PyMC and Liesel: Spike and Slab

PyMC and Liesel: Spike and Slab#

Liesel provides an interface for PyMC, a popular Python library for Bayesian Models. In this tutorial, we see how to specify a model in PyMC and then fit it using Liesel.

Be sure that you have pymc installed. If that’s not the case, you can install Liesel with the optional dependency PyMC.

pip install liesel[pymc]

We will build a Spike and Slab model, a Bayesian approach that allows for variable selection by assuming a mixture of two distributions for the prior distribution of the regression coefficients: a point mass at zero (the “spike”) and a continuous distribution centered around zero (the “slab”). The model assumes that each coefficient \(\beta_j\) has a corresponding indicator variable \(\delta_j\) that takes a value of either 0 or 1, indicating whether the variable is included in the model or not. The prior distribution of the indicator variables is a Bernoulli distribution, with a parameter \(\theta\) that controls the sparsity of the model. When the parameter is close to 1, the model is more likely to include all variables, while when it is close to 0, the model is more likely to select only a few variables. In our case, we assign a Beta hyperprior to \(\theta\):

\[\begin{split}\begin{aligned} \mathbf{y} &\sim \mathcal{N} \left( \mathbf{X}\boldsymbol{\beta}, \sigma^2 \mathbf{I} \right)\\ \boldsymbol{\beta}_j &\sim \mathcal{N}\left(0, (1 - \delta_j)\nu + \delta_j\tau^2_j / \sigma^2 \right)\\ \tau^2_j &\sim \mathcal{IG}(\text{a}_{\tau}, \text{b}_{\tau})\\ \delta_j &\sim\text{Bernoulli}(\theta)\\ \theta &\sim\text{Beta}(\text{a}_\theta, \text{b}_\theta)\\ \sigma^2 &\sim \mathcal{IG}(\text{a}_{\sigma^2}, \text{b}_{\sigma^2}) \end{aligned}.\end{split}\]

where \(\nu\) is a hyperparameter that we set to a fixed small value. That way, when \(\delta_j = 0\), the prior variance for \(\beta_j\) is extremely small, practically forcing it to be close to zero.

First, we generate the data. We use a model with four coefficients but assume that only two variables are relevant, namely the first and the third one.

RANDOM_SEED = 123
rng = np.random.RandomState(RANDOM_SEED)

n = 1000
p = 4

sigma_scalar = 1.0
beta_vec = np.array([3.0, 0.0, 4.0, 0.0])

X = rng.randn(n, p).astype(np.float32)

errors = rng.normal(size=n).astype(np.float32)

y = X @ beta_vec + sigma_scalar * errors

Then, we can specify the model using PyMC.

spike_and_slab_model = pm.Model()

mu = 0.0

alpha_tau = 1.0
beta_tau = 1.0

alpha_sigma = 1.0
beta_sigma = 1.0

alpha_theta = 8.0
beta_theta = 8.0

nu = 0.1

with spike_and_slab_model:
    # priors
    sigma2 = pm.InverseGamma("sigma2", alpha=alpha_sigma, beta=beta_sigma)

    theta = pm.Beta("theta", alpha=alpha_theta, beta=beta_theta)
    delta = pm.Bernoulli("delta", p=theta, size=p)
    tau = pm.InverseGamma("tau", alpha=alpha_tau, beta=beta_tau)

    beta = pm.Normal(
        "beta",
        mu=0.0,
        sigma=nu * (1 - delta) + delta * pm.math.sqrt(tau / sigma2),
        shape=p,
    )

    # make a data node
    Xx = pm.Data("X", X)

    # likelihood
    pm.Normal("y", mu=Xx @ beta, sigma=pm.math.sqrt(sigma2), observed=y)

Let’s take a look at our model:

spike_and_slab_model
\[\begin{split} \begin{array}{rcl} \text{X} &= &\operatorname{Data}(\text{<shared>})\\\text{sigma2} &\sim & \operatorname{InverseGamma}(1,~1)\\\text{theta} &\sim & \operatorname{Beta}(8,~8)\\\text{delta} &\sim & \operatorname{Bernoulli}(\text{theta})\\\text{tau} &\sim & \operatorname{InverseGamma}(1,~1)\\\text{beta} &\sim & \operatorname{Normal}(0,~f(\text{delta},~\text{sigma2},~\text{tau}))\\\text{y} &\sim & \operatorname{Normal}(f(\text{X},~\text{beta}),~f(\text{sigma2})) \end{array} \end{split}\]

The class PyMCInterface offers an interface between PyMC and Goose. By default, the constructor of PyMCInterface keeps track only of a representation of random variables that can be used in sampling. For example, theta is transformed to the real-numbers space with a log-odds transformation, and therefore the model only keeps track of theta_log_odds__. However, we would like to access the untransformed samples as well. We can do this by including them in the additional_vars argument of the constructor of the interface.

