DeepSeek R1 On Naive Bayes and Logistic Regression

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Introduction

本文利用 DeepSeek R1 和 ChatGPT (4o or o1?) 解釋 Naive Bayes (generative) 在什麽情況下可以得到 Logistic Regression (discriminative). DeepSeek R1 和 ChatGPT 都做出正確的解釋。但是 R1 給出的思考過程非常有趣，而且有用！

這個問題：如何從 Naive Bayes 推出 Logistic Regression. 是我在看 Stanford CS236, Ermon 教授在 Lecture 2 的結尾比較 generative AI vs. discriminative AI 的一個 hand-waving 的説法。

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Naive Bayes (Generative)

Naive Bayes 是生成式 AI，因爲可以從 $Y$ (class) 產生 $X_i$ (features)。這種模型基於貝葉斯定理，假設特徵之間是條件獨立的，這使得計算變得簡單而高效。具體來說，給定一個類別 $Y$，我們可以計算每個特徵 $X_i$ 的條件概率 $P(X_i Y)$。然後，我們可以使用這些條件概率來估計給定觀察到的特徵 $X = (X_1, X_2, \ldots, X_n)$ 時某一類別的後驗概率 $P(Y X)$。

在實際應用中，Naive Bayes 常被用於文本分類任務，如垃圾郵件檢測和情感分析。由於其簡單性和高效性，即使在特徵之間存在一定的相關性時，它仍然能夠提供不錯的分類性能。此外，Naive Bayes 模型只需要少量的訓練數據就能夠進行有效的預測，使其在數據稀缺或維度高的情況下尤為有用。

然而，由於 Naive Bayes 假設特徵獨立，因此在某些情況下可能會導致性能下降，例如當特徵之間存在強相關性時。

Logistic Regression (Discriminative)

Regression 問題的 $X_i$ 是 given. 從已知的 $(Y, X_i)$ 訓練出 $p(Y=1 \vert X_i)$.
重點是 $X_i$ given, 不需要 generative model 產生。 如果假設 output feature: $z = \alpha_0 + \sum \alpha_i x_i$ 是綫性關係。 如果是 linear regression, 就是找出 $z$ 和 $x_i$ 的 bias and weight. 如果是 logistic regression, 只要把 $z$ 經過 nonlinear function 變成 probability 即可。 $p(Y=1 \vert X_i) = \sigma(z(X_i)) = \sigma(\alpha_0 + \sum \alpha_i x_i))$.

很明顯 logistic regression 不是 conditional independence! 因爲 $X_i$ 之間有 (linear) dependence. 事實上 logistic regression 比 naive Bayes 是更好 (no conditional independwence) 而且更簡單 (discriminative) 的方法。有趣的是在某一些情況下，naive Bayes 可以得到 logistic regression form, 反之不真。

  Type Assumption Focus
Naived Bayes Generative Strong (conditional independence) Prior
Logistic Regression Discriminative Weak (linear dependence) Evidence

Ermon 並沒有給出詳細的條件和推導。

先説問題的答案：假設 $X_i$ and $Y$ 是 naive Bayian.

  1. $X_i$ with Gaussian distribution to derive the logistic regression
  2. $X_i$ given Y=0 or Y=1 有相同的 variance.

其實 2. 可以放寬成最後的 variance 是一樣即可。不用每一個 variance 都相等。

從一個 $X_1$ 想比較簡單。

DeepSeek R1

以下是思考的過程，非常有意思。

Okay, so I need to explain how Naive Bayes with a Gaussian distribution can lead to deriving logistic regression. Hmm, let me start by recalling what I know about Naive Bayes and logistic regression.

Naive Bayes is a probabilistic classifier based on Bayes’ theorem with the “naive” assumption of feature independence given the class label. When dealing with continuous features, one common approach is to assume each feature follows a Gaussian (normal) distribution conditional on the class. So, for each class, we model each feature with its own mean and variance.

Logistic regression, on the other hand, is a discriminative model that directly estimates the probability of a class given the features. It uses the logistic function to map a linear combination of features to a probability between 0 and 1.

The user wants me to connect these two. I remember that under certain conditions, Naive Bayes can be equivalent to logistic regression. Specifically, if the class-conditional densities are in the exponential family and some other conditions hold, the posterior probability can be expressed as a logistic function.

Let me think step by step. Let’s say we have a binary classification problem with classes 0 and 1. For each class, the features are modeled as Gaussians. The Naive Bayes assumption means that the joint probability of features given the class is the product of individual Gaussian probabilities.

