trace_S1_C))
print('#######################################################################################################')
plt.plot(B_x_axis_pdf_S1_C, y_axis_pdf_S1_C, label='%i. step (Correction[S1])' % (m + 1))
corr_pdf_S1 = y_axis_pdf_S1_C
corr_pdf_S1_list.append(corr_pdf_S1)
print('Corrected PDF: ', corr_pdf_S1)
print('Length of Corrected PDF list = ', len(corr_pdf_S1_list))
x_corr_pdf_list.append(B_x_axis_pdf_S1_C)
但是，我不确定我是否正确执行了这些步骤。我是否正确地接近了这个想法？如果没有，我该如何修改代码？另外，如果可能的话，分享一个教程解释了将两个分布相乘并获得组合概率分布的比例和位置的想法？
编辑1：
我正在和某人谈论这个概念，他提到进行贝叶斯更新。如何在我的代码中实现这种方式并使用 PYMC3 包？