Revealing the Proximate Long-Tail Distribution in Compositional Zero-Shot Learning

Considering the two disjoint sets $\mathcal{Y}^{S}$ and $\mathcal{Y}^{U}$, i.e., $\mathcal{Y}^{S}\cap\mathcal{Y}^{U}=\O$. CZSL aims to classify sample $\mathbf{x}\in\mathcal{X}$ into a composition $y=(s,o)\in\mathcal{Y}$, where $\mathcal{Y}=\mathcal{Y}^{S}\cup\mathcal{Y}^{U}$, and samples from $\mathcal{Y}^{U}$ are unseen during training. $y$ is composed by state $s\in\mathcal{S}$ and object $o\in\mathcal{O}$, $\mathcal{S}$ and $\mathcal{O}$ are sets of states and objects. Samples from $\mathcal{Y}^{S}$ and $\mathcal{Y}^{U}$ share the same objects $o$ and states $s$, but their compositions $(s,o)$ are different. Define the visual space $\mathcal{X}\subseteq\mathbb{R}^{d_{x}}$ and $d_{x}$ is the dimension of the space, $\mathcal{X}$ can be divided into $\mathcal{X}^{S}$ and $\mathcal{X}^{U}$ based on whether their samples belong to seen classes. We can define the train set as $\mathcal{D}_{seen}=\{(\mathbf{x},y)|\mathbf{x}\in\mathcal{X}^{S},y\in\mathcal{Y}^{S}\}$ and an unseen set for evaluation of methods which is $\mathcal{D}_{unseen}=\{(\mathbf{x},y)|\mathbf{x}\ \in\mathcal{X}^{U},y\in\mathcal{Y}^{U}\}$. We employ the Generalized ZSL setup defined in Xian, Schiele, and Akata (2017), which requires both seen and unseen classes involves in testing.

For the problem of approximate long-tailed distributions caused by visual bias in CZSL, ensemble-based methods have demonstrated exceptional performance in CZSL (Saini, Pham, and Shrivastava 2022; Wang et al. 2023). Typically, this approach combines the predictions of two models to produce the final prediction. The first model consists of two independent classifiers $C_{o}$ and $C_{s}$ for objects and states. The second model is a composition classifier $C_{y}$. The process of the model can be viewed as inputting the samples into three classifiers to estimate the posterior probabilities:

$$\begin{split}\begin{aligned} &p(s|\mathbf{x})=\operatorname{softmax}[C_{s}(\mathbf{x})],p(o|\mathbf{x})=\operatorname{softmax}[C_{o}(\mathbf{x})],\\ &p(y|\mathbf{x})=\operatorname{softmax}[C_{y}(\mathbf{x})],\\ &\hat{p}(y|\mathbf{x})=\delta p(y|\mathbf{x})+(1-\delta)\left[p(s|\mathbf{x})+p(o|\mathbf{x})\right],\end{aligned}\end{split}$$

(1)

where $p(s|\mathbf{x}),p(o|\mathbf{x})$ and $p(y|\mathbf{x})$ are posterior probability from classifiers, $\hat{p}(y|\mathbf{x})$ is the final posterior probabilities. $\delta$ is a weight factor. $\mathcal{C}_{y}(\mathbf{x})$ denotes the logits for class $y$ based on sample $\mathbf{x}$, and $\mathcal{C}_{s}(\mathbf{x})$, $\mathcal{C}_{o}(\mathbf{x})$ are similarly defined.

As demonstrated in Tab. 1, augmenting two additional posterior estimates $p(o|\mathbf{x})$ and $p(s|\mathbf{x})$ to $p(y|\mathbf{x})$ can significantly enhance CZSL results. However, only relying solely on $p(o|\mathbf{x})$ and $p(s|\mathbf{x})$ does not enable accurate estimation, this suggests the improvement in results is not due to the introduction of superior classifiers. Consequently, we can deduce the subsequent conjectures: The effectiveness of ensemble-based methods emanates from incorporating $\mathcal{C}_{s}$ and $\mathcal{C}_{o}$, aiding in the classification of compositions that encounter a relative disadvantage within $\mathcal{C}_{y}$. While attribute imbalances vary across states, objects, and compositions, all three elements might not simultaneously experience large visual bias for a particular class. Based on these preliminary studies, we can posit that effective classification of classes with large visual bias within common embedding spaces requires information compensation. In our study, we directly estimate visual bias as compensation described above. Considering that the visual bias generated by state-object combination is difficult to eliminate directly, we try to introduce it as an attribute prior into the training process from the classifier. In the following, we detail this process.

