Bidirectional Generative Retrieval with Multi-Modal LLMs for Text-Video Retrieval

3. Mathematical analysis on Prior Normalization and more references.

   We focus on searching the most relevant video given a text query, i.e., text-to-video retrieval by $P(**v**|**t**)$, which can be written as:

   $$$$

$

P(\mathbf{v}|\mathbf{t})= P(\mathbf{t}|\mathbf{v})P(\mathbf{v})/P(**t**).\tag{Eq. (1)}

$

$$

As discussed in Tab. 1 of the main paper and Fig. 6 in the supplement, query generation (or likelihood) $P(**t** | **v**)$  is more accurate than content generation $P(**v**|**t**)$. In other words, even if $P(**t**| **v**)$ predicts accurately, the final result $P(**v**|**t**)$ can lead to incorrect retrievals due to a highly skewed video prior probability $P(\mathbf{v})$, namely, "**prior bias**".

We will discuss the case where a too large prior gap in $P(\mathbf{v})$ leads to incorrect retrieval results in the posterior $P(\mathbf{v}|\mathbf{t})$ when the likelihood $P(\mathbf{t}|\mathbf{v})$ is correct but the gap is relatively small. Specifically, we first assume that the likelihood predicts correctly but the prior gap is too larger than the likelihood gap. Then we prove that the posterior probability results in a wrong prediction. 

Assume that the likelihood predicts correctly, *i.e.*, $P(\mathbf{t}_m|\mathbf{v}_m) > P(\mathbf{t}_m|\mathbf{v}_n)>0, \forall n \neq m$ but the gap is small. Then, it can be written as:

$$

$

0 < \log P(\mathbf{t}_m|\mathbf{v}_m) - \log P(\mathbf{t}_m|\mathbf{v}_n) < \varepsilon, \tag{Eq. (2)}

$

$$

where the $\varepsilon > 0$ is the maximum gap. Eq. (2) means the retrieval score of $m$-th video (positive sample) is larger than the one of $n$-th video (negative sample) given the $m$-th text, and the gap is smaller than the maximum gap $\varepsilon$.

Another assumption is that the strong prior $P(\mathbf{v}_n)$ over video candidates as:

$$

$

\log P(\mathbf{v}_n) - \log P(\mathbf{v}_m) > c\varepsilon \tag{Eq. (3)}

$

$$

where $c > 1$. In other words, the prior gap is larger than the likelihood gap. 

Then, the posterior gap, which is the gap of retrieval scores measured by the posterior probability, is given as:

$$

$

&\log P(\mathbf{v}_m|\mathbf{t}_m) - \log P(\mathbf{v}_n|\mathbf{t}_m) &\\\\
&\\;= \log P(\mathbf{t}_m|\mathbf{v}_m) + \log P(\mathbf{v}_m) - \log P(\mathbf{t}_m|\mathbf{v}_n) - \log P(\mathbf{v}_n) &\text{(by Eq. (1))} \\\\ 
& \\;\\;< \varepsilon + \log P(\mathbf{v}_m) - \log P(\mathbf{v}_n) & \text{(by Eq. (2))} \\\\
& \\;\\; < \varepsilon - c\varepsilon = \varepsilon (1-c) & \text{(by Eq. (3))} \\\\
& \\;\\;< 0 &\text{(by c >1)} 

$

$$
This concludes the proof. "$P(\mathbf{v}_m|\mathbf{t}_m) < P(\mathbf{v}_n|\mathbf{t}_m)$" indicates that the posterior probability $P(\mathbf{v}|\mathbf{t})$ retrieves a wrong $n$-th video $\mathbf{v}_n$ given $m$-th query text $\mathbf{t}_m$ due to the too strong prior $P(\mathbf{v})$. Our Prior Normalization effectively reduces the impact of prior $P(\mathbf{v})$ and improves the retrieval performance. 


Thank you for suggesting references. This paper is the first work studying prior normalization in the context of text-video retrieval with MLLMs by interpreting the likelihood of a single modality as the prior for calculating the joint probability. As suggested, we will provide a broader view of Prior Normalization that adjusts the strength of the prior distribution including the Bayesian approach [1, 2] or maximum a posteriori (MAP) [3, 4]. For example, [1] proposed an elegant prior normalization technique that transforms non-Gaussian prior into standard Gaussian distributions, enabling LIS-based MCMC sampling for high-dimensional Bayesian inverse problems. We have updated our manuscript by including more references.

