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I am reading Ian Goodfellow "Deep Learning" book. At page 128 it says

One way to interpret maximum likelihood estimation is to view it as minimizing the dissimilarity between the empirical distribution $\hat{p}_{\text{data}}$, defined by the training set and the model distribution, with the degree of dissimilarity between the two measured by the KL divergence. The KL divergence is given by

$$ D_{KL} (\hat{p}_{\text{data}} || p_{\text{model}}) = \mathbb{E}_{\mathbf{x} \sim \hat{p}_{\text{data}}} [\log \hat{p}_{\text{data}} - \log p_{\text{model}}(\mathbf{x})]$$

Starting from the definition of maximum likelihood estimator written in the text:

$$\mathbf{\theta}_{ML} = \arg\max_{\theta} p_{\text{model}}(\mathbb{X}; \mathbf{\theta}) $$

Is there a formal proof for this? What is the intuition behind the formulation of maximum likelihood estimator as minimizing the KL divergence between the empirical distribution and the model distribution?

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This is a late response but I hope it may help:

Proof (this proof is basically a summary of the explanations from the author):

To go for the proof, first we can follow the procedure given by the author which allows us to have a more convenient expression:

$$\begin{aligned} \theta_{ML} &=\arg \max_\theta p_{model}(\mathbb{X};\theta)\\ &= \arg\max_\theta\prod_{i=1}^m p_{model}(x^{(i)};\theta) \\ &= \arg\max_\theta\sum_{i=1}^m \log(p_{model}(x^{(i)};\theta)) \\ &= \arg\max_\theta \frac{1}{m}\sum_{i=1}^m \log(p_{model}(x^{(i)};\theta))\\ &= \arg\max_\theta \sum_{i=1}^m \frac{1}{m} \log(p_{model}(x^{(i)};\theta)) \end{aligned}$$

Note this last expression is the expectation of this $\log(p_{model}(x;\theta))$ function with respect to the empirical distribution defined by the training data ($\hat{p}_{data}$) which puts a probability of $1/m$ on each of the $m$ points $x^{(1)},x^{(2)},...,x^{(m)}$. So we can equivalently write this last expression as:

$$ \theta_{ML} = \arg\max_{\theta}\mathbb{E}_{x\sim \hat{p}_{data}} \log(p_{model}(x;\theta))$$

Hence, obtaining the value of $\theta$ which satisfies this expresion will maximize the likelihood of $p_{model}(x,\theta)$ being the statistical model that best fits our set of data samples $\mathbb{X}$.

But we can also get the parameter $\theta$ that maximizes the likelihood using the KL divergence of the probability distributions $\hat{p}_{data}$ (empirical distribution of $\mathbb{X}$) and $p_{model}$ (our statistical model that we are using to fit $\mathbb{X}$):

$$ D_{\text{KL}}(\hat{p}_{data} \parallel p_{model}) = \mathbb{E}_{x\sim \hat{p}_{data}} [ \log(\hat{p}_{data}(x)) - \log(p_{model}(x;\theta))]$$

This is because of the explanation given by the author, which is that $\mathbb{E}_{x\sim \hat{p}_{data}} \log(\hat{p}_{data}(x))$ does not depend on $\theta$ (it only depends on the data generating process), so it can be trated as a constant. Hence we can adress the same problem of finding the value of $\theta$ that maximizes the likelihood by minimizing this KL divergence, because this is the same as minimizing:

$$ \mathbb{E}_{x\sim \hat{p}_{data}} [- \log(p_{model}(x;\theta))]$$

Which is just the negative form of the simplified expression for $\theta_{ML}$ that we have written earlier from the perspective of the likelihood.

So, to sum up, we can also calculate the parameter $\theta_{ML}$ by: $$ \theta_{ML} = \arg\min_{\theta} D_{\text{KL}}(\hat{p}_{data} \parallel p_{model})= \arg\min_{\theta} \mathbb{E}_{x\sim \hat{p}_{data}} [- \log(p_{model}(x;\theta))] $$


Intuition:

With this said, I believe we can think of using the KL divergence for maximizing the likelihood as a way of making the predicted distribution $(p_{model}(\mathbb{X},\theta))$ as close as possible to the empirical distribution.

Thereby with our predicted distribution and by sampling it, we would be able to obtain a set of samples similar to the initial ones ($\mathbb{X}$). So this may mean that we have correctly calculate the true distribution of the data $\mathbb{X}$.

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    $\begingroup$ There's a detailed discussion in the context of parameter survival models in this paper by Nils Lid Hjort: jstor.org/stable/1403683 The formal expression of idea goes back at least to Huber in 1967, but that paper is harder to find. $\endgroup$ – Thomas Lumley Sep 9 '20 at 22:13

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