> For the complete documentation index, see [llms.txt](https://docs.sdv.dev/sdmetrics/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.sdv.dev/sdmetrics/data-metrics/metrics-in-beta/data-likelihood/gmlikelihood.md).

# GMLikelihood

Data Likelihood describes a set of metrics that calculate the likelihood of the synthetic data belonging to the real data. This metric uses Gaussian Mixture Models to make this calculation.

## Data Compatibility

* **Numerical** : This metric is meant for continuous, numerical data

This metric ignores any incompatible column types.

{% hint style="warning" %}
This metric does not accept missing values
{% endhint %}

## Score

**(highest) ∞**: According to the algorithm, the synthetic data has the highest possible likelihood of belonging to the real data

**(lowest) -∞**: According to the algorithm used, the synthetic data has the lowest possible likelihood of belonging to the real data

{% hint style="danger" %}
**There are multiple interpretations of the score.** A high score can indicates high synthetic data quality as well as low privacy. A low score can indicate low synthetic data quality as well as high privacy. 
{% endhint %}

## How does it work?

This metric fits multiple Gaussian mixture models \[1] to learn the distribution of the real data. The model learns to produce a likelihood estimate for every row ranging from -∞ to to +∞, where -∞ means the row is likely not part of the data and +∞ means that it is.

We apply the model to all the synthetic data and return the average likelihood score.

## Usage

Access this metric from the `single_table` module and use the `compute` method.

```python
from sdmetrics.single_table import GMLikelihood

GMLikelihood.compute(
    real_data=real_table,
    synthetic_data=synthetic_table
)
```

**Parameters**

* (required) `real_data`: A pandas.DataFrame containing the real data
* (required) `synthetic_data`: A pandas.DataFrame containing the same columns of synthetic data
* `metadata`: A metadata dictionary describing the columns (see [Metadata](/sdmetrics/getting-started/metadata.md))
* `n_components`: Number of components to use for the mixture model

<table data-header-hidden><thead><tr><th width="220"></th><th></th></tr></thead><tbody><tr><td>(default) <code>(1, 30)</code></td><td>Search for the optimal number of components between 1 and 30</td></tr><tr><td><code>(&#x3C;low integer>, &#x3C;high integer>)</code></td><td>Search for the optimal number of components between the low and high integer</td></tr><tr><td><code>&#x3C;integer></code></td><td>Use exactly the integer number of components provided</td></tr></tbody></table>

* `covariance_type`: A string describing the the covariance type to use for the mixture models. If multiple values are passed, the best one will be searched. Defaults to `'diag'`. See the [sklearn API](https://scikit-learn.org/stable/modules/generated/sklearn.mixture.GaussianMixture.html) for other possible values.
* `iterations`: Number of times that each number of components should be evaluated before averaging the scores. Defaults to 3.
* `retries`: Number of times that each iteration will be retried if the mixture model crashes during fit. Defaults to 3.

## FAQs

{% hint style="info" %}
**This metric is in Beta.** Be careful when using the metric and interpreting its score.

* The score heavily depends on algorithm used to model the data. If the overall distribution of the real data cannot be learned well, then the likelihood estimates of the synthetic data may not be valid.
* There are multiple interpretations for this metric. (See the [Score](#score) section above.) Of course, this is heavily dependent on how well we trust the algorithm to model the real data.
  {% endhint %}

## References

\[1] <https://en.wikipedia.org/wiki/Mixture_model>


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