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Add marginal latent gaussian GP example
Co-authored-by: Adrien Corenflos <adrien.corenflos@aalto.fi>
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--- | ||
jupyter: | ||
jupytext: | ||
text_representation: | ||
extension: .md | ||
format_name: markdown | ||
format_version: '1.3' | ||
jupytext_version: 1.14.1 | ||
kernelspec: | ||
display_name: Python 3.9.15 ('blackjax-env') | ||
language: python | ||
name: python3 | ||
--- | ||
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# Bayesian Regression With Latent Gaussian Sampler | ||
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In this example, we want to illustrate how to use the marginal sampler implementation [`mgrad_gaussian`](https://blackjax-devs.github.io/blackjax/mcmc.html#blackjax.mgrad_gaussian) of the article [Auxiliary gradient-based sampling algorithms](https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssb.12269). We do so by using the simulated data from the example [Gaussian Regression with the Elliptical Slice Sampler](https://blackjax-devs.github.io/blackjax/examples/GP_EllipticalSliceSampler.html). Please also refer to the complementary example [Bayesian Logistic Regression With Latent Gaussian Sampler](https://blackjax-devs.github.io/blackjax/examples/LogisticRegressionWithLatentGaussianSampler.html). | ||
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## Sampler Overview | ||
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In section we give a brief overview of the idea behind this particular sampler. For more details please refer to the original paper [Auxiliary gradient-based sampling algorithms](https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssb.12269) ([here](https://arxiv.org/abs/1610.09641) you can access the arXiv preprint). | ||
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### Motivation: Auxiliary Metropolis-Hastings samplers | ||
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Let us recall how to sample from a target density $\pi(\mathbf{x})$ using a Metropolis-Hasting sampler trough a *marginal scheme process*. The main idea is to have a mechanism that generate proposals $y$ which we then accept or reject according to a specific criterion. Concretely, suppose that we have an *auxiliary* scheme given by | ||
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1. Sample $\mathbf{u}|\mathbf{x} \sim \pi(\mathbf{u}|\mathbf{x}) = q(\mathbf{u}|\mathbf{x})$. | ||
2. Generate proposal $\mathbf{y}|\mathbf{u}, \mathbf{x} \sim q(\mathbf{y}|\mathbf{x}, \mathbf{u})$ | ||
3. Compute the Metropolis-Hasting ratio | ||
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$$ | ||
\tilde{\varrho} = \frac{\pi(\mathbf{y}|\mathbf{u})q(\mathbf{x}|\mathbf{y}, \mathbf{u})}{\pi(\mathbf{x}|\mathbf{u})q(\mathbf{y}|\mathbf{x}, \mathbf{u})} | ||
$$ | ||
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4. Accept proposal $y$ with probability $\min(1, \tilde{\varrho})$ and reject it otherwise. | ||
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This scheme targets the auxiliary distribution $\pi(\mathbf{x}, \mathbf{u}) = \pi(\mathbf{x}) q(\mathbf{u}|\mathbf{x})$ in two steps. | ||
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Now, suppose we can instead compute the *marginal* proposal distribution $q(\mathbf{y}|\mathbf{x}) = \int q(\mathbf{y}|\mathbf{x}, \mathbf{u}) q(\mathbf{u}|\mathbf{x}) \mathrm{d}u$ in closed form, then an alternative scheme is given by: | ||
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1. We draw a proposal $y \sim q(\mathbf{y}\mid\mathbf{x})$. | ||
2. Then we compute the Metropolis-Hasting ratio | ||
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$$ | ||
\varrho = \frac{\pi(\mathbf{y})q(\mathbf{x}|\mathbf{y})}{\pi(\mathbf{x})q(\mathbf{y}|\mathbf{x})} | ||
$$ | ||
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3. Accept proposal $y$ with probability $\min(1, \varrho)$ and reject it otherwise. | ||
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### Example: Auxiliary Metropolis-Adjusted Langevin Algorithm (MALA) | ||
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Let's consider the case of an auxiliary random walk proposal $q(\mathbf{u}|\mathbf{x}) = N(\mathbf{u}|\mathbf{x}, (\delta /2) \mathbf{I})$ for $\delta > 0$ as in [[Section 2.2] Auxiliary gradient-based sampling algorithms](https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssb.12269), it is shown that one can use a first order approximation to sample from the (intractable) $\pi(\mathbf{x}|\mathbf{u})$ density by choosing | ||