[1] Prior normalization for certified likelihood-informed subspace detection of Bayesian inverse problems

[2] Normalized Power Prior Bayesian Analysis

[3] Marginal maximum a posteriori estimation using Markov chain Monte Carlo

[4] Maximum a posteriori probability estimates in infinite-dimensional Bayesian inverse problem

We appreciate Reviewer ESdv for valuable feedback and constructive comments. We will address all the concerns raised by the reviewer, hoping for more vigorous support for our paper.

1. Weak connection between Prior Normalization and Bidirectional Generative Retrieval.

   For the first time in the context of text-video retrieval with MLLMs, we observe that the model often fails to retrieve the most relevant content given the query due to prior bias where the posterior is overwhelmed by the prior of the content. To alleviate the prior bias, we propose both Bidirectional Generative Retrieval (BGR) and Prior Normalization. Specifically, we apply Prior Normalization with respect to each direction in BGR, i.e., content-generation and query-generation. As a result, alleviating the prior bias in both generative directions leads to outstanding performance across all the text-video retrieval benchmarks.

2. Clarify the query-generation model.

   The text-video retrieval task assumes the content candidates are given and requires the model to find optimal content among the content candidates. Both the content-generation and query-generation models are used to find optimal content among the content candidates. In the content-generation model, the optimal content with the maximum likelihood is chosen by replacing the content candidates one by one in the output of the model, which is most likely to be generated by the query. On the other hand, in the query-generation model, the optimal content with the maximum likelihood is chosen by replacing the content candidates one by one in the input of the model, which is most likely to generate the query. The overall inference procedure is illustrated in Fig. 5 of the supplement. We also have updated Fig. 1 of the main paper and the related description by additionally illustrating content candidates.

Clarification on the query-generation model and its computational cost.

Thank you for identifying the novelty of our work. We believe that such novelty could be seen as a strength rather than a basis for rejection. Our proposed query-generation model does not lead to tremendous computational costs compared to conventional text-video retrieval models. Conventional text-video retrieval models (e.g., UMT [1], InternVideo [2]) have adopted video-text matching (VTM) loss which estimates joint probability $P(\mathbf{t}, \mathbf{v})$ to measure the relevance score between text and video. On the other hand, in the context of generative retrieval, we propose a content-generation model to estimate the posterior $P(\mathbf{v}|\mathbf{t})$ and a query-generation model to estimate the likelihood $P(\mathbf{t}|\mathbf{v})$ in text-to-video retrieval. The computational cost of the query-generation model is similar to that of the conventional VTM approach, differing only in how the relevance score between text and video is measured. The table below also demonstrates that our query-generation model surpasses the VTM by 2.1 in R@1 despite the comparable inference speed on DiDeMo. We have updated the relevant discussion in Sec. G of the supplement.

[1] Unmasked teacher: Towards training-efficient video foundation models

[2] InternVideo: Video Foundation Models for Multimodal Understanding

We appreciate Reviewer Kmgg for valuable feedback and constructive comments. We will address all the concerns raised by the reviewer, hoping for more vigorous support for our paper.

1. Questions on definitions.

   We provide the definition of the conditional probability $P_\text{vtg}(\mathbf{t}|\mathbf{v})$ in Eq. (3) and $P_\text{tvg}(\mathbf{v}|\mathbf{t})$ in Eq. (5) of the main paper. Specifically, $P_\text{vtg}(\mathbf{t}|\mathbf{v})$ is defined by the product of LLM’s output probabilities and $P_\text{tvg}(\mathbf{v}|\mathbf{t})$ is calculated by the cosine similarity between the LLM output tokens and the video vocabularies. Also, thank you for pointing out the confusing abbreviations and typos. We will update our manuscript with less confusing abbreviations in our final version.

2. Question on the normalization strength $\alpha$.

   $\alpha$ is a hyperparameter to adjust the normalization strength in our Prior Normalization. It can be different from each dataset but we observe that $\alpha=1$ generally yields the best performance for all the datasets so we fix $\alpha$ as 1.

3. Add text-image retrieval result.

   Our BGR with Prior Normalization is a generic framework that can be applied to text-image retrieval in a generative manner. The below table shows the performance of BGR on text-image retrieval (Flick3r and COCO). Our BGR outperforms all the text-image generative retrieval baselines.

