Bayes Librarian

import torch
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline
# Retina display
%config InlineBackend.figure_format = 'retina'

from tueplots import bundles
plt.rcParams.update(bundles.beamer_moml())


# Also add despine to the bundle using rcParams
plt.rcParams['axes.spines.right'] = False
plt.rcParams['axes.spines.top'] = False

# Increase font size to match Beamer template
plt.rcParams['font.size'] = 16
# Make background transparent
plt.rcParams['figure.facecolor'] = 'none'

p_l = 1.0/21.0
p_f = 1- p_l
prior = torch.distributions.Categorical(probs=torch.tensor([p_l, p_f]))
prior

Categorical(probs: torch.Size([2]))

pd.Series(index=["Librarian", "Farmer"], data=prior.probs).plot(kind='bar', rot=0)
# Write the values on top of the bar
for i, v in enumerate(prior.probs):
    plt.text(i - 0.1, v + 0.01, f"{v.item():0.3f}")
plt.title("Prior")

Text(0.5, 1.0, 'Prior')

# Get 210 samples from prior given a seed

torch.manual_seed(3)
samples = prior.sample(torch.Size([210, ]))
print(samples)
print(samples.sum())

tensor([0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
        1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
        1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1,
        1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
        1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
        1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
        1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
        1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0,
        1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1])
tensor(199)

p_d_l = torch.tensor(0.4)
p_d_f = torch.tensor(0.1)

p_l_d = (p_d_l * p_l) / (p_d_l * p_l + p_d_f * p_f)
p_l_d

tensor(0.1667)

p_d_l *p_l

tensor(0.0190)

p_d_f*p_f

tensor(0.0952)

## Electricity consumption

prior_theta_0 = torch.distributions.Uniform(80, 100)
prior_theta_1 = torch.distributions.Uniform(1, 2)

# Plottting PDFs

eps = 1e-3
theta_0_lin = torch.linspace(80+eps, 100.0-eps, 100)
plt.plot(theta_0_lin, torch.exp(prior_theta_0.log_prob(theta_0_lin)))
plt.xlabel(r'$\theta_0$')
plt.ylabel(r'$p(\theta_0)$')
plt.title(r'Prior PDF for $\theta_0$')
# Fill between 
plt.fill_between(theta_0_lin, torch.exp(prior_theta_0.log_prob(theta_0_lin)), color='C0', alpha=0.2)

<matplotlib.collections.PolyCollection at 0x7f67945e25b0>

theta_1_lin = torch.linspace(1+eps, 2.0-eps, 100)
plt.plot(theta_1_lin, torch.exp(prior_theta_1.log_prob(theta_1_lin)))
plt.xlabel(r'$\theta_1$')
plt.ylabel(r'$p(\theta_1)$')
plt.title(r'Prior PDF for $\theta_1$')
# Fill between 
plt.fill_between(theta_1_lin, torch.exp(prior_theta_1.log_prob(theta_1_lin)), color='C0', alpha=0.2)

<matplotlib.collections.PolyCollection at 0x7f67923c04c0>

# plotting samples from prior

n_samples = 1000
theta_0s = prior_theta_0.sample(torch.Size([n_samples,]))
theta_1s = prior_theta_1.sample(torch.Size([n_samples,]))

# Plot the samples
fig, ax = plt.subplots(figsize=(8, 2), ncols=2)
ax[0].plot(theta_0_lin, torch.exp(prior_theta_0.log_prob(theta_0_lin)), label='prior')

# plot theta_0 samples
ax[0].scatter(theta_0s, torch.zeros_like(theta_0s),  label='samples',c='r', marker='x')

ax[1].plot(theta_1_lin, torch.exp(prior_theta_1.log_prob(theta_1_lin)), label='prior')
ax[1].scatter(theta_1s, torch.zeros_like(theta_1s), label='samples',c='r', marker='x')

ax[1].set_title(r'$\theta_1$')
ax[0].set_title(r'$\theta_0$')

Text(0.5, 1.0, '$\\theta_0$')

# Generating draws from prior

xs = torch.linspace(1, 1000, 100)
def forward(x, theta_0, theta_1):
    return theta_0 + theta_1 * x

preds = []

for theta_0, theta_1 in zip(theta_0s, theta_1s):
    print(theta_0.item(), theta_1.item())
    preds.append(forward(xs, theta_0, theta_1))
    plt.plot(xs, preds[-1], color='C1', alpha=0.5)

plt.xlabel('Number of Occupants')
plt.ylabel('Electricity Consumption')

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Text(0, 0.5, 'Electricity Consumption')

torch.distributions.Uniform(0, 1).support

Interval(lower_bound=0.0, upper_bound=1.0)

torch.distributions.LogNormal(0, 1).support

GreaterThan(lower_bound=0.0)

forward(xs, theta_0, theta_1)

tensor([83.7934])

torch.stack(preds).max(axis=0).values

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