Ratio matching is applied to discrete data specifically to binary data. It avoids partition function by incorporating ratio of probabilities which leads to the cancellation of partition function.
Ration matching minimizes the following objective function

$L^{(RM)}(x,\theta) = \sum_{j=1}^n(\frac{1}{1+ \frac{p_{model}(x:\theta)}{p_{model}(f(x,j);\theta)}})^2$

Ratio Matching

Score matching helps to train a neural network model without evaluating the partition function , $Z$ or its derivative.
The strategy used is to minimize the expected squared difference between the derivatives of the model's log density with respect to the input and the derivatives of the data's log density with respect to the input.

$L(x,\theta) = \frac{1}{2} \| \nabla_x \log p_{model}(x;\theta),- \nabla_x \log p_{data}(x) \|_{2}^{2}$

$ J(\theta) = \frac{1}{2} E_{p_{data}}(x)L(x,\theta)$

$\theta^* = \min\limits_{\theta} J(\theta)$


Score Matching

There are several methods to train and evaluate undirected graphical models with intractable partition function.

One way is to avoid using partition function and this can be done by following the techniques mentioned below.

 - Log-Likelihood Gradient
 - Stochastic Maximum Likelihood and ContrastiveDivergence
 - Pseudolikelihood
 - Score Matching and Ratio Matching
 - Denoising Score Matching
 - Noise-Contrastive Estimation

San Jose State University

Each of the other models (state machines, formal rule systems, and logic) can be augmented with probabilities. For example, the state machine can be augmented with probabilities to become the weighted automaton, or Markov model. The key advantage of probabilistic models is their ability to solve the many kinds of ambiguity problems that can be recast as “given N choices for some ambiguous input, choose the most probable one”.

Probabilistic models

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

Deep Learning

- Spelling correction or grammatical error correction
- Speech recognition
- Machine translation

Applications of probabilistic models

- Language Models (LMs)
- Hidden Markov Models (HMMs)

Types of probabilistic models

The deep learning approach is to figure out the minimum amount of information that is absolutely needed and find out how to get a reasonable approximation of this information in the shortest amount of time. 

The latent variables are designed differently in which the training algorithm can invent the concepts it needs to model a particular dataset freely and deal with complex problems and are reusable in more different contexts.


The Deep Learning Approach to Structured Probabilistic Models

The partition function is an integral or sum over unnormalized probabilities. This function, often denoted by Z, is used to normalize an unnormalized distribution into a valid distribution.
$$ p(\textbf{x}) =  \frac{1}{Z}  \tilde{p}(\textbf{x}) .  $$

where Z is: 

$$ Z =  \int   \tilde{p}(\textbf{x}) d\textbf{x}.  $$ 
or
$$ Z =  \sum_{x}  \tilde{p}(\textbf{x}) d\textbf{x}.  $$

Z = partition function.  
$$ \tilde{p}(\textbf{x})$$ = Unnormalized distribution. 
$$p(\textbf{x}) $$= Normalized distribution. 
In the context of deep learning, Z is usually intractable, meaning that it often needs to be approximated. 


The Partition Function (introduction).

Probabilistic models use graphs to show interactions between random variables. A graph is a set of nodes connected by vertices. A probabilistic model represents random variables with nodes and their relationships with vertices between them.

Graph Model Structure

- Dramatically reduce the cost of representing probability distributions as well as learning and inference
- In the case of a directed model, sampling is accelerated
- Allow us to explicitly separate representation of knowledge from learning of knowledge or inference given existing knowledge

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