   R@1	Flickr30K (V2T)	COCO (V2T)	Flickr30K (T2V)	COCO (T2V)
   GRACE	-	-	68.4	41.5
   TIGeR	-	-	71.7	46.1
   VTM (ours)	95.9	78.5	86.3	62.0
   VTG (ours)	95.1	80.2	87.5	63.4
   TVG (ours)	97.1	79.6	80.4	56.5
   BGR (ours)	96.8	82.6	88.8	64.6
   BGR$^*$ (ours)	98.7	85.1	89.3	65.9

   Furthermore, we compare our model with non-generative text-image retrieval models in the below table and our BGR surpasses BLIP-2 both on Flickr30K and COCO in V2T. Interestingly, our Prior Normalization can also be applied to BLIP-2 enhancing performance across all settings and achieving a new state-of-the-art result.

   R@1	Flickr30K (V2T)	COCO (V2T)	Flickr30K (T2V)	COCO (T2V)
   ALBEF	94.1	77.6	82.8	60.7
   BLIP	96.7	82.4	86.7	65.1
   BLIP-2	96.9	83.5	89.7	68.3
   BLIP-2 + Prior Normalization (ours)	99.4	89.2	90.7	68.9
   VTM (ours)	95.9	78.5	86.3	62.0
   VTG (ours)	95.1	80.2	87.5	63.4
   TVG (ours)	97.1	79.6	80.4	56.5
   BGR (ours)	96.8	82.6	88.8	64.6
   BGR$^*$ (ours)	98.7	85.1	89.3	65.9

   These results signify the effectiveness of our BGR and Prior Normalization even in text-image retrieval.

4. Discuss computational cost.

   Since traversing all video candidates requires intensive computational cost, we employ a re-ranking approach to efficiently obtain the retrieval scores among top-$k$ candidates. Specifically, as discussed in Sec. 2.1 of the main paper, we adopt a hybrid approach which efficiently retrieves the top-$k$ candidates by VTC loss, and then computes more accurate scores using VTG (or TVG) for all pairs of the top-$k$ candidates. This re-ranking approach among top-$k$ candidates significantly reduces the inference time complexity of the single-tower model from $O(mn)$ to $O(k(m+n))$ given $m$ videos and $n$ texts, where $k \ll m, n$. As a result, the inference time is 307 times faster than traversing all candidate videos on ActivityNet (97 hours → 19 minutes).

5. Performance comparison with the same vision encoder.

   In Tab. 3 of the main paper, the baseline MLLM, referred to as VTM, shares the exact same architecture (including the vision encoder) as our BGR, i.e., UMT + Q-Former + LLM, but is trained with a conventional retrieval approach, video-text matching loss described in Sec. 3.1. While the average R@1 of the VTM baseline is lower than the one of UMT by 2.2, our BGR with Prior Normalization outperforms VTM by 6.6 in R@1 on average. This implies that the performance improvement mainly comes from our BGR and Prior Normalization framework not the model architecture with the strong vision encoder.

Discussion on Prior Normalization and group queries.

Similar to the re-ranking approach that uses only $k$ candidates for traversing, we also sample $k$ queries to calculate coefficients for Prior Normalization which significantly reduces the computational cost compared to considering all the queries. We do understand that group query methods are superior to single queries, but we highlight that our model surpasses all the baselines even without Prior Normalization (i.e., single query setting), and the performance is further improved by applying it. The table below demonstrates the performance of UMT and BGR$^*$ w/o Prior Normalization on five retrieval datasets including MSVD, one-to-many retrieval setting, for a fair comparison. Our BGR$^*$ significantly outperforms UMT by a margin of 5.3 in R@1 on average across all the datasets, achieving a new state-of-the-art performance.

Furthermore, by applying Prior Normalization to both BGR$^*$ and UMT, the performances are increased in both models and our BGR$^*$ still surpasses UMT on average, as discussed in Tab. 5 and 6 of the supplement. The table below summarizes the performance of BGR$^*$ with Prior Normalization as well as other group query methods QB-Norm and DSL. The table shows our BGR$^*$ w/ Prior Normalization outperforms UMT w/ Prior Normalization, while showing superior performance compared to other group query methods.

1. Performance of UMT on MSVD. (Q1)

   We observe that the UMT paper was updated on March 11, 2024, due to the bug reported in its GitHub, and the performance on MSVD is 58.2 in T2V. 56.6 in our table is reproduced performance by us. Please refer to UMT GitHub and Tab. 14 of the paper for more details.