Let us first consider the problem from a simple CZSL approach based on common embedding spaces like (Mancini et al. 2021; Naeem et al. 2021). The optimization objective of these methods can be viewed as the maximum likelihood:

$$\operatorname{argmin}_{\theta}\mathbb{E}_{(\mathbf{x},y)\sim\mathcal{D}_{seen}}[-logp(y|\mathbf{x})],$$

(2)

where $p(y|\mathbf{x})$ is defined in Eq. 1, which denotes the distribution of compositions predicted by the model. $\theta$ denotes the model parameters.

Given the characteristics of CZSL, where each sample is associated with two labels, $s$ and $o$, there is a conditionality between the two in the setup of the dataset, i.e.,

$$p(y)=p(s,o)=p(o)p(s|o),$$

(3)

$p(y)$ and $p(o)$ denotes class prior of class $y$ and object $o$, and $p(s|o)$ is conditional class prior of $s$ and $o$.

Inspired by Su (2020), we look at the above issues through the perspective of mutual information (Kraskov, Stögbauer, and Grassberger 2004), we have:

$$\begin{split}&I(Y;X)\approx\mathbb{E}_{y}D_{KL}[p(y|\mathbf{x})\|p(y)]\\ =&\sum_{\mathbf{x},y}p(\mathbf{x},y)log\frac{p(y|\mathbf{x})}{p(y)}\\ =&\sum_{\mathbf{x},y}p(\mathbf{x},y)log\frac{p(y|\mathbf{x})}{p(o)p(s|o)},\end{split}$$

(4)

where $X$ and $Y$ are discrete random variables corresponding to x and $y$, respectively, and $S$ and $O$ are similarly defined. $D_{KL}$ represents the Kullback-Leibler divergence, while $p(\mathbf{x},y)$ represents the joint probability of the class $y$ and the visual feature x. Due to the real posterior probability between $y$ and $\mathbf{x}$ is unknown, we use $p(y|\mathbf{x})$ as an approximation. We can interpret the optimization of maximum likelihood as follows, based on the posterior term in Eq. 4,

$$log\frac{p(y|\mathbf{x})}{p(o)p(s|o)}\sim\mathcal{C}_{y}(\mathbf{x}),$$

(5)

which can be transfer to:

$$\begin{split}logp(y|\mathbf{x})\sim\mathcal{C}_{y}(\mathbf{x})+log{p(o)p(s|o)},\end{split}$$

(6)

here, $\mathcal{C}_{y}(\mathbf{x})$ represents the logits for class $y$, defined in Eq. 1, $\sim$ denotes approximately equal. The expression on the right-hand side is re-normalized using the $\operatorname{softmax}$ function, i.e.,

$\begin{split}&-logp(y|\mathbf{x})\\ &\sim log\left[1+\sum_{o_{i}\neq o}\sum_{s_{j}\neq s}\frac{p(o_{i})p(s_{j}|o_{i})}{p(o)p(s|o)}e^{\mathcal{C}_{\hat{y}}(\mathbf{x})-\mathcal{C}_{y}(\mathbf{x})}\right]\\ &\sim log\left[1+\sum_{o_{i}\neq o}\sum_{s_{j}\neq s}\left(\frac{p(o_{i})p(s_{j}|o_{i})}{p(o)p(s|o)}\right)^{\eta}e^{\mathcal{C}_{\hat{y}}(\mathbf{x})-\mathcal{C}_{y}(\mathbf{x})}\right],\end{split}$

(7)

where $\hat{y}=(s_{i},o_{i})$, and $\eta$ is an adjustment factor. Eq. 7 demonstrates that by incorporating the class prior $p(s|o)$ and $p(o)$ for state $s$ and object $o$, we can optimize the model’s mutual information. Consequently, we approach the CZSL problem from the perspective of mutual information.

The above idea comes from the logits adjustment (Menon et al. 2020) introduced to address class imbalance (Johnson and Khoshgoftaar 2019; Japkowicz and Stephen 2002), which demonstrate that the inclusion of a class prior enhances the maximization of mutual information, and we generalize it to CZSL task.

