# Multimodal Shot Prediction Based on Spatial-Temporal Interaction between Players in Soccer Videos
## 2 Proposed Multimodal Shot Prediction in Soccer Videos
### Uncertainty-Based Event Prediction
#### 2.3.1 Construction with Bayesian Neural Networks  
The proposed model finally predicts the shot occurrence with BNNs. The Bayesian approach estimates the posterior distribution of the weights by integrating prior knowledge about the weights and information learned from the data. In this way, BNNs can calculate both model and data-dependent uncertainty about inference results and provide confidence in model decisions. In traditional evaluation methods [12; 16] in soccer, inference results calculated from deterministic machine learning models are used, yet users have to be cautious about its use due to output with unknown confidence levels. Especially, since soccer matches do not always follow the same patterns as training data, uncertainty-based analysis and evaluation are needed for performance improvement. Therefore, this study tackles such a problem by not using evaluations from inference results, but predicting based on BNNs. In the proposed method, given the latent feature \(\mathbf{\tilde{F}}_{t}\) of unknown data, the probability of a shot occurrence is calculated according to a predictive probability distribution [53]. This predictive probability distribution \(p(\mathbf{\tilde{y}}_{t}|\mathbf{\tilde{F}}_{t},\mathcal{D})\) is given as follows:  
\[p(\mathbf{\tilde{y}}_{t}|\mathbf{\tilde{F}}_{t},\mathcal{D})=\int p(\mathbf{\tilde{y}}_{t} |\mathbf{\tilde{F}}_{t},\mathbf{\theta})p(\mathbf{\theta}|\mathcal{D})d\mathbf{\theta}, \tag{10}\]  
where \(p(\mathbf{\tilde{y}}_{t}|\mathbf{\tilde{F}}_{t},\mathbf{\theta})\) is the conditional probability distribution of the event occurrence probability \(\mathbf{\tilde{y}}_{t}=(\mathbf{\tilde{y}}_{t}^{(n)},\mathbf{\tilde{y}}_{t}^{(p)})^{\top}\) given the model parameter \(\mathbf{\theta}\). Here, \(\mathbf{\tilde{y}}_{t}^{(n)}\) and \(\mathbf{\tilde{y}}_{t}^{(p)}\) are the negative and positive probability predictions of the unknown data at time step \(t\), respectively. \(p(\mathbf{\theta}|\mathcal{D})\) is the posterior probability distribution of \(\mathbf{\theta}\) given the training data \(\mathcal{D}\) including \(V\) soccer videos as follows:  
\[\mathcal{D}=(\mathcal{D}_{x},\mathcal{D}_{y})=(\{\mathbf{F}_{t,1},\dots,\mathbf{F}_{t,V}\},\{\mathbf{y}_{t,1},\dots,\mathbf{y}_{t,V}\}), \tag{11}\]  
where \(\mathcal{D}_{x}\) and \(\mathcal{D}_{y}\) are the input features and target labels of the training data, respectively. The estimation of \(p(\mathbf{\theta}|\mathcal{D})\) based on [PERSON] theorem is calculated from the following equation:  
\[p(\mathbf{\theta}|\mathcal{D})=\frac{p(\mathcal{D}_{y}|\mathcal{D}_{x},\mathbf{\theta })p(\mathbf{\theta})}{\int_{\theta}p(\mathcal{D}_{y}|\mathcal{D}_{x},\mathbf{\theta})p (\mathbf{\theta})d\mathbf{\theta}}. \tag{12}\]  
Since neural networks have large number of parameters and non-convex probability distributions due to their complexity [54], it is difficult to compute the marginal likelihood \(\int_{\theta}p(\mathcal{D}_{y}|\mathcal{D}_{x},\mathbf{\theta})p(\mathbf{\theta})d\bm {\theta}\). To handle this problem, it is common to avoid direct computation of the marginal likelihood and instead use approximation techniques such as Markov chain Monte Carlo (MCMC) and variational inference (VI). Since MCMC is also computationally expensive and appropriate sampling becomes challenging in high-dimensional spaces, the proposed method adopts Bayes by Backprop [55], which is one of the VI methods.