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$$ | ||
q(\mathbf{y}|\mathbf{u}, \mathbf{x}) \propto N(\mathbf{y}|\mathbf{u} + (\delta/2)\nabla \log \pi(\mathbf{x}), (\delta/2) I). | ||
$$ | ||
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The resulting marginal sampler can be shown to correspond to the Metropolis-adjusted Langevin algorithm (MALA) with | ||
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$$ | ||
q(\mathbf{y}| \mathbf{x}) = N(\mathbf{y}|\mathbf{x} + (\delta/2)\nabla \log \pi(\mathbf{x}), \delta I). | ||
$$ | ||
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### Latent Gaussian Models | ||
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A particular case of interest is the latent Gaussian model where the target density has the form | ||
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$$ | ||
\pi(\mathbf{x}) \propto \overbrace{\exp\{f(\mathbf{x})\}}^{\text{likelihood}} \underbrace{N(\mathbf{x}|\mathbf{0}, \mathbf{C})}_{\text{Gaussian Prior}} | ||
$$ | ||
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In this case, instead of linearising the full log density $\log \pi(\mathbf{x})$, we can linearise $f$ only, which, when combined with a random walk proposal $N(\mathbf{u}|\mathbf{x}, (\delta /2) \mathbf{I})$, recovers to the following auxiliary proposal | ||
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$$ | ||
q(\mathbf{y}|\mathbf{x}, \mathbf{u}) \propto N\left(\mathbf{y}|\frac{2}{\delta} \mathbf{A}\left(\mathbf{u} + \frac{\delta}{2}\nabla f(\mathbf{x})\right), \mathbf{A}\right), | ||
$$ | ||
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where $\mathbf{A} = \delta / 2(\mathbf{C} + (\delta / 2)\mathbf{I})^{-1}\mathbf{C}$. The corresponding marginal density is | ||
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$$ | ||
q(\mathbf{y}|\mathbf{x}) \propto N\left(\mathbf{y}|\frac{2}{\delta} \mathbf{A}\left(\mathbf{x} + \frac{\delta}{2}\nabla f(\mathbf{x})\right), \frac{2}{\delta}\mathbf{A}^2 + \mathbf{A}\right). | ||
$$ | ||
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Sampling from $\pi(\mathbf{x}, \mathbf{u})$ (and therefore from $\pi(\mathbf{x})$) is done via Hastings-within-Gibbs as above. | ||
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A crucial point of this algorithm is the fact that $\mathbf{A}$ can be precomputed and afterward modified cheaply when $\delta$ varies. This makes it easy to calibrate the step-size $\delta$ at low cost. | ||
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--- | ||
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Now that we have a high-level understanding of the algorithm, let's see how to use it in `blackjax`. | ||
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```python | ||
import jax | ||
import jax.numpy as jnp | ||
import jax.random as jrnd | ||
import matplotlib.pyplot as plt | ||
import numpy as np | ||
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from blackjax import mgrad_gaussian | ||
``` | ||
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We generate data through a squared exponential kernel as in the example [Gaussian Regression with the Elliptical Slice Sampler](https://blackjax-devs.github.io/blackjax/examples/GP_EllipticalSliceSampler.html). | ||
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```python | ||
def squared_exponential(x, y, length, scale): | ||
dot_diff = jnp.dot(x, x) + jnp.dot(y, y) - 2 * jnp.dot(x, y) | ||
return scale**2 * jnp.exp(-0.5 * dot_diff / length**2) | ||
``` | ||
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```python | ||
n, d = 2000, 2 | ||
length, scale = 1.0, 1.0 | ||
y_sd = 1.0 | ||
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# fake data | ||
rng = jrnd.PRNGKey(10) | ||
kX, kf, ky = jrnd.split(rng, 3) | ||
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X = jrnd.uniform(kX, shape=(n, d)) | ||
Sigma = jax.vmap( | ||
lambda x: jax.vmap(lambda y: squared_exponential(x, y, length, scale))(X) | ||
)(X) + 1e-3 * jnp.eye(n) | ||
invSigma = jnp.linalg.inv(Sigma) | ||
f = jrnd.multivariate_normal(kf, jnp.zeros(n), Sigma) | ||
y = f + jrnd.normal(ky, shape=(n,)) * y_sd | ||
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# conjugate results | ||
posterior_cov = jnp.linalg.inv(invSigma + 1 / y_sd**2 * jnp.eye(n)) | ||
