pymc3：如何在多级线性回归中建模相关截距和斜率

Question

在多级线性回归的 Pymc3 示例中（示例是 here，使用来自 Gelman 等人 (2007) 的 radon 数据集），截距（对于不同的县) 和斜坡（对于带地下室和不带地下室的公寓）都有一个正常先验。我怎样才能将它们与多变量正态先验一起建模，以便我可以检查它们之间的相关性？

例子中给出的层级模型是这样的：

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)

我正在尝试对先验进行一些更改

    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)

这是我收到的错误消息：

  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

很明显，我在协方差矩阵方面犯了一些错误，但我是 pymc3 的新手，也是 theano 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，所以也许有一些例子吗？我只是找不到它们。

完整的可复制代码和数据可以在示例页面上看到（link 以上）。我没有在这里包括它，因为它太长了，而且我认为那些熟悉 pymc3 的人很可能已经非常熟悉它了:)

Answer 1

您在创建协方差矩阵时忘记添加一行，您未指定 MvNormal 的形状。您的模型应如下所示：

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)

请注意，您也可以从 hierarchical_model 的后部评估截距和斜率的相关性。您可以使用频率论方法或构建另一个贝叶斯模型，将 hierarchical_model 的结果作为观察数据。可能这样会更快。

编辑

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

然后你可以运行以下模型：

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())

您可以根据需要替换x_0和x_1。例如你可能想做：

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

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