# Efficient Estimation of Generative Models Using Tukey Depth
###### Abstract  
Generative models have recently received a lot of attention. However, a challenge with such models is that it is usually not possible to compute the likelihood function, which makes parameter estimation or training of the models challenging. The most commonly used alternative strategy is called likelihood-free estimation, based on finding values of the model parameters such that a set of selected statistics have similar values in the dataset and in samples generated from the model. However, a challenge is how to select statistics that are efficient in estimating unknown parameters. The most commonly used statistics are the mean vector, variances, and correlations between variables, but they may be less relevant in estimating the unknown parameters. We suggest utilizing Tukey depth contours (TDCs) as statistics in likelihood-free estimation. TDCs are highly flexible and can capture almost any property of multivariate data, in addition, they seem to be as of yet unexplored for likelihood-free estimation. We demonstrate that TDC statistics are able to estimate the unknown parameters more efficiently than mean, variance, and correlation in likelihood-free estimation. We further apply the TDC statistics to estimate the properties of requests to a computer system, demonstrating their real-life applicability. The suggested method is able to efficiently find the unknown parameters of the request distribution and quantify the estimation uncertainty. generative models; Tukey depth; likelihood-free estimation; computer resource management 20242  
to use a classifier to compare data and generated samples [6]. The level of confusion when the classifier tries to separate data and generated samples is used as a measure for the similarity between the generated samples and the data. The classifier must be trained in tandem with the generator, using so-called adversarial training, making the training procedure challenging. GANs are usually used for data of high dimensions, such as images. For data of lower dimensions, the most common approach is to use the moments of the data (mean and variance) and dependency between the variables in the form of the correlation/covariance matrix as statistics [7]. However, a challenge is that there might be properties of the data that are not efficiently captured by these statistics, such as asymmetries, and that can be important for the efficient estimation of the unknown parameters. However, it is not obvious how to select a set of statistics that are sufficiently flexible to capture almost any property of multivariate data. To address these issues, in this paper, we suggest using so-called Tukey depth contours (TDCs) [8] as statistics to summarise the properties of multivariate data and generated samples. To the best of our knowledge, TDCs have never been used in likelihood-free estimation. Tukey depth contours are highly flexible and can capture almost any property of multivariate data, and have proven efficient to detect events in complex multivariate data [9]. This paper demonstrates that TDCs are also useful in likelihood-free estimation. Tukey depth falls within the field of data depth. Data depth measures how deep an arbitrary point is positioned in a dataset. The concept has received a lot of attention in the statistical literature since [PERSON] first introduced the idea of Tukey depth and illustrated its application in ordering/ranking multivariate data [10]. It is also worth mentioning that there are many other variations of data depth introduced and investigated in the literature, notably Mahalanobis depth [11], convex hull peeling depth [12], Oja depth [13], simplicial depth [14], zonoid depth [15] and spatial depth [16]. Data depth has proven to offer powerful tools in both descriptive statistical problems, such as data visualisation [17; 18; 19; 20] and quality control [21], and in inferential ones, such as outlier identification [22; 23; 24], estimation [25; 26] and hypothesis testing [19; 27; 28; 29].