问题描述
我正在学习使用 PyMC3 的统计反思课程。在第 4 章的末尾,他们要求原始 (!Kung) 数据集中没有的数据点的单个值的 HDI。在 PyMC3 中可以这样做吗?
在 scikit-learn 中,您有 fit()
和 predict()
,您可以预测全新输入的输出。
使用 PyMC3,您可以 sample()
获取您的跟踪,并且您可以要求进行后验预测检查,但我无法将我感兴趣的值的任何参数传递给它。我做到了设法使用共享 theano 变量以迂回的方式完成,也可以手动完成。
编辑:我在最后添加了一个 pm.Data()
和 pm.set_data()
示例。我认为这可能就是答案,但我正在等待其他人确认,然后才将其标记为已回答。
这就是我所做的。
weight_s
是标准化的重量数据。标准化是通过这个函数完成的:
def standardize(array,reference=None):
if reference is None:
reference = array
return (array - reference.mean()) / reference.std()
这是 PyMC3 模型:
import pymc3 as pm
with pm.Model() as m_adult:
a = pm.normal("α",mu=155,sd=20)
b = pm.Lognormal("β",mu=0,sd=1)
mu = pm.Deterministic("μ",a + b * adults.weight_s)
sigma = pm.Uniform("σ",50)
height = pm.normal("height",mu=mu,sd=sigma,observed=adults.height)
trace_adult = pm.sample()
这是数据的样子(注意 HDI 为 89%):
height_pred = pm.fast_sample_posterior_predictive(trace_adult,model=m_adult)["height"]
fig,ax = plt.subplots()
ax.plot(adults.weight,adults.height,".")
ax.plot(adults.weight,trace_adult.μ.mean(axis=0),color="black")
az.plot_hdi(adults.weight,trace_adult.μ,ax=ax,height_pred,ax=ax)
ax.set(xlabel="weight",ylabel="height")
fig.tight_layout()
首先,我将向您展示手动版本:
missing_weights = np.array([45,40,65,31,53])
expected_height = np.array([
(trace_adult.α + trace_adult.β * standardize(weight,adults.weight)).mean()
for weight in missing_weights
])
hdis = np.array([
az.hdi(np.random.normal(
trace_adult.α + trace_adult.β * standardize(weight,adults.weight),trace_adult.σ,)) for weight in missing_weights
])
data = np.vstack((missing_weights,expected_height,hdis.T)).T
missing_df = pd.DataFrame(,columns=["weight","expected_height","hdi_lower","hdi_upper"])
print(missing_df)
这给了我们:
weight expected_height hdi_lower hdi_upper
0 45 154.603176 146.981285 163.149938
1 40 150.105295 142.095583 158.474277
2 65 172.594698 164.401102 180.786641
3 31 142.009110 134.163952 150.233028
4 53 161.799785 153.881956 170.209779
如果你看图表,这些数字是有意义的。
from theano import shared
shared_weights_s = shared(adults.weight_s.values)
with pm.Model() as m_adult:
a = pm.normal("α",a + b * shared_weights_s)
sigma = pm.Uniform("σ",observed=adults.height)
trace_adult = pm.sample()
现在,对于 shared_weights 的新值,我们有三个选择:
- 一件一件地做事情
- 替换为未知权重
- 附加到末尾
对于一对一的情况:
missing_weights = np.array([45,53])
rows = []
for weight in missing_weights:
row = [weight]
shared_weights_s.set_value(standardize(np.array([weight]),adults.weight))
height_pred_single = pm.fast_sample_posterior_predictive(trace_adult,model=m_adult)["height"]
row.append(height_pred_single.mean())
row.extend(list(az.hdi(height_pred_single).mean(axis=0)))
rows.append(row)
missing_df = pd.DataFrame(rows,"hdi_upper"])
print(missing_df)
为他们所有人做这件事给了我们:
weight expected_height hdi_lower hdi_upper
0 45 154.604520 146.485327 162.713345
1 40 150.113378 142.001151 158.263953
2 65 172.580212 164.357970 180.843184
3 31 142.010954 133.786200 150.142080
4 53 161.792962 153.651266 169.926615
您可以一次完成所有这些:
missing_weights = np.array([45,53])
shared_weights_s.set_value(standardize(missing_weights,adults.weight))
height_pred_replace = pm.fast_sample_posterior_predictive(trace_adult,model=m_adult)["height"]
