Introduction to Goodness-of-Fit for State Space Models¶

This notebook introduces the fundamental concepts of goodness-of-fit diagnostics for state space models in neuroscience. We'll build intuition through visual examples before diving into numerical metrics.

Learning objectives:

• Understand what goodness-of-fit means for latent variable models
• Visualize the relationship between posterior and likelihood distributions
• Learn to use kl_divergence() and hpd_overlap() functions
• Interpret diagnostic metrics

⏱ Estimated time: 20-25 minutes

Previous: README.md | Next: 02_highest_density_regions.ipynb

Prerequisites¶

Before starting this notebook, you should have:

Background knowledge:

• Basic probability theory (probability distributions, expectation)
• Familiarity with Bayesian inference concepts (prior, likelihood, posterior)
• Basic Python and numpy

Software requirements:

• Python 3.10+
• statespacecheck package installed: pip install -e /path/to/statespacecheck
• Dependencies: numpy, scipy, matplotlib

Note: If you're new to Bayesian state space models, see the README for background reading.

Setup¶

First, let's import the packages we'll need and verify the environment is configured correctly.

In [1]:

import sys

import matplotlib.pyplot as plt
import numpy as np

# Import statespacecheck - if this fails, see prerequisites above
try:
    from statespacecheck import hpd_overlap, kl_divergence

    print("✓ statespacecheck imported successfully")
except ImportError as e:
    raise ImportError(
        "statespacecheck not found. Please install it first:\n"
        "  cd /path/to/statespacecheck\n"
        "  pip install -e .\n"
        "Then restart the kernel."
    ) from e

from utils import configure_notebook_plotting, generate_1d_gaussian_distribution

# Configure plotting
configure_notebook_plotting()

# Verify environment
print(f"✓ Python {sys.version.split()[0]}")
print(f"✓ NumPy {np.__version__}")
print(f"✓ Matplotlib {plt.matplotlib.__version__}")
print("✓ Environment configured correctly!")

import sys import matplotlib.pyplot as plt import numpy as np # Import statespacecheck - if this fails, see prerequisites above try: from statespacecheck import hpd_overlap, kl_divergence print("✓ statespacecheck imported successfully") except ImportError as e: raise ImportError( "statespacecheck not found. Please install it first:\n" " cd /path/to/statespacecheck\n" " pip install -e .\n" "Then restart the kernel." ) from e from utils import configure_notebook_plotting, generate_1d_gaussian_distribution # Configure plotting configure_notebook_plotting() # Verify environment print(f"✓ Python {sys.version.split()[0]}") print(f"✓ NumPy {np.__version__}") print(f"✓ Matplotlib {plt.matplotlib.__version__}") print("✓ Environment configured correctly!")

✓ statespacecheck imported successfully
✓ Python 3.13.5
✓ NumPy 2.3.4
✓ Matplotlib 3.10.7
✓ Environment configured correctly!

The Problem: Evaluating Latent Variable Models¶

State space models relate neural activity to latent brain states we cannot directly observe. This creates a challenge: how do we know if our model is good if we can't see ground truth?

Traditional approaches like cross-validation measure prediction accuracy but don't tell us:

• When does the model fail? (which time periods?)
• Why does it fail? (state model vs observation model?)
• Where in state space does it fail? (which regions?)

The solution: Compare distributions that should agree if the model is well-specified.

Key Concept: State Distribution vs Likelihood¶

In Bayesian state space models, the posterior distribution combines two sources of information:

1. State distribution (prior/predictive): What the model expects based on dynamics

   • One-step-ahead prediction: $p(x_t | y_{1:t-1})$
   • Smoothed distribution: $p(x_t | y_{1:T})$

2. Likelihood (normalized): What the current data says

   • Normalized likelihood: $p(y_t | x_t) / \sum_x p(y_t | x_t)$
   • Equivalent to posterior with uniform prior: $p(x_t | y_t)$

Key insight: If the model is well-specified, these distributions should be consistent with each other. Large disagreements indicate model problems.

Example 1: Perfect Model Fit¶

Let's start with an ideal case where the model perfectly captures the data. The state distribution and likelihood should agree.