So, the posterior probability P(Y=1 X) would be proportional to P(X Y=1) * P(Y=1). Similarly for Y=0. Taking the ratio of these two posteriors and applying Bayes’ theorem, we can get the log odds.
If I take the log of the ratio P(Y=1 X)/P(Y=0 X), that should equal the log of [P(X Y=1)P(Y=1)/P(X Y=0)P(Y=0)]. Because of the Naive assumption, the joint probabilities factor into products, so each feature’s contribution can be summed up.

Since each feature is Gaussian, the log of the ratio for each feature would involve the difference between the Gaussian densities for Y=1 and Y=0. The Gaussian density has the form (1/(sqrt(2πσ²))) * exp(-(x-μ)²/(2σ²)). Taking the log of this would give a quadratic term, but if we assume that the variances are the same for both classes, maybe the quadratic terms cancel out? Wait, in Naive Bayes, each class has its own parameters, so maybe we need to assume that the variances are equal across classes for each feature?

Alternatively, if we assume that the covariance matrices are the same between the two classes, but in Naive Bayes, the covariance is diagonal because of independence. Wait, but in the case of Gaussian Naive Bayes, each feature is independent given the class, so the covariance matrix is diagonal, but each class can have different variances. But if we further assume that the variances for each feature are the same across classes, then maybe the quadratic terms would disappear when taking the log-ratio.

Let me try to write this out.

For a single feature x_i, the likelihood ratio for class 1 vs class 0 would be:

P(x_i Y=1) / P(x_i Y=0) = [exp(-(x_i - μ₁i)²/(2σ_i²)) / sqrt(2πσ_i²)] / [exp(-(x_i - μ₀i)²/(2σ_i²)) / sqrt(2πσ_i²)] = exp[ ((x_i - μ₀i)² - (x_i - μ₁i)² ) / (2σ_i²) ) ]

Expanding the squares in the exponent:

= exp[ (x_i² - 2x_i μ₀i + μ₀i² - x_i² + 2x_i μ₁i - μ₁i²) / (2σ_i²) ) ]

Simplify terms:

= exp[ (2x_i (μ₁i - μ₀i) + (μ₀i² - μ₁i²)) / (2σ_i²) ) ]

Factor out the numerator:

= exp[ ( (μ₁i - μ₀i)(2x_i - (μ₀i + μ₁i)) ) / (2σ_i²) ) ]

But this seems a bit complicated. Wait, let’s see:

Alternatively, after simplifying the numerator:

( (μ₀i² - μ₁i²) + 2x_i(μ₁i - μ₀i) ) / (2σ_i²)

= [ - (μ₁i² - μ₀i²) + 2x_i(μ₁i - μ₀i) ] / (2σ_i² )

Factor the μ terms:

= [ - (μ₁i - μ₀i)(μ₁i + μ₀i) + 2x_i(μ₁i - μ₀i) ] / (2σ_i² )

Factor out (μ₁i - μ₀i):

= (μ₁i - μ₀i) [ - (μ₁i + μ₀i) + 2x_i ] / (2σ_i² )

= (μ₁i - μ₀i)(2x_i - μ₁i - μ₀i) / (2σ_i² )

Which can be written as:

(μ₁i - μ₀i)/σ_i² * (x_i - (μ₁i + μ₀i)/2 )

So the log of the likelihood ratio for feature i is:

(μ₁i - μ₀i)/σ_i² * (x_i - (μ₁i + μ₀i)/2 )

Therefore, when we take the log of the posterior ratio for all features (since they are independent), we sum over all features:

log(P(Y=1 X)/P(Y=0 X)) = log(P(Y=1)/P(Y=0)) + Σ_i [ (μ₁i - μ₀i)/σ_i² * (x_i - (μ₁i + μ₀i)/2 ) ]
This expression is linear in x_i! So the log odds are a linear function of the features x_i. That means the posterior probability P(Y=1 X) can be written as a logistic function of a linear combination of the features, which is exactly what logistic regression models.

Therefore, under the assumptions of Gaussian Naive Bayes (features are Gaussian given the class, independent, and with class-specific means but shared variances for each feature), the posterior probabilities take the form of logistic regression. Thus, Gaussian Naive Bayes leads to a logistic regression model.

Wait, but in the derivation above, I assumed that the variances σ_i² are the same for both classes? Because I used σ_i² for both class 0 and 1. In standard Gaussian Naive Bayes, each class can have its own variance for each feature. So if the variances are different, then the quadratic terms would not cancel out, and the log odds would include quadratic terms, making the decision boundary quadratic instead of linear. Therefore, to get logistic regression (which has a linear decision boundary), we need to assume that the variances are equal across classes for each feature.