2. Hyper-parameter settings and the issue of group query failure. (Q2 and Q3)

   Our Prior Normalization does not assume the one-to-one correspondence of the dataset. Unlike previous approaches like QB-Norm and DSL, which explicitly enforce one-to-one correspondence through group queries, we use other queries to calibrate the posterior probability, preventing the posterior probability from being overwhelmed by the prior probability. Therefore, our Prior Normalization successfully handles the issue of group query failure caused by the many-to-one correspondences in MSVD. Furthermore, although we have used the default settings of QB-Norm and DSL without using temperature in the Softmax operation, we observe that our BGR is robust to such group query methods by considering the posterior and likelihood at the same time in a bidirectional manner, already alleviating the overwhelming problem of the prior probability to some extent.

We appreciate Reviewer xKPZ for valuable feedback and constructive comments. We will address all the concerns raised by the reviewer, hoping for more vigorous support for our paper.

1. Compare with text-image generative retrieval.

   Our BGR with Prior Normalization is a generic framework that can be applied to text-image retrieval in a generative manner. The below table shows the performance of BGR on text-image retrieval (Flick3r and COCO). Our BGR outperforms all the text-image generative retrieval baselines.

   R@1	Flickr30K (V2T)	COCO (V2T)	Flickr30K (T2V)	COCO (T2V)
   GRACE	-	-	68.4	41.5
   TIGeR	-	-	71.7	46.1
   VTM (ours)	95.9	78.5	86.3	62.0
   VTG (ours)	95.1	80.2	87.5	63.4
   TVG (ours)	97.1	79.6	80.4	56.5
   BGR (ours)	96.8	82.6	88.8	64.6
   BGR$^*$ (ours)	98.7	85.1	89.3	65.9

   Furthermore, we compare our model with non-generative text-image retrieval models in the below table and our BGR surpasses BLIP-2 both on Flickr30K and COCO in V2T. Interestingly, our Prior Normalization can also be applied to BLIP-2 enhancing performance across all settings and achieving a new state-of-the-art result.

   R@1	Flickr30K (V2T)	COCO (V2T)	Flickr30K (T2V)	COCO (T2V)
   ALBEF	94.1	77.6	82.8	60.7
   BLIP	96.7	82.4	86.7	65.1
   BLIP-2	96.9	83.5	89.7	68.3
   BLIP-2 + Prior Normalization (ours)	99.4	89.2	90.7	68.9
   VTM (ours)	95.9	78.5	86.3	62.0
   VTG (ours)	95.1	80.2	87.5	63.4
   TVG (ours)	97.1	79.6	80.4	56.5
   BGR (ours)	96.8	82.6	88.8	64.6
   BGR$^*$ (ours)	98.7	85.1	89.3	65.9

   These results signify the effectiveness of our BGR and Prior Normalization even in text-image retrieval.

2. Questions on definitions.

   We defined $v_1$, $v_2$, $t_1$, and $t_2$ in Line 221 of Sec. 3.1 of the main paper, which denote the features of the input video and text tokens. Furthermore, $\tilde{t}_1$ and $\tilde{t}_2$ are the output features of the LLM from $t_1$ and $t_2$, respectively, which are defined in Line 243. In Fig. 3, due to the auto-regressive property of the LLM, the output feature of the LLM predicts the next token, i.e., $\tilde{t}_1$ predicts $t_2$, and similarly in $\tilde{v}_1$. Also, we interpret the likelihood of a single-modality as the prior probability for calculating the joint probability of a multi-modality. The definition of the prior probability $P(\mathbf{t})$ is provided in Line 310 and 321 of Sec. 3.3.

   Specifically, in text-video retrieval, we marginalize the text-video correspondences $P(\mathbf{v}, \mathbf{t})$ across $\mathbf{v}$ to obtain $P(\mathbf{t})$, i.e., $P(\mathbf{t}) \triangleq \sum_\mathbf{v} P(\mathbf{v}, \mathbf{t})$

3. Discuss time complexity between single-tower and dual-tower models.

   Since the single-tower model generally requires further inference time than the dual-tower model, we employ a re-ranking approach to efficiently obtain the retrieval scores among top-$k$ candidates. Specifically, as discussed in Sec. 2.1 of the main paper, we adopt a hybrid approach which first retrieves the top-$k$ candidates by VTC loss from the dual-tower model, and then computes more accurate scores using VTG (or TVG) from the single-tower model for all pairs of the top-$k$ candidates. This re-ranking approach among top-$k$ candidates significantly reduces the inference time complexity of the single-tower model from $O(mn)$ to $O(k(m+n))$ given $m$ videos and $n$ texts, where $k \ll m, n$. The below table shows the performance comparison between the single-tower approach with the re-ranking strategy and the dual-tower approach of our BGR. With the efficient re-ranking strategy, the R@1 is remarkably increased by 22.9 on ActivityNet.