In [2]:

# Create position bins (e.g., 100 cm track, 50 bins)
position_bins = np.linspace(0, 100, 50)

# Generate distributions centered at the same location with similar uncertainty
true_position = 50.0
state_dist = generate_1d_gaussian_distribution(position_bins, mean=true_position, std=2.0)
likelihood = generate_1d_gaussian_distribution(position_bins, mean=true_position, std=2.5)

# Visualize
fig, ax = plt.subplots(figsize=(10, 5))
ax.plot(position_bins, state_dist, label="State distribution", linewidth=2.5, color="#1f77b4")
ax.plot(
    position_bins, likelihood, label="Likelihood", linewidth=2.5, color="#ff7f0e", linestyle="--"
)
ax.fill_between(position_bins, state_dist, alpha=0.3, color="#1f77b4")
ax.fill_between(position_bins, likelihood, alpha=0.3, color="#ff7f0e")
ax.axvline(
    true_position, color="black", linestyle=":", linewidth=2, label="True position", alpha=0.5
)
ax.set_xlabel("Position (cm)")
ax.set_ylabel("Probability Density")
ax.set_title("Perfect Model Fit: State and Likelihood Agree")
ax.legend(frameon=True, loc="upper right")
plt.tight_layout()
plt.show()

# Create position bins (e.g., 100 cm track, 50 bins) position_bins = np.linspace(0, 100, 50) # Generate distributions centered at the same location with similar uncertainty true_position = 50.0 state_dist = generate_1d_gaussian_distribution(position_bins, mean=true_position, std=2.0) likelihood = generate_1d_gaussian_distribution(position_bins, mean=true_position, std=2.5) # Visualize fig, ax = plt.subplots(figsize=(10, 5)) ax.plot(position_bins, state_dist, label="State distribution", linewidth=2.5, color="#1f77b4") ax.plot( position_bins, likelihood, label="Likelihood", linewidth=2.5, color="#ff7f0e", linestyle="--" ) ax.fill_between(position_bins, state_dist, alpha=0.3, color="#1f77b4") ax.fill_between(position_bins, likelihood, alpha=0.3, color="#ff7f0e") ax.axvline( true_position, color="black", linestyle=":", linewidth=2, label="True position", alpha=0.5 ) ax.set_xlabel("Position (cm)") ax.set_ylabel("Probability Density") ax.set_title("Perfect Model Fit: State and Likelihood Agree") ax.legend(frameon=True, loc="upper right") plt.tight_layout() plt.show()

Interpretation¶

Notice how the two distributions:

• Peak at the same location
• Have similar spread
• Overlap substantially

This indicates the model's predictions (state distribution) align well with what the data says (likelihood). This is what we expect from a well-specified model.

Example 2: Poor Model Fit¶

Now let's see what happens when the model fails to capture the data. The distributions should disagree.

In [3]:

# Generate distributions centered at different locations
state_mean = 30.0
likelihood_mean = 70.0  # Large spatial offset!

state_dist_poor = generate_1d_gaussian_distribution(position_bins, mean=state_mean, std=2.0)
likelihood_poor = generate_1d_gaussian_distribution(position_bins, mean=likelihood_mean, std=3.0)

# Visualize
fig, ax = plt.subplots(figsize=(10, 5))
ax.plot(position_bins, state_dist_poor, label="State distribution", linewidth=2.5, color="#1f77b4")
ax.plot(
    position_bins,
    likelihood_poor,
    label="Likelihood",
    linewidth=2.5,
    color="#ff7f0e",
    linestyle="--",
)
ax.fill_between(position_bins, state_dist_poor, alpha=0.3, color="#1f77b4")
ax.fill_between(position_bins, likelihood_poor, alpha=0.3, color="#ff7f0e")
ax.axvline(state_mean, color="#1f77b4", linestyle=":", alpha=0.5, linewidth=2)
ax.axvline(likelihood_mean, color="#ff7f0e", linestyle=":", alpha=0.5, linewidth=2)
ax.set_xlabel("Position (cm)")
ax.set_ylabel("Probability Density")
ax.set_title("Poor Model Fit: State and Likelihood Disagree")
ax.legend(frameon=True, loc="upper right")
plt.tight_layout()
plt.show()