So in summary, if we use Gaussian Naive Bayes with the assumption that the variances of each feature are the same between the two classes, then the log posterior odds become a linear function of the features, leading to the same form as logistic regression. This shows that under these specific assumptions, Naive Bayes leads to logistic regression.

Another point is the prior term log(P(Y=1)/P(Y=0)), which becomes the intercept term in logistic regression. The coefficients for each feature in logistic regression would then be [(μ₁i - μ₀i)/σ_i²], and the features are centered around (μ₁i + μ₀i)/2. But in logistic regression, we usually have a weight multiplied by x_i plus an intercept. So the terms here can be rearranged to match the standard logistic regression formulation.

Let me check the standard logistic regression model:

log(P(Y=1 X)/P(Y=0 X)) = β₀ + Σ β_i x_i

Comparing this to our derived expression:

log-odds = log(P(Y=1)/P(Y=0)) + Σ [ (μ₁i - μ₀i)/σ_i² * x_i - (μ₁i - μ₀i)(μ₁i + μ₀i)/(2σ_i²) ]

So combining the terms, this can be rewritten as:

β₀ + Σ β_i x_i

where β₀ = log(P(Y=1)/P(Y=0)) - Σ [ (μ₁i - μ₀i)(μ₁i + μ₀i) / (2σ_i²) ]

and β_i = (μ₁i - μ₀i)/σ_i²

Therefore, the coefficients in logistic regression are determined by the difference in means between the classes divided by the variance of the feature, and the intercept incorporates the prior class probabilities and the adjustment from the means.

This shows that when the Gaussian Naive Bayes model assumes equal variances for each feature across classes, the resulting classifier is equivalent to logistic regression with parameters derived from the means, variances, and class priors.

Therefore, the connection is that under the assumptions of Gaussian Naive Bayes with equal variances for each feature across classes, the posterior log odds are linear in the features, leading to the same functional form as logistic regression. This is an example of how a generative model (Naive Bayes) can lead to a discriminative model (logistic regression) under specific conditions.

To explain the connection between Naive Bayes with Gaussian distributions and logistic regression, we start by analyzing their probabilistic foundations and assumptions:

1. Gaussian Naive Bayes Model

  • Assumptions:
    • Binary classification (classes $Y=0$ and $Y=1$).
    • Features $X = (X_1, X_2, \dots, X_n)$ are conditionally independent given $Y$.
    • Each feature $X_i$ follows a Gaussian distribution for each class: \(P(X_i | Y=k) = \mathcal{N}(\mu_{ki}, \sigma_i^2), \quad k \in \{0, 1\},\) where $\mu_{ki}$ is the mean of $X_i$ for class $k$, and variances $\sigma_i^2$ are shared across classes (critical assumption).

2. Posterior Probability via Bayes’ Theorem

The posterior probability for $Y=1$ given $X$ is: \(P(Y=1 | X) = \frac{P(X | Y=1)P(Y=1)}{P(X)}.\)

Similarly, for $Y=0$: \(P(Y=0 | X) = \frac{P(X | Y=0)P(Y=0)}{P(X)}.\)

The log-odds ratio (key to logistic regression) is: \(\log \frac{P(Y=1 | X)}{P(Y=0 | X)} = \log \frac{P(X | Y=1)}{P(X | Y=0)} + \log \frac{P(Y=1)}{P(Y=0)}.\)

3. Simplifying the Log-Odds

Using the Naive Bayes assumption of conditional independence: \(\log \frac{P(X | Y=1)}{P(X | Y=0)} = \sum_{i=1}^n \log \frac{P(X_i | Y=1)}{P(X_i | Y=0)}.\)

For Gaussian features, substitute the PDFs: \(\log \frac{\mathcal{N}(X_i; \mu_{1i}, \sigma_i^2)}{\mathcal{N}(X_i; \mu_{0i}, \sigma_i^2)} = \frac{(\mu_{1i} - \mu_{0i})}{\sigma_i^2}X_i - \frac{\mu_{1i}^2 - \mu_{0i}^2}{2\sigma_i^2}.\)

Summing over all features and adding the prior term: \(\log \frac{P(Y=1 | X)}{P(Y=0 | X)} = \underbrace{\log \frac{P(Y=1)}{P(Y=0)} - \sum_{i=1}^n \frac{\mu_{1i}^2 - \mu_{0i}^2}{2\sigma_i^2}}_{\text{Intercept } \beta_0} + \sum_{i=1}^n \underbrace{\frac{\mu_{1i} - \mu_{0i}}{\sigma_i^2}}_{\text{Coefficient } \beta_i} X_i.\)

4. Connection to Logistic Regression

The log-odds expression matches the linear form of logistic regression: \(\log \frac{P(Y=1 | X)}{P(Y=0 | X)} = \beta_0 + \sum_{i=1}^n \beta_i X_i,\)

where:

  • $\beta_0 = \log \frac{P(Y=1)}{P(Y=0)} - \sum_{i=1}^n \frac{\mu_{1i}^2 - \mu_{0i}^2}{2\sigma_i^2}$,
  • $\beta_i = \frac{\mu_{1i} - \mu_{0i}}{\sigma_i^2}$.

