Logistic regression

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Run in Google Colab

Tutorial: Bayesian Logistic Regression

The purpose of this tutorial is to show how to build a simple Bayesian model to deduce the line which separates two categories of points.

Problem

A logistic model is a statistical model where we have a collection of things that can be divided into two categories — pictures of cats or dogs, patients who are immune or susceptible, students who pass or fail, and so on. An assumption of the model is: the probability that a given thing is in a category can be computed by taking a linear combination of characteristics of the thing.

The problem we seek to solve in this tutorial is: when given a set of examples of each category, can we infer the boundary between the categories?

Prerequisites

We will be using the following packages within this tutorial.

• arviz and bokeh for interactive visualizations.

Let's code this in Bean Machine! Import the Bean Machine library and some fundamental PyTorch classes.

# Install Bean Machine in Colab if using Colab.
import sys


if "google.colab" in sys.modules and "beanmachine" not in sys.modules:
    !pip install beanmachine

import os
import warnings

import arviz as az
import beanmachine.ppl as bm
import torch
import torch.distributions as dist
from beanmachine.ppl.inference.monte_carlo_samples import MonteCarloSamples
from beanmachine.tutorials.utils import plots
from bokeh.io import output_notebook
from bokeh.models import ColumnDataSource, MultiLine, Span
from bokeh.plotting import show
from IPython.display import Markdown

The next cell includes convenient configuration settings to improve the notebook presentation as well as setting a manual seed for reproducibility.

# Eliminate excess UserWarnings from Python.
warnings.filterwarnings("ignore")

# Plotting settings
az.rcParams["plot.backend"] = "bokeh"
az.rcParams["stats.hdi_prob"] = 0.89

# Manual seed
bm.seed(0)

# Other settings for the notebook.
smoke_test = "SANDCASTLE_NEXUS" in os.environ or "CI" in os.environ

Example: orange and blue points

For the purposes of this tutorial our "things" will be two-dimensional points between $(-10,-10)$ and $(10,10)$. Our two categories will be represented by $0.0$, which we'll render as blue, and $1.0$, which we'll render as orange.

The probability that a given point $(x,y)$ will be orange is

$P((x,y)\text{ is orange})=\Large{\frac{1}{1+e^{-b_0-b_1x-b_2y}}}$

for some coefficients $b_0,b_1,b_2$.

That equation is seen to be a linear combination of properties of the point if we express the probability as log odds:

$\text{logit}(P((x,y)\text{ is orange}))=b_0+b_1x+b_2y$

The boundary between the two categories is the set of points where the probability is $0.5$ that the point is orange; the log odds of $0.5$ is $0$, so the equation of the set of $(x, )$ points separating the categories is a straight line:

$b_0+b_1x+b_2y=0$

Or, expressed as slope and intercept:

$y=-\frac{b_1}{b_2}x-\frac{b_0}{b_2}$

Our goal therefore is to infer possible values for $b_0,b_1,b_2$ from which we can compute the line separating the categories.

Creating sample data

We start by creating a data set to illustrate the model. In order to make computations faster and easier, we will express the entire data set as a single tensor representing 200 points. Each row of the tensor will be in the form $\begin{bmatrix}1.0&x&y\end{bmatrix}$. That way we can compute the linear combination by matrix-multiplying each row by $\begin{bmatrix}b_0\\b_1\\b_2\end{bmatrix}$ to obtain the log odds that the point is orange: $b_0+b_1x+b_2y$.

For our synthetic dataset, we will assume the following parameters.

N = 200
low = -10.0
high = 10.0
uniform = dist.Uniform(
    low=torch.tensor([1.0, low, low]),
    high=torch.tensor([1.0, high, high]),
)
points = torch.tensor([uniform.sample().tolist() for i in range(N)]).view(N, 3)

# Required for visualizing in Colab.
output_notebook(hide_banner=True)

x = points[:, 1]
y = points[:, 2]
cds = ColumnDataSource({"x": x.tolist(), "y": y.tolist()})
tips = [("y", "@y{0.000}"), ("x", "@x{0.000}")]
synthetic_data_plot = plots.scatter_plot(
    plot_sources=cds,
    tooltips=tips,
    figure_kwargs={
        "title": "Synthetic data",
        "x_axis_label": "x",
        "y_axis_label": "y",
    },
    plot_kwargs={"fill_color": "black"},
)
show(synthetic_data_plot)

loading...

We now assign points to categories $0.0$ (blue) and $1.0$ (orange).