# Generate distributions centered at different locations state_mean = 30.0 likelihood_mean = 70.0 # Large spatial offset! state_dist_poor = generate_1d_gaussian_distribution(position_bins, mean=state_mean, std=2.0) likelihood_poor = generate_1d_gaussian_distribution(position_bins, mean=likelihood_mean, std=3.0) # Visualize fig, ax = plt.subplots(figsize=(10, 5)) ax.plot(position_bins, state_dist_poor, label="State distribution", linewidth=2.5, color="#1f77b4") ax.plot( position_bins, likelihood_poor, label="Likelihood", linewidth=2.5, color="#ff7f0e", linestyle="--", ) ax.fill_between(position_bins, state_dist_poor, alpha=0.3, color="#1f77b4") ax.fill_between(position_bins, likelihood_poor, alpha=0.3, color="#ff7f0e") ax.axvline(state_mean, color="#1f77b4", linestyle=":", alpha=0.5, linewidth=2) ax.axvline(likelihood_mean, color="#ff7f0e", linestyle=":", alpha=0.5, linewidth=2) ax.set_xlabel("Position (cm)") ax.set_ylabel("Probability Density") ax.set_title("Poor Model Fit: State and Likelihood Disagree") ax.legend(frameon=True, loc="upper right") plt.tight_layout() plt.show()

Interpretation¶

The distributions:

• Peak at very different locations (40 cm apart!)
• Have minimal overlap
• Indicate fundamental disagreement between model and data

What does this tell us?

• The model's predictions don't match observations
• Could indicate: wrong dynamics model, wrong observation model, or insufficient model capacity
• We need to investigate and revise the model

Data Format: Understanding Array Shapes¶

Important: All statespacecheck functions expect distributions with time as the first dimension:

• 1D spatial: shape (n_time, n_bins)
• 2D spatial: shape (n_time, n_x_bins, n_y_bins)

Why time-first?

• Enables vectorized operations over time
• Consistent with time-series analysis conventions
• Allows functions to return time-resolved diagnostics

For single time points (like our examples so far), we add a time dimension using [np.newaxis, :].

Quantifying Agreement: KL Divergence¶

Visual inspection is helpful, but we need quantitative metrics. Kullback-Leibler (KL) divergence measures information divergence between distributions.

$$D_{KL}(P || Q) = \sum_x P(x) \log\frac{P(x)}{Q(x)}$$

Interpretation thresholds:

• Low divergence (< 0.1): Distributions agree well
• Moderate divergence (0.1 - 1.0): Some disagreement, worth investigating
• High divergence (> 1.0): Substantial mismatch, model problems likely

About these thresholds: These are empirical guidelines based on experience with neuroscience data. In practice:

• KL < 0.1 indicates distributions differ by less than 10% in information content
• KL > 1.0 represents substantial information loss if using one distribution instead of the other
• Your data may differ: Validate these thresholds on known-good and known-bad models in your domain

Let's compute KL divergence for our examples:

In [4]:

# For single time point, need to add time dimension
state_dist_2d = state_dist[np.newaxis, :]  # Shape: (1, n_bins) - time dimension added
likelihood_2d = likelihood[np.newaxis, :]

state_dist_poor_2d = state_dist_poor[np.newaxis, :]
likelihood_poor_2d = likelihood_poor[np.newaxis, :]

# Compute KL divergence
kl_good = kl_divergence(state_dist_2d, likelihood_2d)[0]
kl_poor = kl_divergence(state_dist_poor_2d, likelihood_poor_2d)[0]

print("KL Divergence Results:")
print(f"  Perfect fit:  {kl_good:.4f} (low - distributions agree)")
print(f"  Poor fit:     {kl_poor:.4f} (high - distributions disagree)")

# For single time point, need to add time dimension state_dist_2d = state_dist[np.newaxis, :] # Shape: (1, n_bins) - time dimension added likelihood_2d = likelihood[np.newaxis, :] state_dist_poor_2d = state_dist_poor[np.newaxis, :] likelihood_poor_2d = likelihood_poor[np.newaxis, :] # Compute KL divergence kl_good = kl_divergence(state_dist_2d, likelihood_2d)[0] kl_poor = kl_divergence(state_dist_poor_2d, likelihood_poor_2d)[0] print("KL Divergence Results:") print(f" Perfect fit: {kl_good:.4f} (low - distributions agree)") print(f" Poor fit: {kl_poor:.4f} (high - distributions disagree)")

KL Divergence Results:
  Perfect fit:  0.0431 (low - distributions agree)
  Poor fit:     89.0166 (high - distributions disagree)

The KL divergence quantifies what we saw visually:

• Perfect fit has very low divergence
• Poor fit has much higher divergence (over 100x larger!)

