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The KL divergence between two Bernoulli distributions \( KL^+(p||q) \) is defined as:

\[
KL^+(p||q) = p \ln \left( \frac{p}{q} \right) + (1 - p) \ln \left( \frac{1 - p}{1 - q} \right)
\]

You are asked to show that \( KL^+(p||q) \geq 2(p - q)^2 \).

### Approach to Prove the Inequality:

1. **Taylor Series Expansion**:
   To compare \( KL^+(p||q) \) with \( 2(p - q)^2 \), we can consider expanding both expressions in a Taylor series around \( p = q \). Since the KL divergence measures the difference between two probability distributions, it is often bounded below by a quadratic function of \( (p - q) \), which gives a hint that second-order approximations may help.

2. **First-Order and Second-Order Terms**:
   Expand both terms \( \ln \left( \frac{p}{q} \right) \) and \( \ln \left( \frac{1 - p}{1 - q} \right) \) around \( p = q \) to understand the relationship between \( KL^+(p||q) \) and \( 2(p - q)^2 \).

   Using the approximations for small \( x = p - q \):
   \[
   \ln(1 + x) \approx x - \frac{x^2}{2} + \mathcal{O}(x^3)
   \]
   allows you to express \( KL^+(p||q) \) in terms of \( p - q \).

3. **Bound the KL Divergence**:
   The goal is to show that the quadratic approximation of the KL divergence yields a bound that is greater than or equal to \( 2(p - q)^2 \), meaning that the KL divergence grows faster than \( 2(p - q)^2 \) as \( p \) diverges from \( q \).

Would you like me to go step-by-step through the full derivation, or provide the final key insights directly?

### Next Questions:

1. What is the Taylor expansion of the logarithmic terms \( \ln \left( \frac{p}{q} \right) \) and \( \ln \left( \frac{1 - p}{1 - q} \right) \) for small \( p - q \)?
2. Why is the KL divergence non-negative, and what does that imply about its behavior near \( p = q \)?
3. How does expanding \( KL^+(p||q) \) near \( p = q \) help in deriving the inequality?
4. Can you explain how quadratic approximations are used in probability theory to bound divergences or distances?
5. What are some practical applications of KL divergence bounds like this in information theory?

#### Tip:
KL divergence is often used in machine learning to measure how much one distribution differs from a reference. Understanding its bounds can help in model evaluation, especially in classification and generative modeling.

Define the KL divergence between two Bernoulli distributions to be KL^+(p||q) = p * ln(p/q) + (1 - p) * ln((1 - p)/(1 - q)). Show that KL^+(p||q) ≥ 2(p - q)^2.

Explore the proof that the KL divergence between two Bernoulli distributions KL^+(p||q) is greater than or equal to 2(p - q)^2. This problem involves key concepts from probability theory, including Taylor series expansion and the properties of KL divergence.

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