## 5 Mixture Density Network  
In this section, we present our algorithm for predicting the probability distribution of incompleteness ratios (RISs) using Neural Networks. The conventional neural network technique of minimization of sum-of-squared error or the cross-entropy error function tries to approximate the conditional average of the target data. Supervised machine learning models try to learn the mapping between input features and target values. For problems in which the objective is the predicting of continuous variables such as predicting the incompleteness score (RIS) of new data, the conditional average cannot describe the statistical properties of the target data well. The regression models can predict the continuous output. In most of the problems, we consider that the target values are following Gaussian distribution. However, working with real experimental data shows that most of the time this is not the case. If the distribution of data has multiple modalities, we cannot predict only the target values. A simple model can only learn the linear mapping between the input and output values. In order to avoid this issue in the case of non-Gaussian distribution, we should learn the distribution instead of directly mapping the input and output values. In our approach, we use a neural network model called Mixture Density Networks (MDNs) which overcomes this limitation and provides a suitable framework for modeling conditional density functions [32,33]. This model is a combination of a conventional neural network with a mixture density model and uses the conventional least-squares technique as well. MDNs can predict the expected value of a target and the probability distribution [30]. Assume that x is the input and f (x) is a deterministic output. In order to have a better prediction, we add some normally distributed noise to f (x). Let f (x) be f (x) with the noise added. Then, we train a simple neural network that learns f (x). In this paper, we consider x and f (x) as the incompleteness scores and their frequencies, respectively. Figure 12: Empirical CDF on the fitted beta distribution on Provisional Counts of Death Dataset. Figure 13: quantile–quantile plot for Provisional Counts of Death Dataset. The output layer of the network gives the fitted parameters of a normal distribution. Then, our model maximizes the values of the probability density function (PDF) of the normal distribution of the dataset. After training the network, we can obtain the predicted values. Figure 14 shows the network that we have designed. We use three hidden dense layers, each including 12 nodes. Table 4 shows the variables used in the network. The output layer has two nodes denoted by \(\mu\) and \(\sigma\) corresponding to the parameters of a normal distribution. We apply this method to the BCG Strains Data set. We consider the incompleteness ratios of the columns of the data set as the input. Our model predicts the frequency of any random incompleteness ratio. Figure 15a shows the fitted distribution and the predicted values for incompleteness ratios using a regular neural network. As it is shown in the figure, red dots show the predicted frequencies for each random incompleteness ratio. The new step is to predict not only the frequencies but the distribution of each frequency as well. For doing so, we apply MDN to the network which is shown in Figure 15b. The output layer will have two nodes rather than one corresponding to the mean and the variance of the distribution. We use the following loss function and follow a similar approach in [30], where \(\mu\) and \(\sigma\) are the mean and variance of the distribution, respectively. \[Cost=-\log(\frac{1}{\sqrt{2\pi\sigma}}.exp[-\frac{(y-\mu)^{2}}{2\sigma^{2}}])\]  
We should note that as the result of our experiment, with the current information that we have from the dataset, there is nothing to be learned from our neural network since we can easily obtain the expected values of incompleteness ratios computationally.