This metric gives us an objective measure of model-data agreement.

Quantifying Agreement: HPD Overlap¶

Another way to assess agreement is highest posterior density (HPD) region overlap. This measures spatial overlap between high-probability regions.

Key idea:

• Identify the 95% HPD region for each distribution
• Compute how much they overlap
• Normalize by the smaller region size

Interpretation:

• High overlap (> 0.7): Distributions concentrate mass in similar regions
• Moderate overlap (0.3 - 0.7): Partial agreement
• Low overlap (< 0.3): Distributions are spatially inconsistent

In [5]:

# Compute HPD overlap
overlap_good = hpd_overlap(state_dist_2d, likelihood_2d, coverage=0.95)[0]
overlap_poor = hpd_overlap(state_dist_poor_2d, likelihood_poor_2d, coverage=0.95)[0]

print("HPD Overlap Results (95% coverage):")
print(f"  Perfect fit:  {overlap_good:.4f} (high - good spatial agreement)")
print(f"  Poor fit:     {overlap_poor:.4f} (low - poor spatial agreement)")

# Compute HPD overlap overlap_good = hpd_overlap(state_dist_2d, likelihood_2d, coverage=0.95)[0] overlap_poor = hpd_overlap(state_dist_poor_2d, likelihood_poor_2d, coverage=0.95)[0] print("HPD Overlap Results (95% coverage):") print(f" Perfect fit: {overlap_good:.4f} (high - good spatial agreement)") print(f" Poor fit: {overlap_poor:.4f} (low - poor spatial agreement)")

HPD Overlap Results (95% coverage):
  Perfect fit:  1.0000 (high - good spatial agreement)
  Poor fit:     0.0000 (low - poor spatial agreement)

Again, the metric confirms our visual intuition:

• Perfect fit shows high overlap (distributions concentrate in similar regions)
• Poor fit shows low overlap (distributions are in different regions)

HPD overlap provides a complementary view to KL divergence, focusing on spatial consistency.

Progressive Disagreement¶

Let's explore the full spectrum from perfect agreement to complete disagreement by varying the spatial offset between distributions.

In [6]:

# Generate a range of offsets
offsets = np.linspace(0, 50, 20)  # 0 to 50 cm offset
kl_values = []
overlap_values = []

for offset in offsets:
    # Generate distributions with increasing offset
    state_test = generate_1d_gaussian_distribution(position_bins, mean=40.0, std=2.0)
    likelihood_test = generate_1d_gaussian_distribution(position_bins, mean=40.0 + offset, std=2.5)

    # Add time dimension
    state_test_2d = state_test[np.newaxis, :]
    likelihood_test_2d = likelihood_test[np.newaxis, :]

    # Compute metrics
    kl_values.append(kl_divergence(state_test_2d, likelihood_test_2d)[0])
    overlap_values.append(hpd_overlap(state_test_2d, likelihood_test_2d, coverage=0.95)[0])

# Visualize
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

# KL divergence
ax1.plot(offsets, kl_values, linewidth=2.5, marker="o", markersize=6, color="#1f77b4")
ax1.axhline(0.1, color="green", linestyle="--", alpha=0.5, label="Low threshold")
ax1.axhline(1.0, color="orange", linestyle="--", alpha=0.5, label="High threshold")
ax1.fill_between(offsets, 0, 0.1, alpha=0.2, color="green", label="Good fit")
ax1.fill_between(offsets, 1.0, ax1.get_ylim()[1], alpha=0.2, color="red", label="Poor fit")
ax1.set_xlabel("Spatial Offset (cm)")
ax1.set_ylabel("KL Divergence")
ax1.set_title("KL Divergence vs Spatial Offset")
ax1.legend(frameon=True, loc="upper left")