missing_df = pd.DataFrame(missing_weights,columns=["weight"])
missing_df["expected_height"] = height_pred_replace.mean(axis=0)
missing_df[["hdi_lower","hdi_upper"]] = az.hdi(height_pred_replace)
print(missing_df)
这给了我们:
weight expected_height hdi_lower hdi_upper
0 45 154.578096 147.066342 163.069805
1 40 150.042506 141.561599 158.120596
2 65 172.568430 164.079591 180.536870
3 31 142.080048 134.173959 150.345556
4 53 161.830472 153.327694 169.717058
最后,我们可以将其添加到之前的共享权重变量的末尾并取尾部:
missing_weights = np.array([45,53])
shared_weights_s.set_value(np.append(adults.weight_s.values,standardize(missing_weights,adults.weight)))
height_pred_append = pm.fast_sample_posterior_predictive(trace_adult,columns=["weight"])
missing_df["expected_height"] = height_pred_append.mean(axis=0)[-len(missing_weights):]
missing_df[["hdi_lower","hdi_upper"]] = az.hdi(height_pred_append)[-len(missing_weights):]
print(missing_df)
这给了我们:
weight expected_height hdi_lower hdi_upper
0 45 154.640287 146.093825 162.477313
1 40 150.088713 142.168331 158.314038
2 65 172.633776 164.086280 180.483805
3 31 142.019331 133.516545 150.491937
4 53 161.880175 153.530868 169.771088
如您所见,所有这些方法最终都会给出相同的结果。有没有官方/最好的方法来做到这一点?可以不设置全局共享变量并修改它吗? PyMC3 有没有这样的功能,或者是未来可能会添加的东西? (如果足够简单,我可能可以为此提出拉取请求;我还是 PyMC3 的新手。)
编辑:我想我找到了答案:使用 pm.Data()
。
with pm.Model() as m_adult:
weight_s = pm.Data("weight_s",adults.weight_s.values)
a = pm.normal("α",a + b * weight_s)
sigma = pm.Uniform("σ",observed=adults.height)
trace_adult = pm.sample()
然后,在尝试时,我们pm.set_data()
:
missing_weights = np.array([45,53])
with m_adult:
pm.set_data({"weight_s": standardize(missing_weights,adults.weight)})
height_pred_data = pm.fast_sample_posterior_predictive(trace_adult)["height"]
missing_df = pd.DataFrame(missing_weights,columns=["weight"])
missing_df["expected_height"] = height_pred_data.mean(axis=0)
missing_df[["hdi_lower","hdi_upper"]] = az.hdi(height_pred_data)
print(missing_df)
给出:
weight expected_height hdi_lower hdi_upper
0 45 154.584063 145.828088 162.512174
1 40 150.184853 142.272258 158.451555
2 65 172.662069 164.522903 180.803430
3 31 141.949137 133.310865 149.811098
4 53 161.719867 153.848599 169.638495
解决方法
我想我找到了答案:使用 pm.Data()
。
with pm.Model() as m_adult:
weight_s = pm.Data("weight_s",adults.weight_s.values)
a = pm.Normal("α",mu=155,sd=20)
b = pm.Lognormal("β",mu=0,sd=1)
mu = pm.Deterministic("μ",a + b * weight_s)
sigma = pm.Uniform("σ",50)
height = pm.Normal("height",mu=mu,sd=sigma,observed=adults.height)
trace_adult = pm.sample()
然后,在尝试时,我们pm.set_data()
:
missing_weights = np.array([45,40,65,31,53])
with m_adult:
pm.set_data({"weight_s": standardize(missing_weights,adults.weight)})
height_pred_data = pm.fast_sample_posterior_predictive(trace_adult)["height"]
missing_df = pd.DataFrame(missing_weights,columns=["weight"])
missing_df["expected_height"] = height_pred_data.mean(axis=0)
missing_df[["hdi_lower","hdi_upper"]] = az.hdi(height_pred_data)
print(missing_df)
给出:
weight expected_height hdi_lower hdi_upper
0 45 154.584063 145.828088 162.512174
1 40 150.184853 142.272258 158.451555
2 65 172.662069 164.522903 180.803430
3 31 141.949137 133.310865 149.811098
4 53 161.719867 153.848599 169.638495