As stated in Introduction, we undertake the transformation of CZSL into an approximate long-tailed distribution issue caused by visual bias from state-object combinations. Our argument centers on the proposition that attribute imbalance within CZSL contributes to an approximate form of class imbalance, since visual bias hinders reduces the distinguishability of some of the samples. Therefore, exclusive reliance on the class prior is inadequate. Building upon this rationale, we propose to use the attribute prior to assume the function of the class prior within the long-tailed distribution, serving as an approximation.

We propose incorporating the model’s conditional posterior probabilities as an approximation for this scenario. We continue to denote it as the ‘prior’ due to its function as a prior probability during the training process, despite being computed using posterior probability. Since attribute imbalance cannot be directly quantified from the dataset, we simulate it by utilizing the posterior probability of the additional classifiers, for $\mathbf{x}$ and its corresponding $s,o$, we have:

$$\begin{split}\hat{p}(s)=\mathbb{E}_{\mathbf{x}\sim p(\mathbf{x})}[p(s|\mathbf{x})],\hat{p}(o)=\mathbb{E}_{\mathbf{x}\sim p(\mathbf{x})}[p(o|\mathbf{x})],\end{split}$$

(8)

where $\mathbf{x}\in\mathcal{D}_{seen}$, $p(s|\mathbf{x})$ and $p(o|\mathbf{x})$ are defined in Eq. 1, which are posterior probabilities from $\mathcal{C}_{s}$ and $\mathcal{C}_{o}$, we use their predicted expectations for all training samples as a special attribute prior. From this we can replace the class prior in Eq. 7 with following item:

$$k(s,o)=\operatorname{softmax}\left[\sigma(s,o)\hat{p}(s)\hat{p}(o)\right],$$

(9)

where $\sigma(s,o)$ is a function used to model the conditional nature of the composition, i.e.,

$$\sigma(s,o)=\left\{\begin{array}[]{rcl}&1&(s,o)\in\mathcal{Y}^{S}\cup\mathcal{Y}^{U},\\ &0&else.\\ \end{array}\right.$$

(10)

From this we obtain the final objective function according to Eq. 7:

$$\mathcal{L}_{cls}=log\left[1+\sum_{o_{i}\neq o}\sum_{s_{j}\neq s}\left(\frac{k(s_{j},o_{i})}{k(s,o)}\right)^{\eta}e^{\mathcal{C}_{\hat{y}}(\mathbf{x})-\mathcal{C}_{y}(\mathbf{x})}\right].$$

(11)

Due to the introduction of unseen classes in the inference phase we need to make additional adjustments. CZSL usually measures model performance in terms of $\mathcal{A}^{H}$ which denotes Harmonic Mean (HM) accuracy:

$$\mathcal{A}^{H}=2/(\frac{1}{\mathcal{A}^{S}}+\frac{1}{\mathcal{A}^{U}}),$$

(12)

where $\mathcal{A}^{S}$, $\mathcal{A}^{U}$ denote seen and unseen accuracy. Chen et al. (2022) provides a lower bound of HM, below we briefly describe its conclusions. For HM’s lower bound we have:

$$\mathcal{A}^{H}\geq 1/\mathbb{E}_{\mathbf{x}\sim p(\mathbf{x})}\frac{|\mathcal{Y}|p(\mathcal{Y})p(y|y\in\mathcal{Y})}{q(\mathcal{C}_{out}=y|\mathbf{x})p(y|\mathbf{x})},$$

(13)

where $q(\mathcal{C}_{out}=y|\mathbf{x})$ represents the probability of predicting class $y$ using our model. The set $\mathcal{Y}$ can be either $\mathcal{Y}^{S}$ or $\mathcal{Y}^{U}$, $p(y|y\in\mathcal{Y})$ represents the conditional class prior, and $|\mathcal{Y}|p(\mathcal{Y})$ can be seen as a hyper-parameter that quantifies the differences between seen and unseen classes. Considering that the gap between the domains of seen and unseen classes in CZSL is not significant, we can simply treat $|\mathcal{Y}|p(\mathcal{Y})$ as an ignorable constant in the following process.

