Show code cell source
MAKE_BOOK_FIGURES=True
import numpy as np
import scipy.stats as st
import matplotlib as mpl
import matplotlib.pyplot as plt
%matplotlib inline
import matplotlib_inline
matplotlib_inline.backend_inline.set_matplotlib_formats('svg')
import seaborn as sns
sns.set_context("paper")
sns.set_style("ticks")
def set_book_style():
plt.style.use('seaborn-v0_8-white')
sns.set_style("ticks")
sns.set_palette("deep")
mpl.rcParams.update({
# Font settings
'font.family': 'serif', # For academic publishing
'font.size': 8, # As requested, 10pt font
'axes.labelsize': 8,
'axes.titlesize': 8,
'xtick.labelsize': 7, # Slightly smaller for better readability
'ytick.labelsize': 7,
'legend.fontsize': 7,
# Line and marker settings for consistency
'axes.linewidth': 0.5,
'grid.linewidth': 0.5,
'lines.linewidth': 1.0,
'lines.markersize': 4,
# Layout to prevent clipped labels
'figure.constrained_layout.use': True,
# Default DPI (will override when saving)
'figure.dpi': 600,
'savefig.dpi': 600,
# Despine - remove top and right spines
'axes.spines.top': False,
'axes.spines.right': False,
# Remove legend frame
'legend.frameon': False,
# Additional trim settings
'figure.autolayout': True, # Alternative to constrained_layout
'savefig.bbox': 'tight', # Trim when saving
'savefig.pad_inches': 0.1 # Small padding to ensure nothing gets cut off
})
def set_notebook_style():
plt.style.use('seaborn-v0_8-white')
sns.set_style("ticks")
sns.set_palette("deep")
mpl.rcParams.update({
# Font settings - using default sizes
'font.family': 'serif',
'axes.labelsize': 10,
'axes.titlesize': 10,
'xtick.labelsize': 9,
'ytick.labelsize': 9,
'legend.fontsize': 9,
# Line and marker settings
'axes.linewidth': 0.5,
'grid.linewidth': 0.5,
'lines.linewidth': 1.0,
'lines.markersize': 4,
# Layout settings
'figure.constrained_layout.use': True,
# Remove only top and right spines
'axes.spines.top': False,
'axes.spines.right': False,
# Remove legend frame
'legend.frameon': False,
# Additional settings
'figure.autolayout': True,
'savefig.bbox': 'tight',
'savefig.pad_inches': 0.1
})
def save_for_book(fig, filename, is_vector=True, **kwargs):
"""
Save a figure with book-optimized settings.
Parameters:
-----------
fig : matplotlib figure
The figure to save
filename : str
Filename without extension
is_vector : bool
If True, saves as vector at 1000 dpi. If False, saves as raster at 600 dpi.
**kwargs : dict
Additional kwargs to pass to savefig
"""
# Set appropriate DPI and format based on figure type
if is_vector:
dpi = 1000
ext = '.pdf'
else:
dpi = 600
ext = '.tif'
# Save the figure with book settings
fig.savefig(f"{filename}{ext}", dpi=dpi, **kwargs)
def make_full_width_fig():
return plt.subplots(figsize=(4.7, 2.9), constrained_layout=True)
def make_half_width_fig():
return plt.subplots(figsize=(2.35, 1.45), constrained_layout=True)
if MAKE_BOOK_FIGURES:
set_book_style()
else:
set_notebook_style()
make_full_width_fig = make_full_width_fig if MAKE_BOOK_FIGURES else lambda: plt.subplots()
make_half_width_fig = make_half_width_fig if MAKE_BOOK_FIGURES else lambda: plt.subplots()
Visualizing Monte Carlo Uncertainty#
In the last two lectures, we repeatedly used the law of large numbers to estimate expectations using samples. In particular, we studied this integral:
where \(X\sim p(x)\) and \(g(x)\) is a function of \(x\). The sampling-based approximation required \(X_1,X_2,\dots\) be independent copies of \(X\). Then, we considered the random variables \(Y_1 = g(X_1), Y_2 = g(X_2), \dots\), which are also independent and identically distributed. The law of large states that their sampling average converges to their mean:
This is the Monte Carlo way of estimating integrals. If you played with the hands-on activities, you noticed that for small \( N \), we could get very different answers. Here we will build some intuition about this epistemic uncertainty induced by finite samples.
Example: 1D expectation#
Let’s try it out with the same test function we used before (Example 3.4 of [Robert and Casella, 2004]). Assume that \(X\sim\mathcal{U}([0,1])\) and pick:
The correct value for the expectation is:
Let’s calculate the Monte Carlo estimate a few times and visualize its uncertainty:
Show code cell source
# The function of x we would like to consider
g = lambda x: (np.cos(50 * x) + np.sin(20 * x)) ** 2
# How many times do you want to run MC
num_mc = 2
# Number of samples to take
N = 100
# A common plot for all estimates
fig, ax = plt.subplots()
# So do it ``num_mc`` times:
for i in range(num_mc):
# Generate samples from X
x_samples = np.random.rand(N)
# Get the corresponding Y's
y_samples = g(x_samples)
# Evaluate the sample average for all sample sizes
I_running = np.cumsum(y_samples) / np.arange(1, N + 1)
ax.plot(np.arange(1, N+1), I_running, 'b', lw=0.5)
# The true value
ax.plot(np.arange(1, N+1), [0.965] * N, color='r')
# and the labels
ax.set_xlabel('$N$')
ax.set_ylabel(r'$\bar{I}_N$')
sns.despine(trim=True);
Questions#
Run the code 2-3 times to observe that you get a slightly different answer every time.
Set the number of Monte Carlo samples
num_mcto 100 (or higher). Observe how different MC runs envelop the correct answer. This is epistemic uncertainty. How can we get it without running this repeatedly?Now increase
Nto 10000 and see how the uncertainty disappears.