   ActivityNet	training	inference	R@1
   dual-tower	$O(m + n)$	$O(m + n)$	53.0
   single-tower	$O(m + n)$	$O(k(m+n))$	75.9

We appreciate Reviewer XdBX for valuable feedback and constructive comments. We will address all the concerns raised by the reviewer, hoping for more vigorous support for our paper.

1. Discuss the asymmetric information problem between text and video modalities.

   As noted by Reviewer XdBX, we also observe that some captions are overly simple and relatively ambiguous compared to videos. These captions often exhibit high prior probabilities and are easily retrieved by most videos. However, the query-generation model mitigates this issue by alleviating prior bias and encouraging one-to-one correspondence in retrieval results. We have updated Fig. 6a in Sec. E of the supplement visualizing that one caption is predominantly retrieved by most videos with the content-generation model. On the other hand, in Fig. 6b of the supplement, adopting the query-generation model alleviates this issue resulting in the optimal retrieval result where each caption is retrieved by its own paired video despite the imbalanced quantity of information in each modality. Exploring training frameworks to address the issue of asymmetric information will be an intriguing direction for future research.

2. Does the proposed approach exhibit overfitting to visually striking videos?

   In fact, the baseline MLLM easily retrieves the visually “common” video (not visually striking) due to the video prior bias problem, and our goal is to prevent the model from such overfitting (bias towards common videos). Interestingly, the video prior bias is relatively less pronounced compared to the text prior bias as shown in Fig. 6a on V2T and 6c on T2V of the supplement. A distinct vertical line is shown in Fig. 6a, absent in Fig. 6c, implying that the baseline MLLM exhibits greater overfitting to text than video.

3. Discuss performance-time trade-off.

   Although our final model, BGR$^*$, requires an ensemble of output scores optimized for three different objectives, the overall inference process remains efficient due to the shared feature extraction stage across all objectives. In the below table, ensembling VTM, VTG, and TVG boosts R@1 significantly to 82.4, while achieving a 19% improvement in inference speed compared to processing each objective separately.

   DiDeMo	R@1	Inference time (sec/sample)
   VTM	74.6	127
   VTG	76.7	120
   TVG	56.5	78
   BGR (VTG + TVG)	78.6	167
   BGR$^*$ (VTM + VTG + TVG)	82.4	262

4. Compare with relevant works.

   GLSCL proposed a simple yet effective parameter-free global interaction module for efficient text-video matching and DiffusionRet introduced a novel approach by formulating the retrieval task as a process of gradually generating joint distribution from noise. The below table shows that our BGR$^*$ significantly outperforms both GLSCL and DiffusionRet across all the datasets.

   R@1 on T2V	DiDeMo	ActivityNet	LSMDC	MSRVTT
   GLSCL	46.7	41.4	23.4	50.5
   DiffusionRet	48.9	48.1	24.4	49.0
   UMT	70.4	66.8	43.0	58.8
   BGR$^*$ (ours)	82.4	75.9	51.1	62.0

3. Analyze the inference time of Prior Normalized Decoding.

   Since the number of newly generated tokens is less than 10 in our experiment in Tab. 2, the additional overhead in inference time is marginal. Specifically, all the benchmarks consist of multi-choice questions, requiring the model to generate a short sequence. This results in only 4.9% overhead in the inference time while achieving a 16.3 improvement in the performance on average. Note that a greater number of token generations may lead to an increase in inference time.

   	avg. $\Delta$	MME	MMBench	MVBench	VideoMME	MLVU	NExT-QA	SeedBench
   VideoChat2 w/o Prior Normalized Decoding	-	1505.7 (1.5 sec/sample)	63.9 (1.2 sec/sample)	60.1 (2.4 sec/sample)	42.2 (4.1 sec/sample)	45.8 (6.9 sec/sample)	78.9 (1.4 sec/sample)	61.2 (0.9 sec/sample)
   VideoChat2 w/ Prior Normalized Decoding	+16.3 (+4.9%)	1607.0 (2.0 sec/sample)	66.2 (1.2 sec/sample)	62.3 (2.4 sec/sample)	47.1 (4.1 sec/sample)	48.5 (7.1 sec/sample)	79.4 (1.5 sec/sample)	61.7 (1.0 sec/sample)

We appreciate Reviewer hsa3 for valuable feedback and constructive comments. We will address all the concerns raised by the reviewer, hoping for more vigorous support for our paper.

