# PyMC and Liesel: Spike and Slab Liesel provides an interface for [PyMC](https://www.pymc.io/welcome.html), 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. ``` bash 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 $\theta$: $$ \begin{aligned} \mathbf{y} &\sim \mathcal{N} \left( \mathbf{X}\boldsymbol{\beta}, \sigma^2 \mathbf{I} \right)\\ \boldsymbol{\beta}_j &\sim \mathcal{N}\left(0, (1 - \delta_j)\nu + \delta_j\tau^2_j / \sigma^2 \right)\\ \tau^2_j &\sim \mathcal{IG}(\text{a}_{\tau}, \text{b}_{\tau})\\ \delta_j &\sim\text{Bernoulli}(\theta)\\ \theta &\sim\text{Beta}(\text{a}_\theta, \text{b}_\theta)\\ \sigma^2 &\sim \mathcal{IG}(\text{a}_{\sigma^2}, \text{b}_{\sigma^2}) \end{aligned}. $$ 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. ``` python 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. ``` python 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: ``` python spike_and_slab_model ``` $$ \begin{array}{rcl} \text{sigma2} &\sim & \operatorname{InverseGamma}(1,~1)\\\text{theta} &\sim & \operatorname{Beta}(8,~8)\\\text{delta} &\sim & \operatorname{Bernoulli}(\text{theta})\\\text{tau} &\sim & \operatorname{InverseGamma}(1,~1)\\\text{beta} &\sim & \operatorname{Normal}(0,~f(\text{delta},~\text{sigma2},~\text{tau}))\\\text{y} &\sim & \operatorname{Normal}(f(\text{beta}),~f(\text{sigma2})) \end{array} $$ The class {class}`.PyMCInterface` offers an interface between PyMC and Goose. By default, the constructor of {class}`.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 {meth}`.get_initial_state`. The model state is represented as a `Position`. ``` python 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: ``` python 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 {class}`~.goose.GibbsKernel` for $\boldsymbol{\delta}$ and a {class}`~.goose.NUTSKernel` both for the remaining parameters. ``` python 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... /Users/johannesbrachem/Documents/git/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 0%| | 0/3 [00:00