Finding the Bayesian optimum for $\mathcal{A}^{H}$ is difficult. However, it is possible to maximize its lower bound, which is equal to minimizing the upper bound of its inverse, i.e., the denominator term of Eq. 13 is minimized if:

$$\tilde{y}=\operatorname{argmax}_{y}\left[\mathcal{C}_{y}(\mathbf{x})+\eta logp(y|y\in\mathcal{Y})\right],$$

(14)

where $\eta$ is from Eq. 11, $\tilde{y}$ is the predicted label for sample $\mathbf{x}$. For conditional class prior $p(y|y\in\mathcal{Y}^{S})$, which represents the true class frequency when $y$ belong to seen classes. Following Eq. 11, we similarly replace the prior with the attribute prior estimate in Eq. 9 here, which is:

$$p(y|(s,o)\in\mathcal{Y}^{S}):=k(s,o),\quad(s,o)\in\mathcal{Y}^{S},$$

(15)

and the attribute prior of unseen classes are not available to the model, we model it here using a combination of the estimation from Importance Sampling (Neal 2001) with the attribute prior from seen samples, which can be denoted as:

$$p(y|y\in\mathcal{Y}^{U}):=k(s,o)+\frac{\hat{k}_{\mathbf{x}}(s,o)}{\lambda k(s,o)},\quad(s,o)\in\mathcal{Y}^{U},$$

(16)

where $\frac{1}{\lambda}$ is a hyper-parameter denotes the distribution of $\mathbf{x}$. The above results are re-transformed into probability distributions in the actual calculation. And $\hat{k}_{\mathbf{x}}(s,o)$ is instance-based conditional posterior probability:

$$\hat{k}_{\mathbf{x}}(s,o)=\operatorname{softmax}\left[\sigma(s,o)p(s|\mathbf{x})p(o|\mathbf{x})\right],$$

(17)

the aforementioned setup arises because during testing, we are unable to provide posterior probabilities $\hat{k}_{\mathbf{x}}(s,o)$ from multiple samples simultaneously. Furthermore, Importance Sampling results in significant variance when the number of samples is insufficient. To address this, we attempt to augment it by leveraging seen attribute prior.

ProLT makes inferences during testing phase based on Eq. 14, our aim is to integrate local information during testing with the prior derived from seen classes, to address the disparities between seen and unseen classes. With Eq. 14 ProLT theoretically achieves the best overall accuracy.

This section provides a concise summary of the aforementioned methods. Our approach, illustrated in Fig. 2, involves training two independent classifiers denoted as $\mathcal{C}_{s}$ and $\mathcal{C}_{o}$. These classifiers are implemented using prototype learners, namely $\mathcal{P}_{s}$ and $\mathcal{P}_{o}$, and visual embedders $\mathcal{V}_{o}$, $\mathcal{V}_{s}$, to determine the prototypes of states and objects, i.e.,

$$\mathcal{C}_{s}(\mathbf{x}):=\frac{cos(\mathcal{V}_{s}(\mathbf{x}),\mathcal{P}_{s}(s))}{\tau},\mathcal{C}_{o}(\mathbf{x}):=\frac{cos(\mathcal{V}_{o}(\mathbf{x}),\mathcal{P}_{o}(o))}{\tau},$$

(18)

where $\tau$ is the temperature. These classifiers are trained with vanilla cross-entropy loss:

$$\mathcal{L}_{ic}=log[1+\sum_{s^{\prime}\neq s}e^{\mathcal{C}_{s^{\prime}}(\mathbf{x})-\mathcal{C}_{s}(\mathbf{x})}][1+\sum_{o^{\prime}\neq o}e^{\mathcal{C}_{o^{\prime}}(\mathbf{x})-\mathcal{C}_{o}(\mathbf{x})}].$$

(19)

Once the classifiers reach a specific training stage, we calculate the attribute prior using Eq. 8, and employ the loss function $\mathcal{L}_{cls}$ from Eq. 11 for training the classifier $\mathcal{C}_{y}$ for compositions:

$$\mathcal{C}_{y}(\mathbf{x}):=\frac{cos(\mathcal{V}_{y}(\mathbf{x}),\mathcal{P}_{y}(y))}{\tau},$$

(20)

where $\mathcal{P}_{y}$ is the prototype learner for compositions and $\mathcal{V}_{y}$ is a visual embedder. After training, the model uses Eq. 14 for inference.