# HPD overlap
ax2.plot(offsets, overlap_values, linewidth=2.5, marker="o", markersize=6, color="#ff7f0e")
ax2.axhline(0.7, color="green", linestyle="--", alpha=0.5, label="High threshold")
ax2.axhline(0.3, color="orange", linestyle="--", alpha=0.5, label="Low threshold")
ax2.fill_between(offsets, 0.7, 1.0, alpha=0.2, color="green", label="Good fit")
ax2.fill_between(offsets, 0, 0.3, alpha=0.2, color="red", label="Poor fit")
ax2.set_xlabel("Spatial Offset (cm)")
ax2.set_ylabel("HPD Overlap")
ax2.set_title("HPD Overlap vs Spatial Offset")
ax2.legend(frameon=True, loc="upper right")

plt.tight_layout()
plt.show()

# Generate a range of offsets offsets = np.linspace(0, 50, 20) # 0 to 50 cm offset kl_values = [] overlap_values = [] for offset in offsets: # Generate distributions with increasing offset state_test = generate_1d_gaussian_distribution(position_bins, mean=40.0, std=2.0) likelihood_test = generate_1d_gaussian_distribution(position_bins, mean=40.0 + offset, std=2.5) # Add time dimension state_test_2d = state_test[np.newaxis, :] likelihood_test_2d = likelihood_test[np.newaxis, :] # Compute metrics kl_values.append(kl_divergence(state_test_2d, likelihood_test_2d)[0]) overlap_values.append(hpd_overlap(state_test_2d, likelihood_test_2d, coverage=0.95)[0]) # Visualize fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5)) # KL divergence ax1.plot(offsets, kl_values, linewidth=2.5, marker="o", markersize=6, color="#1f77b4") ax1.axhline(0.1, color="green", linestyle="--", alpha=0.5, label="Low threshold") ax1.axhline(1.0, color="orange", linestyle="--", alpha=0.5, label="High threshold") ax1.fill_between(offsets, 0, 0.1, alpha=0.2, color="green", label="Good fit") ax1.fill_between(offsets, 1.0, ax1.get_ylim()[1], alpha=0.2, color="red", label="Poor fit") ax1.set_xlabel("Spatial Offset (cm)") ax1.set_ylabel("KL Divergence") ax1.set_title("KL Divergence vs Spatial Offset") ax1.legend(frameon=True, loc="upper left") # HPD overlap ax2.plot(offsets, overlap_values, linewidth=2.5, marker="o", markersize=6, color="#ff7f0e") ax2.axhline(0.7, color="green", linestyle="--", alpha=0.5, label="High threshold") ax2.axhline(0.3, color="orange", linestyle="--", alpha=0.5, label="Low threshold") ax2.fill_between(offsets, 0.7, 1.0, alpha=0.2, color="green", label="Good fit") ax2.fill_between(offsets, 0, 0.3, alpha=0.2, color="red", label="Poor fit") ax2.set_xlabel("Spatial Offset (cm)") ax2.set_ylabel("HPD Overlap") ax2.set_title("HPD Overlap vs Spatial Offset") ax2.legend(frameon=True, loc="upper right") plt.tight_layout() plt.show()

Key Observations¶

KL Divergence (left panel):

• Increases rapidly with spatial offset
• Stays low (< 0.1) when offset is small
• Becomes very large when distributions are far apart

HPD Overlap (right panel):

• Decreases as distributions separate
• Stays high (> 0.7) when distributions are close
• Drops to zero when distributions don't overlap at all

Notice: The two metrics provide complementary information. KL divergence is sensitive to the entire distribution shape, while HPD overlap focuses on where the high-probability mass is located.

Effect of Uncertainty¶

Let's examine how the width/uncertainty of distributions affects our metrics.

In [7]:

# Generate distributions with varying uncertainty
uncertainties = np.linspace(1.0, 10.0, 20)
kl_values_unc = []
overlap_values_unc = []

# Fixed offset
fixed_offset = 10.0

for unc in uncertainties:
    # State distribution with fixed std, likelihood with varying std
    state_test = generate_1d_gaussian_distribution(position_bins, mean=40.0, std=2.0)
    likelihood_test = generate_1d_gaussian_distribution(
        position_bins, mean=40.0 + fixed_offset, std=unc
    )

    state_test_2d = state_test[np.newaxis, :]
    likelihood_test_2d = likelihood_test[np.newaxis, :]

    kl_values_unc.append(kl_divergence(state_test_2d, likelihood_test_2d)[0])
    overlap_values_unc.append(hpd_overlap(state_test_2d, likelihood_test_2d, coverage=0.95)[0])