For this example we will assign categories to points using $\begin{bmatrix}-2.0\\0.3\\-0.5\end{bmatrix}$ as our coefficients, which makes the line separating the categories $y=0.6x-4$.

true_coefficients = torch.tensor([-2.0, 0.3, -0.5]).view(3, 1)
true_slope = -float(true_coefficients[1] / true_coefficients[2])
true_intercept = -float(true_coefficients[0] / true_coefficients[2])


def log_odds(point):
    return point.view(1, 3).mm(true_coefficients)


observed_categories = torch.tensor(
    [dist.Bernoulli(logits=log_odds(point)).sample() for point in points]
)

Data visualization methods

It is useful to have helper methods to visualize the synthetic data set and the line which separates the two categories.

def categorize_points(points, categories):
    orange_x = []
    orange_y = []
    blue_x = []
    blue_y = []
    for point, category in zip(points, categories):
        if category == 1:
            orange_x.append(float(point[1]))
            orange_y.append(float(point[2]))
        else:
            blue_x.append(float(point[1]))
            blue_y.append(float(point[2]))
    return {
        "orange": {"x": orange_x, "y": orange_y, "label": ["orange"] * len(orange_x)},
        "blue": {"x": blue_x, "y": blue_y, "label": ["blue"] * len(blue_x)},
    }


def plot_line(slope, intercept, high=10, low=-10):
    if intercept > high or intercept < low:
        return
    xs = [low, high]
    ys = [slope * low + intercept, slope * high + intercept]
    if ys[0] > high:
        xs[0] = (high - intercept) / slope
        ys[0] = high
    elif ys[0] < low:
        xs[0] = (low - intercept) / slope
        ys[0] = low
    if ys[1] > high:
        xs[1] = (high - intercept) / slope
        ys[1] = high
    elif ys[1] < low:
        xs[1] = (low - intercept) / slope
        ys[1] = low
    return xs, ys

# Required for visualizing in Colab.
output_notebook(hide_banner=True)

points_with_categories = categorize_points(points, observed_categories)
orange_cds = ColumnDataSource(
    {
        "x": points_with_categories["orange"]["x"],
        "y": points_with_categories["orange"]["y"],
        "label": points_with_categories["orange"]["label"],
    }
)
orange_tips = [("Category", "@label"), ("y", "@y{0.000}"), ("x", "@x{0.000}")]
blue_cds = ColumnDataSource(
    {
        "x": points_with_categories["blue"]["x"],
        "y": points_with_categories["blue"]["y"],
        "label": points_with_categories["blue"]["label"],
    }
)
blue_tips = [("Category", "@label"), ("y", "@y{0.000}"), ("x", "@x{0.000}")]
synthetic_data_with_categories_plot = plots.scatter_plot(
    plot_sources=[orange_cds, blue_cds],
    tooltips=[orange_tips, blue_tips],
    figure_kwargs={
        "title": "Synthetic data with categories",
        "x_axis_label": "x",
        "y_axis_label": "y",
    },
    legend_items=["Category orange", "Category blue"],
    plot_kwargs={"fill_color": "label"},
)
# Add the separating line.
x, y = plot_line(true_slope, true_intercept)
synthetic_data_with_categories_plot.line(
    x=x,
    y=y,
    legend_label="Separating line",
    line_color="black",
    line_width=3,
    line_alpha=1,
)
show(synthetic_data_with_categories_plot)

loading...

Model

We can now start building our model.

The first thing we need is a prior distribution for our three coefficients.

We have no reason to believe that the coefficients will be either positive or negative, so we should choose a prior distribution that is centered on zero. We use a matrix multiplication to compute the linear combination, and therefore make the prior a column vector of samples from a normal distribution:

@bm.random_variable
def coefficients():
    mean = torch.zeros(3, 1)
    sigma = torch.ones(3, 1)
    return dist.Normal(mean, sigma)

Our model for categories is now straightforward: each category is chosen by matrix-multiplying the point by the prior distribution of coefficients, and we get a set of categories from the Bernoulli distribution; either $0.0$ (blue) or $1.0$ (orange):

@bm.random_variable
def categories():
    return dist.Bernoulli(logits=points.mm(coefficients()))

Inference

We can now infer the posterior distribution of the coefficients given the observations:

num_samples = 2 if smoke_test else 2000
num_adaptive_samples = 0 if smoke_test else num_samples // 2
num_chains = 1 if smoke_test else 4

observations = {categories(): observed_categories.view(N, 1)}
queries = [coefficients()]
mc = bm.GlobalNoUTurnSampler()

%%time
samples = mc.infer(
    queries=queries,
    observations=observations,
    num_samples=num_samples,
    num_chains=num_chains,
    num_adaptive_samples=num_adaptive_samples,
)

Out:

Samples collected:   0%|          | 0/3000 [00:00<?, ?it/s]
Samples collected:   0%|          | 0/3000 [00:00<?, ?it/s]
Samples collected:   0%|          | 0/3000 [00:00<?, ?it/s]
Samples collected:   0%|          | 0/3000 [00:00<?, ?it/s]

Out:

CPU times: user 33.2 s, sys: 107 ms, total: 33.3 s
Wall time: 33.2 s

sampled_coefficients = samples.get_chain()[coefficients()]

The slopes and intercepts are computed from the sampled coefficients:

slopes = [-float(s[1] / s[2]) for s in sampled_coefficients if float(s[2]) != 0.0]
intercepts = [-float(s[0] / s[2]) for s in sampled_coefficients if float(s[2]) != 0.0]

Posterior results

A histogram of the inferred slopes should cluster near the true value, marked in red. A slight deviation is to be expected since there is noise in our data.

# Required for visualizing in Colab.
output_notebook(hide_banner=True)

slope_hist_plot = plots.histogram_plot(slopes)
# Add a line showing the true slope.
span = Span(
    location=true_slope,
    dimension="height",
    line_color="red",
    line_width=3,
)
slope_hist_plot.add_layout(span)
show(slope_hist_plot)

loading...

And similarly for the intercepts. Our model actually predicts a higher median intercept to capture the orange points above the slope in our ground truth.

# Required for visualizing in Colab.
output_notebook(hide_banner=True)

intercept_hist_plot = plots.histogram_plot(intercepts)
# Add a line showing the true slope.
span = Span(
    location=true_intercept,
    dimension="height",
    line_color="red",
    line_width=3,
)
intercept_hist_plot.add_layout(span)
show(intercept_hist_plot)

loading...

Inference typically finds a reasonable distribution of possible lines that separate these two categories.

Diagnostics

In addition to visualizing the lines directly, we can also use the diagnostics capabilities of Arviz to examine the diagnostics of our variables:

filtered_samples = {k: v for k, v in samples.items() if k == coefficients()}
az_data = MonteCarloSamples(filtered_samples).to_inference_data()
summary_df = az.summary(az_data, round_to=3).to_markdown()
Markdown(summary_df)

	mean	sd	hdi_5.5%	hdi_94.5%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
coefficients()[0,0]	-1.774	0.294	-2.245	-1.313	0.005	0.004	3561.2	5042.12	1.001
coefficients()[1,0]	0.262	0.048	0.185	0.336	0.001	0.001	3543.11	4425.9	1
coefficients()[2,0]	-0.452	0.064	-0.548	-0.342	0.001	0.001	3137.52	4031.13	1.001

As you can see, the average and median (50%) values inferred are reasonably close to the true values but there is a large spread. Remember that any constant multiple of the true values would produce the same line; if we got $-4.0$, $0.6$ and $-1.2$ then the slope and intercept would be the same and only the amount of "mixing" near the line would change, so we can expect that this inference problem may have some fairly large variance.

The n_eff column is the effective sample size, which indicates how correlated the posterior samples are to each other; higher is better. This effective sample size is a little low.

Notice that we generated the data by separating points along a line with slope $0.6$ and intercept $-4.0$, but were we to observe the specific set of 200 points we classified without knowing the true parameters ahead of time, we would deduce that the separating line had a slightly lower slope than $0.6$ and an intercept around $-3.0$.

To more clearly illustrate the accuracy of the inference, we can take a random selection of the inferred lines and plot them on the data set:

# Required for visualizing in Colab.
output_notebook(hide_banner=True)

# Replicate the original data separating plot above.
randomly_selected_lines_plot = plots.scatter_plot(
    plot_sources=[orange_cds, blue_cds],
    tooltips=[orange_tips, blue_tips],
    figure_kwargs={
        "title": "Synthetic data with categories",
        "x_axis_label": "x",
        "y_axis_label": "y",
    },
    legend_items=["Category orange", "Category blue"],
    plot_kwargs={"fill_color": "label"},
)

# Add randomly selected sampled separating lines.
num_lines = 25
sampled_indices = torch.randint(0, len(slopes), (num_lines,)).tolist()
xs = []
ys = []
for sampled_index in sampled_indices:
    sampled_slope = slopes[sampled_index]
    sampled_intercept = intercepts[sampled_index]
    x, y = plot_line(sampled_slope, sampled_intercept)
    xs.append(x)
    ys.append(y)
cds = ColumnDataSource({"xs": xs, "ys": ys})
glyph = MultiLine(xs="xs", ys="ys", line_color="magenta", line_alpha=0.2)
randomly_selected_lines_plot.add_glyph(cds, glyph)

# Add the separating line.
x, y = plot_line(true_slope, true_intercept)
randomly_selected_lines_plot.line(
    x=x,
    y=y,
    legend_label="Separating line",
    line_color="black",
    line_width=3,
    line_alpha=1,
)

show(randomly_selected_lines_plot)

loading...