The initial position can be extracted with get_initial_state(). The model state is represented as a Position.

interface = PyMCInterface(
    spike_and_slab_model, additional_vars=["sigma2", "tau", "theta"]
)
state = interface.get_initial_state()

Since \(\delta_j\) is a discrete variable, we need to use a Gibbs sampler to draw samples for it. Unfortunately, we cannot derive the posterior analytically, but what we can do is use a Metropolis-Hastings step as a transition function:

def delta_transition_fn(prng_key, model_state):
    draw_key, mh_key = jax.random.split(prng_key)
    theta_logodds = model_state["theta_logodds__"]
    p = jax.numpy.exp(theta_logodds) / (1 + jax.numpy.exp(theta_logodds))
    draw = jax.random.bernoulli(draw_key, p=p, shape=(4,))
    proposal = {"delta": jax.numpy.asarray(draw, dtype=np.int64)}
    _, state = gs.mh.mh_step(
        prng_key=mh_key, model=interface, proposal=proposal, model_state=model_state
    )
    return state

Finally, we can sample from the posterior as we do for any other Liesel model. In this case, we use a GibbsKernel for \(\boldsymbol{\delta}\) and a NUTSKernel both for the remaining parameters.

builder = gs.EngineBuilder(seed=13, num_chains=4)
builder.set_model(interface)
builder.set_initial_values(state)
builder.set_duration(warmup_duration=1000, posterior_duration=2000)

builder.add_kernel(
    gs.NUTSKernel(
        position_keys=["beta", "sigma2_log__", "tau_log__", "theta_logodds__"]
    )
)
builder.add_kernel(gs.GibbsKernel(["delta"], transition_fn=delta_transition_fn))

builder.positions_included = ["sigma2", "tau"]

engine = builder.build()

engine.sample_all_epochs()

liesel.goose.builder - WARNING - No jitter functions provided. The initial values won't be jittered
liesel.goose.engine - INFO - Initializing kernels...
/home/runner/work/liesel/liesel/.venv/lib/python3.13/site-packages/jax/_src/numpy/array_methods.py:125: UserWarning: Explicitly requested dtype float64 requested in astype is not available, and will be truncated to dtype float32. To enable more dtypes, set the jax_enable_x64 configuration option or the JAX_ENABLE_X64 shell environment variable. See https://github.com/jax-ml/jax#current-gotchas for more.
  return lax_numpy.astype(self, dtype, copy=copy, device=device)
liesel.goose.engine - INFO - Done
liesel.goose.engine - INFO - Starting epoch: FAST_ADAPTATION, 75 transitions, 25 jitted together

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  proposal = {"delta": jax.numpy.asarray(draw, dtype=np.int64)}

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liesel.goose.engine - WARNING - Errors per chain for kernel_00: 3, 2, 2, 4 / 75 transitions
liesel.goose.engine - INFO - Finished epoch
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liesel.goose.engine - INFO - Finished epoch
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liesel.goose.engine - WARNING - Errors per chain for kernel_00: 2, 1, 2, 1 / 100 transitions
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liesel.goose.engine - INFO - Finished epoch
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liesel.goose.engine - WARNING - Errors per chain for kernel_00: 1, 1, 1, 1 / 500 transitions
liesel.goose.engine - INFO - Finished epoch
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liesel.goose.engine - WARNING - Errors per chain for kernel_00: 1, 1, 1, 1 / 50 transitions
liesel.goose.engine - INFO - Finished epoch
liesel.goose.engine - INFO - Finished warmup
liesel.goose.engine - INFO - Starting epoch: POSTERIOR, 2000 transitions, 25 jitted together

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liesel.goose.engine - INFO - Finished epoch

Now, we can take a look at the summary of the results and at the trace plots.

results = engine.get_results()
print(gs.Summary(results))

/home/runner/work/liesel/liesel/.venv/lib/python3.13/site-packages/arviz_stats/base/diagnostics.py:313: RuntimeWarning: invalid value encountered in scalar divide
  varsd = varvar / evar / 4
/home/runner/work/liesel/liesel/.venv/lib/python3.13/site-packages/arviz_stats/base/diagnostics.py:313: RuntimeWarning: invalid value encountered in scalar divide
  varsd = varvar / evar / 4
/home/runner/work/liesel/liesel/.venv/lib/python3.13/site-packages/arviz_stats/base/diagnostics.py:90: RuntimeWarning: invalid value encountered in scalar divide
  (between_chain_variance / within_chain_variance + num_samples - 1) / (num_samples)

                         var_fqn     kernel var_index  sample_size      mean  \
variable
beta                     beta[0]  kernel_00      (0,)         8000  3.037727
beta                     beta[1]  kernel_00      (1,)         8000 -0.010908
beta                     beta[2]  kernel_00      (2,)         8000  3.955964
beta                     beta[3]  kernel_00      (3,)         8000 -0.001761
delta                   delta[0]  kernel_01      (0,)         8000  1.000000
delta                   delta[1]  kernel_01      (1,)         8000  0.085125
delta                   delta[2]  kernel_01      (2,)         8000  1.000000
delta                   delta[3]  kernel_01      (3,)         8000  0.063125
sigma2                    sigma2          -        ()         8000  1.014129
sigma2_log__        sigma2_log__  kernel_00        ()         8000  0.013033
tau                          tau          -        ()         8000  0.508712
tau_log__              tau_log__  kernel_00        ()         8000  2.156108
theta_logodds__  theta_logodds__  kernel_00        ()         8000  0.036925

                      var        sd      ess_bulk     ess_tail  mcse_mean  \
variable
beta             0.001047  0.032364  12350.724123  6256.921075   0.000292
beta             0.000906  0.030099  13113.375119  6451.783328   0.000263
beta             0.000982  0.031343  14087.219211  5872.803421   0.000265
beta             0.000956  0.030924  13099.915481  5619.069861   0.000270
delta            0.000000  0.000000   8000.000000  8000.000000   0.000000
delta            0.077879  0.279068    373.017695   373.017695   0.014450
delta            0.000000  0.000000   8000.000000  8000.000000   0.000000
delta            0.059140  0.243188    511.668790   511.668790   0.010752
sigma2           0.002056  0.045342  12679.989600  6471.143078   0.000404
sigma2_log__     0.001993  0.044640  12680.000414  6471.143078   0.000397
tau              0.012407  0.111386   6499.235557  4334.338046   0.001376
tau_log__        0.627498  0.792148   7418.998540  4600.136996   0.009974
theta_logodds__  0.219882  0.468916   6499.234703  4334.338046   0.005823

                  mcse_sd      rhat    q_0.05     q_0.5    q_0.95   hdi_low  \
variable
beta             0.000207  1.002090  2.984296  3.037531  3.090531  2.985247
beta             0.000183  1.001970 -0.060500 -0.011142  0.038715 -0.060222
beta             0.000192  1.001343  3.904705  3.956123  4.007814  3.901984
beta             0.000192  1.001467 -0.052818 -0.001802  0.049597 -0.050066
delta                 NaN       NaN  1.000000  1.000000  1.000000  1.000000
delta            0.021481  1.013259  0.000000  0.000000  1.000000  0.000000
delta                 NaN       NaN  1.000000  1.000000  1.000000  1.000000
delta            0.019314  1.007246  0.000000  0.000000  1.000000  0.000000
sigma2           0.000291  0.999936  0.941998  1.012915  1.090738  0.942568
sigma2_log__     0.000281  0.999939 -0.059752  0.012833  0.086855 -0.056601
tau              0.000891  1.000694  0.325165  0.508691  0.692288  0.324628
tau_log__        0.009165  1.000442  1.041645  2.055873  3.599275  0.932972
theta_logodds__  0.004166  1.000686 -0.730136  0.034769  0.810836 -0.732583

                 hdi_high
variable
beta             3.091338
beta             0.038947
beta             4.004807
beta             0.051560
delta            1.000000
delta            0.000000
delta            1.000000
delta            0.000000
sigma2           1.090913
sigma2_log__     0.089524
tau              0.691743
tau_log__        3.418921
theta_logodds__  0.808280

As we can see from the posterior means of the \(\boldsymbol{\delta}\) parameters, the model was able to recognize those variable with no influence on the respose \(\mathbf{y}\):

  1. \(\delta_1\) and \(\delta_3\) (delta[0] and delta[2] in the table) have a posterior mean of \(1\), indicating inclusion.

  2. \(\delta_2\) and \(\delta_4\) (delta[1] and delta[3] in the table) have a posterior mean of \(0.06\), indicating exclusion.

gs.plot_trace(results)

../../_images/results-plot-output-1.png