There are numerous recent approaches to compositionality research, and three datasets have been primarily employed for evaluation: MIT-States (Isola, Lim, and Adelson 2015), UT-Zappos (Yu and Grauman 2014), and C-GQA (Naeem et al. 2021). We utilized a standardized evaluation dataset for a reasonable comparison with previous methods.

MIT-States presents a considerable challenge, consists of 53,753 images. It comprises 115 state classes, 245 object classes, and 1,962 compositions. In the total compositions, there are 1,262 seen compositions, and 700 compositions remain unseen. UT-Zappos is a collection of 50,025 images that focuses on various forms of footwear. It consists of 12 object classes and 16 state classes which is a fine-grained dataset, yielding 116 compositions, of which 83 are seen. C-GQA is introduced by Naeem et al. (2021), which encompasses a wide variety of real-world common objects. It comprises 413 states, 674 objects, and over 27,000 images, along with more than 9,000 compositions, consisting of 5,592 seen and 1,932 unseen compositions.

The setting of GZSL (Xian, Schiele, and Akata 2017) requires both seen and unseen compositions during testing. We report the best accuracy of seen classes (best seen), the unseen class (best unseen), and its harmonic accuracy (HM). In order to measure the performance on attribute learning, we report the best accuracy of states (best sta) and objects (best obj). Building upon the research of (Naeem et al. 2021) and (Wang et al. 2019), we calculate the Area Under the Curve (AUC) by comparing the accuracy on seen and unseen compositions with various bias terms.

Below we present the details of the implementation of ProLT on ResNet-18 (He et al. 2016).

ProLT is mainly compared with recent methods using fixed ResNet-18 as backbone with the same settings. We also compared ProLT with the CLIP-based approaches (Lu et al. 2023; Nayak, Yu, and Bach 2022) after using CLIP (Radford et al. 2021) to learn visual and semantic embeddings. The comparison results are shown in Tab. 2.

The results demonstrate that ProLT achieves a new state-of-the-art performance when using ResNet-18 as backbone on the MIT-States, UT-Zappos, and C-GQA. Specifically, our method achieves the highest AUC of $6.0\%$ on MIT-States, surpassing CANet by $0.6\%$. On the UT-Zappos, we achieve the highest HM of $49.3\%$, outperforming CANet by $2.0\%$. Although ProLT has a slight disadvantage on the C-GQA dataset, it remains competitive with the state-of-the-art methods, achieving an HM of $14.4\%$. As for the CLIP-based approaches, ProLT has produced remarkable outcomes. Unlike DFSP, our method avoids the incorporation of extra self-attention or cross-attention mechanisms. Despite this, we excel across all three datasets, attaining an HM of $38.2\%$ on MIT-States and $49.4\%$ on UT-Zappos. These results underscore the compatibility of ProLT when combined with CLIP.

In this section, we verify that each of these modules plays an active role by ablating each of its parts on UT-Zappos with ResNet-18. The results are shown in Tab. 3 and Tab. 4.

Our method primarily comprises the subsequent hyper-parameters: 1) logit-adjusting factor ($\eta$), and 2) factor about the sample distribution ($\lambda$). We test on the UT-Zappos under various hyper-parameters based on ResNet-18, shown in Fig. 4. For $\eta$, the best AUC are observed when $\eta=1.6$, and the gap between seen and unseen begins to decrease as $\eta$ increases. Concerning $\lambda$, the outcomes are documented within the interval $\lambda\in[1.0,50.0]$ with increments about 5.0. The pinnacle value for the seen class is observed at $20.0$, and $35.0$ for unseen class. Overall, these hyper-parameter settings yield results characterized by minimal fluctuations, thus underscoring the robustness of our methodology.

Qualitative results for unseen compositions, accompanied by the top-3 predictions when we use ResNet-18 as backbone, are displayed in Fig. 3. Concerning MIT-States, we argue that certain erroneous predictions as partially justifiable. For instance, the phrase tiny dog, for which the model’s incorrect predictions involve small dog and tiny animal, exhibits a high degree of semantic similarity. A similar phenomenon can be observed for the brown chair in C-GQA. For UT-Zappos, ProLT’s limitation in fine-grained classification persists. An illustrative example is the outcomes for leather boot.M, our approach encounters challenges in making nuanced differentiations within the category of boots.

In this section, we provide details of the setup of ProLT when using CLIP to learn visual and semantic embeddings.