Since $P(Y=0\vert X) = 1 - P(Y=1\vert X)$

The posterior probability then becomes the logistic function: \(P(Y=1 | X) = \frac{1}{1 + e^{-(\beta_0 + \sum_{i=1}^n \beta_i X_i)}},\)

which is exactly the logistic regression model.

Key Conditions for Equivalence

  1. Gaussian Class-Conditional Distributions: Features are Gaussian given the class.
  2. Shared Variances: Each feature has the same variance $\sigma_i^2$ for both classes (homoscedasticity).
  3. Conditional Independence: Features are independent given $Y$.

Conclusion

Under these assumptions, Gaussian Naive Bayes implicitly learns a linear decision boundary, identical to logistic regression. This shows that:

  • Generative models (like Naive Bayes) can reduce to discriminative models (like logistic regression) under specific conditions.
  • The linearity arises from the cancellation of quadratic terms in the log-odds when variances are shared between classes.

GPT-4o?

Naive Bayes and logistic regression are both fundamental classification algorithms in machine learning, each with distinct approaches to modeling data. Interestingly, under certain conditions, Gaussian Naive Bayes can be shown to have a direct relationship with logistic regression.

Gaussian Naive Bayes (GNB):

GNB is a generative model that assumes features are conditionally independent given the class label and that each feature follows a Gaussian (normal) distribution. For a binary classification problem with classes $y \in {0, 1}$ and a feature vector $\mathbf{x} = (x_1, x_2, \ldots, x_n)$, the model estimates the joint probability $P(\mathbf{x}, y)$ and uses Bayes’ theorem to compute the posterior probability $P(y \mid \mathbf{x})$.

The likelihood of a feature $x_i$ given class $y$ is:

\[P(x_i \mid y) = \frac{1}{\sqrt{2\pi\sigma_{iy}^2}} \exp\left( -\frac{(x_i - \mu_{iy})^2}{2\sigma_{iy}^2} \right)\]

where $\mu_{iy}$ and $\sigma_{iy}^2$ are the mean and variance of feature $x_i$ for class $y$.

Deriving Logistic Regression from GNB:

To explore the connection between GNB and logistic regression, consider the scenario where the variances $\sigma_{iy}^2$ are assumed to be equal across classes for each feature $x_i$, i.e., $\sigma_{i0}^2 = \sigma_{i1}^2 = \sigma_i^2$.

The posterior probability $P(y = 1 \mid \mathbf{x})$ can be expressed using Bayes’ theorem:

\[P(y = 1 \mid \mathbf{x}) = \frac{P(\mathbf{x} \mid y = 1) P(y = 1)}{P(\mathbf{x})}\]

Taking the logarithm of the odds ratio gives:

\[\log \frac{P(y = 1 \mid \mathbf{x})}{P(y = 0 \mid \mathbf{x})} = \log \frac{P(\mathbf{x} \mid y = 1) P(y = 1)}{P(\mathbf{x} \mid y = 0) P(y = 0)}\]

Substituting the Gaussian likelihoods and simplifying, we obtain a linear function of $\mathbf{x}$:

\[\log \frac{P(y = 1 \mid \mathbf{x})}{P(y = 0 \mid \mathbf{x})} = \left( \frac{\mu_1 - \mu_0}{\sigma^2} \right)^T \mathbf{x} + \text{constant}\]

This is precisely the form used in logistic regression, where the log-odds of the probability is modeled as a linear combination of the input features.

Therefore, under the assumption of equal variances for each feature across classes, Gaussian Naive Bayes and logistic regression yield equivalent decision boundaries, despite being derived from different modeling principles—generative for GNB and discriminative for logistic regression.

This relationship highlights how different assumptions and modeling approaches can lead to similar predictive models in machine learning.

Reference

Yang Song, PPDM, ICLR 2021: https://www.youtube.com/watch?v=L9ZegT87QK8&ab_channel=ArtificialIntelligence