posterior_mean = jnp.dot(posterior_cov, y) * 1 / y_sd**2 | ||
``` | ||
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Let's visualize the distribution of the vector `y`. | ||
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```python | ||
plt.figure(figsize=(8, 5)) | ||
plt.hist(np.array(y), bins=50, density=True) | ||
plt.xlabel("y") | ||
plt.title("Histogram of data.") | ||
plt.show() | ||
``` | ||
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## Sampling | ||
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Now we proceed to run the sampler. First, we set the sampler parameters: | ||
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```python | ||
# sampling parameters | ||
n_warm = 2000 | ||
n_iter = 500 | ||
``` | ||
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Next, we define the the log-probability function. For this we need to set the log-likelihood function. | ||
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```python | ||
loglikelihood_fn = lambda f: -0.5 * jnp.dot(y - f, y - f) / y_sd**2 | ||
logprob_fn = lambda f: loglikelihood_fn(f) - 0.5 * jnp.dot(f @ invSigma, f) | ||
``` | ||
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Now we are ready to initialize the sampler. The output is type is a `NamedTuple` with the following fields: | ||
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``` | ||
init: | ||
A pure function which when called with the initial position and the | ||
target density probability function will return the kernel's initial | ||
state. | ||
step: | ||
A pure function that takes a rng key, a state and possibly some | ||
parameters and returns a new state and some information about the | ||
transition. | ||
``` | ||
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```python | ||
init, step = mgrad_gaussian(logprob_fn=logprob_fn, mean=jnp.zeros(n), covariance=Sigma) | ||
``` | ||
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We continue by setting the inference loop. | ||
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```python | ||
def inference_loop(rng, init_state, kernel, n_iter): | ||
keys = jrnd.split(rng, n_iter) | ||
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def step(state, key): | ||
state, info = kernel(key, state) | ||
return state, (state, info) | ||
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_, (states, info) = jax.lax.scan(step, init_state, keys) | ||
return states, info | ||
``` | ||
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We are now ready to run the sampler! The only extra parameters in the `step` function is `delta`, which (as seen in the sampler description) corresponds (in a loose sense) to the step-size of MALA algorithm. | ||
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**Remark:** Note that one can calibrate the `delta` parameter as described in the example [Bayesian Logistic Regression With Latent Gaussian Sampler](https://blackjax-devs.github.io/blackjax/examples/LogisticRegressionWithLatentGaussianSampler.html). | ||
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```python | ||
%%time | ||
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kernel = lambda key, x: step(rng_key=key, state=x, delta=0.5) | ||
initial_state = init(f) | ||
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states, info = inference_loop(jrnd.PRNGKey(0), init(f), kernel, n_warm + n_iter) | ||
samples = states.position[n_warm:] | ||
``` | ||
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## Diagnostics | ||
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Finally we evaluate the results. | ||
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```python | ||
error_mean = jnp.mean((samples.mean(axis=0) - posterior_mean) ** 2) | ||
error_cov = jnp.mean((jnp.cov(samples, rowvar=False) - posterior_cov) ** 2) | ||
print( | ||
f"Mean squared error for the mean vector {error_mean} and covariance matrix {error_cov}" | ||
) | ||
``` | ||
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```python | ||
keys = jrnd.split(rng, 500) | ||
predictive = jax.vmap(lambda k, f: f + jrnd.normal(k, (n,)) * y_sd)( | ||
keys, samples[-1000:] | ||
) | ||
``` | ||
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```python | ||
plt.figure(figsize=(8, 5)) | ||
plt.hist(np.array(y), bins=50, density=True) | ||
plt.hist(np.array(predictive.reshape(-1)), bins=50, density=True, alpha=0.8) | ||
plt.xlabel("y") | ||
plt.title("Predictive distribution") | ||
plt.show() | ||
``` |