Following Sec. 4.5, we conducted an identical experiment on CLIP to validate the effectiveness of ProLT. As demonstrated in Tab. A.5, we compare the outcomes on UT-Zappos (Yu and Grauman 2014) under three scenarios: without incorporating any priors but using a model-ensemble method, with the inclusion of class priors, and with the inclusion of attribute priors. Similarly, the results demonstrate the beneficial impact of incorporating the attribute prior. In comparison to the direct utilization of the class prior, our approach leads to a rise of $1.9\%$ in AUC and $2.0\%$ in HM.

Moreover, we conduct a comparative analysis by removing the attribute prior during both the training and testing phases. Referencing Tab. A.6, when $\eta=0$, indicating our methods is changed to a simple common embedding space method like Naeem et al. (2021), which led to a significant drop in results. A significant enhancement is observed when these are combined, similar to the findings in Tab. 4. Collectively, the aforementioned experiments substantiate the favorable impact of ProLT on CLIP.

This section delves into the analysis of the effects arising from the subsequent hyper-parameter configurations: 1) temperature $\tau$, 2) hidden layer dimensions, and 3) word embeddings. Illustrated in Fig. A.5, we present the outcomes achieved across various $\tau$ values on UT-Zappos. The peak AUC emerges at $\tau=0.14$ and the difference between seen and unseen is minimized at $\tau=0.1$. Likewise, we present outcomes utilizing diverse word embeddings on UT-Zappos, detailed in Tab. A.7. ProLT excels when employing GloVe, yet generally, variations in word vectors exhibit minimal impact. Concerning the varying dimension configurations, we document the outcomes obtained using dimensions ${256,512,1024,2048,4096}$ for the hidden layers in three classifiers, as indicated in Tab. A.8. Notably, we discern that a hidden layer dimension of 1024 consistently yields optimal results. However, when employing dimensions of 2048 or 4096, we posit that the inferior performance could result from the propensity of higher-dimensional hidden layers to manifest overfitting on seen classes.

Eq. 16 employs Importance Sampling to estimate the attribute prior for unseen classes. The specifics of this approach are outlined in this section. During this procedure, we introduce an auxiliary proposal distribution to aid in creating an approximate estimation, i.e., the distribution of the seen attribute prior $k(s,o)$. Therefore, the estimation of the prior for unseen classes can be represented as follows:

$$\frac{1}{n}\sum_{i}^{n}\frac{p(\mathbf{x}_{i})\hat{k}_{\mathbf{x}}(s,o)}{k(s,o)}.$$

(A.21)

In Eq. 16, $p(\mathbf{x}_{i})$ is replaced by $\lambda$, which is a hyper-parameter. This is owing to the unavailability of direct access to the data distribution for the test set. As the posterior can be obtained only for individual samples during testing, we set $n$ to 1 in practice. This approach yields significant variance due to an inadequate sample size. Consequently, we posit that it should be integrated with the another information.

In this section, we validate the effectiveness of the approach on UT-Zappos that combining the seen attribute prior with Importance Sampling, as detailed in Sec. 3.5. Tab. A.9 presents the outcomes where we nullify the probability estimated by Importance Sampling via setting $\lambda\rightarrow\infty$ in Eq. 16, and the results when we set the seen attribute prior to $0$. It is worth noting that with seen attribute prior set to 0, we generalize the probability of Importance Sampling to the seen class for consistency. We conducted experiments on two embedding functions to ensure robustness following Tab. 3. From the table, we can observe that the introduction of the two respectively brings about an improvement in AUC, HM relative to the baseline, while it is not significant in the rest of the metrics. In addition, the combination of the two usually creates complementarities, suggesting that they are not mutually exclusive.

As shown in Fig. A.6, we display the adjusted posterior $p(y|\mathbf{x})$ for the same compositions in Fig. 1. It is evident that ProLT yields a more balanced distribution stemming from $\mathcal{C}_{y}$. Furthermore, it becomes apparent that $\mathcal{C}_{y}$ exhibits a preference for compositions characterized by significant visual bias in Fig. 1. But the incorporation of the prior, as defined in Eq. 14, can mitigates this distinction.

This aspect reflects ProLT: enhancing the model’s grasp of fundamental visual-semantic relationships during the training phase through the maximization of mutual information. Utilizing posterior probability adjustment, ProLT achieves classification by harmonizing both a prior and a posterior during the inference phase.