1. Discuss the novelty of the proposed method.

   To the best of our knowledge, this is the first work that discusses the prior bias in text-video retrieval by MLLMs. We observe that the model often fails to retrieve the most relevant content given the query due to prior bias where the posterior is overwhelmed by the strong prior of the content. To alleviate the prior bias problem, we propose several remedies. First, we investigate that the query-generation model is less biased toward such prior bias and exhibits more powerful performance than the content-generation model. Also, we propose Prior Normalization, a simple yet effective plug-and-play module, to further alleviate the prior bias problem even in the content-generation model. Finally, by integrating the query-generation and content-generation models along with Prior Normalization, we present Bidirectional Generative Retrieval which demonstrates outstanding performance across all the text-video retrieval benchmarks.

2. Provide mathematical analysis on Prior Normalization.

   We focus on searching the most relevant video given a text query, i.e., text-to-video retrieval by $P(**v**|**t**)$, which can be written as:

   $$$$

$

P(\mathbf{v}|\mathbf{t})= P(\mathbf{t}|\mathbf{v})P(\mathbf{v})/P(**t**).\tag{Eq. (1)}

$

$$

As discussed in Tab. 1 of the main paper and Fig. 6 in the supplement, query generation (or likelihood) $P(**t** | **v**)$  is more accurate than content generation $P(**v**|**t**)$. In other words, even if $P(**t**| **v**)$ predicts accurately, the final result $P(**v**|**t**)$ can lead to incorrect retrievals due to a highly skewed video prior probability $P(\mathbf{v})$, namely, "**prior bias**".

We will discuss the case where a too large prior gap in $P(\mathbf{v})$ leads to incorrect retrieval results in the posterior $P(\mathbf{v}|\mathbf{t})$ when the likelihood $P(\mathbf{t}|\mathbf{v})$ is correct but the gap is relatively small. Specifically, we first assume that the likelihood predicts correctly but the prior gap is too larger than the likelihood gap. Then we prove that the posterior probability results in a wrong prediction. 

Assume that the likelihood predicts correctly, *i.e.*, $P(\mathbf{t}_m|\mathbf{v}_m) > P(\mathbf{t}_m|\mathbf{v}_n)>0, \forall n \neq m$ but the gap is small. Then, it can be written as:

$$

$

0 < \log P(\mathbf{t}_m|\mathbf{v}_m) - \log P(\mathbf{t}_m|\mathbf{v}_n) < \varepsilon, \tag{Eq. (2)}

$

$$

where the $\varepsilon > 0$ is the maximum gap. Eq. (2) means the retrieval score of $m$-th video (positive sample) is larger than the one of $n$-th video (negative sample) given the $m$-th text, and the gap is smaller than the maximum gap $\varepsilon$.

Another assumption is that the strong prior $P(\mathbf{v}_n)$ over video candidates as:

$$

$

\log P(\mathbf{v}_n) - \log P(\mathbf{v}_m) > c\varepsilon \tag{Eq. (3)}

$

$$

where $c > 1$. In other words, the prior gap is larger than the likelihood gap. 

Then, the posterior gap, which is the gap of retrieval scores measured by the posterior probability, is given as:

$$

$

&\log P(\mathbf{v}_m|\mathbf{t}_m) - \log P(\mathbf{v}_n|\mathbf{t}_m) &\\\\
&\\;= \log P(\mathbf{t}_m|\mathbf{v}_m) + \log P(\mathbf{v}_m) - \log P(\mathbf{t}_m|\mathbf{v}_n) - \log P(\mathbf{v}_n) &\text{(by Eq. (1))} \\\\ 
& \\;\\;< \varepsilon + \log P(\mathbf{v}_m) - \log P(\mathbf{v}_n) & \text{(by Eq. (2))} \\\\
& \\;\\; < \varepsilon - c\varepsilon = \varepsilon (1-c) & \text{(by Eq. (3))} \\\\
& \\;\\;< 0 &\text{(by c >1)} 

$

$$

This concludes the proof. "$P(\mathbf{v}_m|\mathbf{t}_m) < P(\mathbf{v}_n|\mathbf{t}_m)$" indicates that the posterior probability $P(\mathbf{v}|\mathbf{t})$ retrieves a wrong $n$-th video $\mathbf{v}_n$ given $m$-th query text $\mathbf{t}_m$ due to the too strong prior $P(\mathbf{v})$. Our Prior Normalization effectively reduces the impact of prior $P(\mathbf{v})$ and improves the retrieval performance.