# Visualize
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

ax1.plot(uncertainties, kl_values_unc, linewidth=2.5, marker="o", markersize=6, color="#1f77b4")
ax1.set_xlabel("Likelihood Uncertainty (std, cm)")
ax1.set_ylabel("KL Divergence")
ax1.set_title("Effect of Uncertainty on KL Divergence")

ax2.plot(
    uncertainties, overlap_values_unc, linewidth=2.5, marker="o", markersize=6, color="#ff7f0e"
)
ax2.set_xlabel("Likelihood Uncertainty (std, cm)")
ax2.set_ylabel("HPD Overlap")
ax2.set_title("Effect of Uncertainty on HPD Overlap")

plt.tight_layout()
plt.show()

# Generate distributions with varying uncertainty uncertainties = np.linspace(1.0, 10.0, 20) kl_values_unc = [] overlap_values_unc = [] # Fixed offset fixed_offset = 10.0 for unc in uncertainties: # State distribution with fixed std, likelihood with varying std state_test = generate_1d_gaussian_distribution(position_bins, mean=40.0, std=2.0) likelihood_test = generate_1d_gaussian_distribution( position_bins, mean=40.0 + fixed_offset, std=unc ) state_test_2d = state_test[np.newaxis, :] likelihood_test_2d = likelihood_test[np.newaxis, :] kl_values_unc.append(kl_divergence(state_test_2d, likelihood_test_2d)[0]) overlap_values_unc.append(hpd_overlap(state_test_2d, likelihood_test_2d, coverage=0.95)[0]) # Visualize fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5)) ax1.plot(uncertainties, kl_values_unc, linewidth=2.5, marker="o", markersize=6, color="#1f77b4") ax1.set_xlabel("Likelihood Uncertainty (std, cm)") ax1.set_ylabel("KL Divergence") ax1.set_title("Effect of Uncertainty on KL Divergence") ax2.plot( uncertainties, overlap_values_unc, linewidth=2.5, marker="o", markersize=6, color="#ff7f0e" ) ax2.set_xlabel("Likelihood Uncertainty (std, cm)") ax2.set_ylabel("HPD Overlap") ax2.set_title("Effect of Uncertainty on HPD Overlap") plt.tight_layout() plt.show()

Key Observations¶

As likelihood uncertainty increases:

• KL divergence decreases: The distributions become more similar as the likelihood spreads out
• HPD overlap increases then plateaus: More spread means more spatial overlap, but eventually coverage limits are reached

Important insight: Model uncertainty affects diagnostic metrics. A very uncertain model might appear to "fit well" simply because it doesn't make strong predictions. This is why we need multiple complementary diagnostics!

Summary: What We Learned¶

Core concepts:

1. State space models have latent variables we can't directly observe
2. We assess fit by comparing state distribution (model prediction) vs likelihood (data-driven)
3. Agreement indicates good fit; disagreement indicates problems

Diagnostic metrics:

1. KL Divergence: Measures information divergence

   • Low (< 0.1): Good agreement
   • High (> 1.0): Poor agreement

2. HPD Overlap: Measures spatial overlap of high-probability regions

   • High (> 0.7): Good spatial agreement
   • Low (< 0.3): Poor spatial agreement

Key insights:

• Visual inspection builds intuition
• Quantitative metrics provide objective assessment
• Multiple metrics give complementary views
• Uncertainty affects interpretation

Next steps:

• Next notebook: 02_highest_density_regions.ipynb - Deep dive into HPD regions
• Jump ahead: 03_time_resolved_diagnostics.ipynb - Apply to real time series data

Exercises (Optional)¶

Try modifying the code above to explore these questions:

1. ⭐ Easy: What happens with very peaked distributions (std = 0.5)? How do KL and overlap change?

   • Hint: Modify the std parameter in the distribution generation

2. ⭐⭐ Moderate: How do metrics change if both distributions have high uncertainty (std > 10)?

   • Hint: Does high mutual uncertainty make disagreement harder to detect?

3. ⭐⭐⭐ Advanced: Can you create a case where KL divergence is low but overlap is also low?

   • Hint: Think about multimodal distributions (we'll explore these in notebook 02)

In [ ]: