gauss_infer_1d.ipynb

Kernel: Python [conda env:py3713]

Inference about a random variable given a noisy observation

import jax.numpy as jnp
from jax import random
import matplotlib.pyplot as plt
from scipy.stats import multivariate_normal as mvn
import seaborn as sns
import os

try:
    from probml_utils import savefig, latexify
except ModuleNotFoundError:
    %pip install -qq git+https://github.com/probml/probml-utils.git
    from probml_utils import savefig, latexify

os.environ["LATEXIFY"] = ""
os.environ["FIG_DIR"] = "figures"

latexify(width_scale_factor=2, fig_height=2)

def plot_data(quantiles, pdf_dict, variance_prior, interact_flag, save_name=""):
    """
    Plot data

    Args:
    ----------
    quantiles : JAX array
        Quantiles for normal distribution

    pdf_dict : dictionary
        Data and plotting options for
        all PDF's

    variance_prior : int/float
        Variance of Prior

    save_name : string, default=''
        Filename for the saved graph

    Returns:
    ----------
    None
    """

    # Setup graph and parameters
    fig = plt.figure()

    # Update limits for interactive plot
    if interact_flag:
        plt.ylim(0, 1)
        plt.xlim(-10, 10)
    else:
        plt.ylim(0, 0.6)
        plt.xlim(-7, 7)

    # Plot graph
    for key, value in pdf_dict.items():
        plt.plot(
            quantiles,
            value["pdf"],
            color=value["color"],
            label=key,
            linestyle=value["linestyle"],
            linewidth=1.5,
        )

    # Update labels,legends and title
    plt.title(f"Prior with variance of {variance_prior}")
    plt.xlabel("$x$")
    plt.ylabel("$p(x)$")
    plt.legend(loc="upper left")
    sns.despine()

    # Save figure to files
    if len(save_name) > 0:
        savefig(save_name)

    plt.show()

def generate_PDF(prior_var=[1, 5], prior_mean=[0, 0], observed_var=[1, 1], interact_flag=False):
    """
    Plot PDF for prior, likelihood and posterior of a gaussian
    distribution to gain insight about a random variable given
    a noisy observation under the effect of a strong prior
    versus a weak prior

    Args:
    ----------
    prior_var : list,default=[1, 5]
        List of Variance of priors

    prior_mean : list,default=[0, 0]
        List of Mean of Priors

    observed_var : list, default=[1,1]
        List of Variance of observed
        values

    interact_flag : bool, default=False
        Interactive plot indicator

    Returns:
    ----------
    None
    """

    # The noisy observation
    obsrvd_val = 3

    # Quantiles for data
    quantiles = jnp.arange(-10, 10, 0.10)

    for mean_prior, variance_prior, variance_observed in zip(prior_mean, prior_var, observed_var):

        # Prior variance,mean and PDF
        prior_var = variance_prior
        prior_mu = mean_prior
        prior_pdf = mvn.pdf(quantiles, mean=prior_mu, cov=prior_var)

        # Likelihood variance,mean and PDF
        likelihood_var = variance_observed
        likelihood_mu = jnp.mean(obsrvd_val)
        likelihood_pdf = mvn.pdf(quantiles, mean=likelihood_mu, cov=likelihood_var)

        # Number of noisy observations
        obsrvd_val_count = jnp.size(obsrvd_val)

        # Posterior variance,mean and PDF
        posterior_var = (likelihood_var * prior_var) / ((obsrvd_val_count * prior_var) + likelihood_var)
        posterior_mu = posterior_var * ((prior_mu / prior_var) + ((obsrvd_val_count * likelihood_mu) / likelihood_var))
        posterior_pdf = mvn.pdf(quantiles, mean=posterior_mu, cov=posterior_var)

        # Setup data and graphing options
        pdf_plot_dict = {
            "prior": {"pdf": prior_pdf, "color": "blue", "linestyle": "-"},
            "likelihood": {"pdf": likelihood_pdf, "color": "red", "linestyle": ":"},
            "posterior": {"pdf": posterior_pdf, "color": "black", "linestyle": "-."},
        }

        # Plot
        plot_data(
            quantiles,
            pdf_plot_dict,
            variance_prior,
            interact_flag,
            f"gauss_infer_1d_prior{variance_prior}",
        )

generate_PDF()

WARNING:absl:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)

saving image to figures/gauss_infer_1d_prior1_latexified.pdf
Figure size: [3. 2.]
saving image to figures/gauss_infer_1d_prior5_latexified.pdf
Figure size: [3. 2.]

from ipywidgets import Layout, interact
import ipywidgets as widgets


@interact(
    prior_var=widgets.FloatSlider(
        description="prior variance",
        min=1.0,
        max=5.0,
        value=1.0,
        step=0.1,
        style=dict(description_width="initial"),
        continuous_update=False,
    ),
    prior_mean=widgets.FloatSlider(
        description="prior mean",
        min=-3.0,
        max=3.0,
        value=0.0,
        step=0.1,
        style=dict(description_width="initial"),
        continuous_update=False,
    ),
    observed_var=widgets.FloatSlider(
        description="Variance of Observed values",
        min=1.0,
        max=5.0,
        value=1.0,
        step=0.1,
        style=dict(description_width="initial"),
        layout=Layout(width="40%"),
        continuous_update=False,
    ),
)
def update(prior_var, prior_mean, observed_var):
    generate_PDF(
        prior_var=[prior_var],
        prior_mean=[prior_mean],
        observed_var=[observed_var],
        interact_flag=True,
    )

interactive(children=(FloatSlider(value=1.0, continuous_update=False, description='prior variance', max=5.0, m…