## 1 Introduction  
Travel time is one of the most important metrics to measure the operational efficiency of a transportation network; it is of interest to traffic operators and travelers alike. From an operator's perspective, travel time information is used for better management and control of the traffic system to ease congestion. From a traveler's perspective, travel time information can provide them with better route choice and departure time decisions. Nowadays, most traffic monitoring systems are based on point detectors (e.g., loop detectors) installed along the roadway. Although these systems collect traffic counts and occupancy rates that can be used to estimate point speeds and segment travel times, such estimations are prone to major errors [1; 2; 3]. The recent proliferation of in-vehicle electronics (e.g., smartphones with GPS units and electronic toll collection transponders) provides new possible solutions for estimating travel time. These devices on the vehicles are used to measure actual travel times between distant locations; the actual travel time can be calculated after the vehicle has passed a road segment [4; 5; 6; 7; 8; 9]. Although estimation-based probe vehicles provide more reliable information to capture actual travel time, there are still some issues that remain to be resolved. For instance, there are issues with the privacy and low penetration rate of on-board devices [10; 11; 12; 13; 14; 15]. If a sufficient number of samples cannot be obtained to estimate travel times within a time interval, biased travel times can be estimated, with higher variance [12; 13; 16]. Given this situation, several different statistical methods have been proposed over the past few years. The regression model is proposed, which makes use of correlations between links to generalize low-frequency probe vehicle data and captures the underlying factors behind spatial and temporal variations in velocity [17]. The Fuzzy logic is also proposed to assign vehicle trajectories to a certain degree of membership. In order to estimate the mean travel time for the entire population, different traffic conditions are matched according to the degree of membership [18; 19; 20]. Other previous studies have focused on the effect of small samples from probes on the estimation of mean travel time [13; 21; 22; 23]. These studies show that simple travel time averages based on low-frequency probes cannot approach the overall mean travel time due to correlations between samples. It must be said that it is still common to obtain a sample of probe vehicles that is small and not representative of the entire population. As traffic becomes non-stationary, the variability in estimated travel times indicates that the uncertainty and changing characteristics of travel times tend to increase during peak periods [24; 25; 26]. Since the distribution of travel time with the variability during peak periods does not follow a single model, mean travel time using point estimation is not meaningful as a statistical inference about the estimator of travel time. In order to be meaningful as a statistical inference, interval estimation of travel time can be used as an alternative to overcome the limitations of the point estimation. However, there are few previous studies on interval estimation of travel time [16; 27; 28; 29], and it is still difficult to accurately estimate travel time using interval estimation. The objective of this study is to overcome the aforementioned problems in estimating travel time under low-frequency probe data during peak periods. To this end, this paper develops a novel travel time estimation model using a Bayesian approach. The rest of this article is organized as follows: The next section provides background information on the Bayesian approach and introduces the framework of the novel estimation model. This is followed by a description of the test data for the case study and a comparison of results using different estimation methods. The last section provides the conclusions of the paper and discusses their implications.