# Outlier Likelihood

To detect out-of-distribution (OOD) samples in a multi-modal setting, we can leverage the alignment score $$s(\mathbf{x}, \mathbf{y})$$ between a data point $$\mathbf{x}$$ and its corresponding instruction $$\mathbf{y}$$. Since the Multi-modal alignment agents (e.g. Visual Language Models, Speech Language Models) are trained to embed semantically similar pairs close together, so a low alignment score reflects a divergence in semantic coherence between the two, indicating a possible OOD sample. 

We formalize this detection process within a probabilistic framework, defining the likelihood of a sample being OOD as follows:

$$
\mathcal{L}\_{\tau, k}(s(\mathbf{x}, \mathbf{y})) = 1 - \frac{1}{1 + \exp(-k(s(\mathbf{x}, \mathbf{y}) - \tau))}
$$

We then simply apply a threshold to assign an outlier flag:

$$
\mathbf{I\_{ood}}(\mathbf{x}\_i, \mathbf{x}*j) = \begin{cases} 1 & \text{if } \mathcal{L}*{\tau, k}(s(\mathbf{x}, \mathbf{y})) < \epsilon\_o \ 0 & \text{otherwise} \end{cases}
$$

<figure><img src="/files/6GN1MMA75r829QhKybMM" alt="" width="563"><figcaption><p>Outlier Score</p></figcaption></figure>

In practice, threshold values $$\tau, \epsilon\_o$$ are determined through precision-recall (PR) analysis on a development set, calibrating boundary control to maximize OOD detection accuracy. This approach enables precise, probabilistic separation of ID and OOD samples by aligning the boundaries with the semantic coherence objectives of the alignment model.


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