To make use of the expression of the negative phase as an expectation we can use Monte Carlo methods to approximately maximise the likelihood of models with intractible partition functions. The naive way of implemnting this would be to burn in a set of Markov chains from a random initialization every time the gradient is needed. However, this approach is infeasible when learning using stochastic gradient descent as this means the chains must be burned in once per gradient step. 

Stochastic Maximum Likelihood

$$\nabla_{\boldsymbol \theta} \log Z = \mathbb{E}_{\mathbf{x} \sim p(\mathbf{x})} \nabla_{\boldsymbol \theta} \log \tilde{p}(\mathbf{x})$$

New York University

The gradient of the log-likelihood which is used in Maximum Likelihood Estimation can be decomposed into the following:
$$\nabla_{\boldsymbol \theta} \log p(\mathbf{x}; \boldsymbol \theta) = \nabla_{\boldsymbol \theta} \log \tilde{p}(\mathbf{x}; \boldsymbol \theta) - \nabla_{\boldsymbol \theta } \log Z(\boldsymbol \theta)$$
Where $\tilde{p}(\mathbf{x}; \boldsymbol \theta)$ is the unnormalized probabilioty density and $Z$ is the partition function. This is well-known as the decomposition into the positive phase and negative phase of learning. Due to the reliance of the partition function on the parameters, learning models by maximum likelihood is particularly difficult. 

Log-Likelihood Gradient

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

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