Question

在多级线性回归的Pymc3示例中（示例为here，使用Gelman等人（2007）的DoubleAnimationUsingKeyFrames数据集），拦截（针对不同的县）和斜坡（有和没有地下室的公寓）每个都有一个普通的先前。如何将它们与多元正态先验模型一起建模，以便我可以检查它们之间的相关性？

示例中给出的层次模型如下：

radon

我正试图对先辈们进行一些改变

with pm.Model() as hierarchical_model:
    # Hyperpriors for group nodes
    mu_a = pm.Normal('mu_a', mu=0., sd=100**2)
    sigma_a = pm.HalfCauchy('sigma_a', 5)
    mu_b = pm.Normal('mu_b', mu=0., sd=100**2)
    sigma_b = pm.HalfCauchy('sigma_b', 5)

    # Intercept for each county, distributed around group mean mu_a
    # Above we just set mu and sd to a fixed value while here we
    # plug in a common group distribution for all a and b (which are
    # vectors of length n_counties).
    a = pm.Normal('a', mu=mu_a, sd=sigma_a, shape=n_counties)
    # Intercept for each county, distributed around group mean mu_a
    b = pm.Normal('b', mu=mu_b, sd=sigma_b, shape=n_counties)

    # Model error
    eps = pm.HalfCauchy('eps', 5)

    radon_est = a[county_idx] + b[county_idx] * data.floor.values

    # Data likelihood
    radon_like = pm.Normal('radon_like', mu=radon_est, sd=eps, observed=data.log_radon)
    hierarchical_trace = pm.sample(2000)

以下是我收到的错误消息：

    with pm.Model() as correlation_model:
        # Hyperpriors for group nodes
        mu_a = pm.Normal('mu_a', mu=0., sd=100**2)
        mu_b = pm.Normal('mu_b', mu=0., sd=100**2)

        # here I want to model a and b together
        # I borrowed some code from a multivariate normal model
        # but the code does not work
        sigma = pm.HalfCauchy('sigma', 5, shape=2)

        C_triu = pm.LKJCorr('C_triu', n=2, p=2)
        C = T.fill_diagonal(C_triu[np.zeros((2,2), 'int')], 1)
        cov = pm.Deterministic('cov', T.nlinalg.matrix_dot(sigma, C, sigma))
        tau = pm.Deterministic('tau', T.nlinalg.matrix_inverse(cov))

        a, b = pm.MvNormal('mu', mu=(mu_a, mu_b), tau=tau,
                           shape=(n_counties, n_counties))

        # Model error
        eps = pm.HalfCauchy('eps', 5)
        radon_est = a[county_idx] + b[county_idx] * data.floor.values

        # Data likelihood
        radon_like = pm.Normal('radon_like', mu=radon_est, sd=eps, observed=data.log_radon)
        correlation_trace = pm.sample(2000)

显然，我对协方差矩阵犯了一些错误，但我是File "<ipython-input-108-ce400c54cc39>", line 14, in <module> tau = pm.Deterministic('tau', T.nlinalg.matrix_inverse(cov)) File "/home/olivier/anaconda3/lib/python3.5/site-packages/theano/gof/op.py", line 611, in __call__ node = self.make_node(*inputs, **kwargs) File "/home/olivier/anaconda3/lib/python3.5/site-packages/theano/tensor/nlinalg.py", line 73, in make_node assert x.ndim == 2 AssertionError的新手，而且是pymc3的新手，所以不知道如何修复它。我认为这应该是一个相当常见的用例，所以可能有一些例子吗？我找不到它们。

可以在示例页面上看到完整的可复制代码和数据（上面给出的链接）。我没有把它包括在这里，因为它太长了，而且我认为那些熟悉theano的人很可能已经非常熟悉它了。）

Answer 1

在创建错误指定MvNormal形状的协方差矩阵时，您忘记添加一行。你的模型应该是这样的：

with pm.Model() as correlation_model:
    mu = pm.Normal('mu', mu=0., sd=10, shape=2)
    sigma = pm.HalfCauchy('sigma', 5, shape=2)

    C_triu = pm.LKJCorr('C_triu', n=2, p=2)
    C = tt.fill_diagonal(C_triu[np.zeros((2,2), 'int')], 1.)
    sigma_diag = tt.nlinalg.diag(sigma) # this line
    cov = tt.nlinalg.matrix_dot(sigma_diag, C, sigma_diag)
    tau = tt.nlinalg.matrix_inverse(cov)

    ab = pm.MvNormal('ab', mu=mu, tau=tau, shape=(n_counties, 2))

    eps = pm.HalfCauchy('eps', 5)
    radon_est = ab[:,0][county_idx] + ab[:,1][county_idx] * data.floor.values

    radon_like = pm.Normal('radon_like', mu=radon_est, sd=eps, observed=data.log_radon)
    trace = pm.sample(2000)

注意，或者，您可以从hierarchical_model的后验评估截距与斜率的相关性。您可以使用频率论方法或构建另一个贝叶斯模型，该模型将hierarchical_model的结果作为观察数据。可能这可能会更快。

修改

如果你想从后验评估两个变量的相关性，你可以做类似的事情。

chain = hierarchical_trace[100:]
x_0 = chain['mu_a']
x_1 = chain['mu_b']
X = np.vstack((x_0, x_1)).T

然后您可以运行以下模型：

with pm.Model() as correlation:
    mu = pm.Normal('mu', mu=0., sd=10, shape=2)
    sigma = pm.HalfCauchy('sigma', 5, shape=2)

    C_triu = pm.LKJCorr('C_triu', n=2, p=2)
    C = tt.fill_diagonal(C_triu[np.zeros((2,2), 'int')], 1.)
    sigma_diag = tt.nlinalg.diag(sigma)
    cov = tt.nlinalg.matrix_dot(sigma_diag, C, sigma_diag)
    tau = tt.nlinalg.matrix_inverse(cov)

    yl = pm.MvNormal('yl', mu=mu, tau=tau, shape=(2, 2), observed=X)
    trace = pm.sample(5000, pm.Metropolis())

您可以根据需要更换x_0和x_1。例如，您可能想要这样做：

x_0 = np.random.normal(chain['mu_a'], chain['sigma_a'])
x_1 = np.random.normal(chain['mu_b'], chain['sigma_b'])

pymc3：如何在多级线性回归中建模相关截距和斜率

1 个答案: