A Comprehensive Review and Application of Bayesian Methods in Hydrological Modelling: Past, Present, and Future Directions

Abstract

Bayesian methods have revolutionised hydrological modelling by providing a framework for managing uncertainty, improving model calibration, and enabling more accurate predictions. This paper reviews the evolution of Bayesian methods in hydrology, from their initial applications in flood-frequency analysis to their current use in streamflow forecasting, flood risk assessment, and climate-change adaptation. It discusses the development of key Bayesian techniques, such as Markov Chain Monte Carlo (MCMC) methods, hierarchical models, and approximate Bayesian computation (ABC), and their integration with remote sensing and big data analytics. The paper also presents simulated examples demonstrating the application of Bayesian methods to flood, drought, and rainfall data, showcasing the potential of these methods to inform water-resource management, flood risk mitigation, and drought prediction. The future of Bayesian hydrology lies in expanding the use of machine learning, improving computational efficiency, and integrating large-scale datasets from remote sensing. This review serves as a resource for hydrologists seeking to understand the evolution and future potential of Bayesian methods in addressing complex hydrological challenges.

1. Introduction

1.1. Overview

Bayesian methods, based on Bayes’ theorem, have become a cornerstone of modern hydrological modelling due to their capacity to handle uncertainty in data and model parameters. Hydrology, the study of water in the environment, is inherently uncertain due to the complex and dynamic interactions between climatic, topographic, and hydrological variables. Understanding the behaviour of water bodies, whether rivers, lakes, or groundwater, requires models that can predict future events such as floods, droughts, and rainfall, while accounting for the uncertainty in the data and parameters. Bayesian methods provide a probabilistic framework to incorporate this uncertainty into hydrological models, making them more transparent and reliable.

The origins of Bayesian methods in hydrology can be traced back to the late 20th century, where early applications focused on flood risk analysis and hydrological parameter estimation. Over the years, the adoption of computational advancements such as Markov Chain Monte Carlo (MCMC) methods has allowed for more complex and robust modelling techniques. Today, Bayesian methods are widely used in areas such as flood forecasting, drought prediction, and rainfall modelling, where traditional deterministic models often fall short in handling the inherent variability and uncertainty.

The need for Bayesian methods in hydrology has grown as climate change, urbanisation, and other global changes introduce new challenges to water management. These changes have made hydrological systems more unpredictable, requiring models that can not only forecast and estimate hydrological extremes but also quantify the associated uncertainties. This paper seeks to explore the historical development, current applications, and future directions of Bayesian methods in hydrology, with a focus on flood, drought, rainfall modelling and other hybrid approaches. The next section provides an outline of the key equations used in Bayesian analysis, as applied in this study. These equations form the foundation for the Bayesian methods implemented in the various hydrological applications discussed in this manuscript.

1.2. Fundamental Bayes’ Theorem

At its core, Bayes’ theorem provides a way to update the probability for a hypothesis (or parameter) θ in light of new data D. The theorem is written as:

$$P\left(\theta |D\right)={\displaystyle \frac{P\left(D|\theta \right)P\left(\theta \right)}{P\left(D\right)}}$$

(1)

where

• $PP\left(\theta |D\right)$ is the posterior probability distribution of the parameter $\theta$ after observing the data D.
• $PP\left(D|\theta \right)$ is the likelihood, which represents the probability of observing data D given the parameter $\theta$.
• P(θ) is the prior probability distribution of θ before the data are observed.
• P(D) is the marginal likelihood (or evidence), acting as a normalising constant, calculated by integrating over all possible parameter values:

  $$P\left(D\right)=\int P\left(D|\theta \right)P\left(\theta \right)d\theta$$

  (2)

Because P(D) does not depend on θ, it is common to write the relationship in proportional form:

$$P\left(\theta |D\right)\propto P\left(D|\theta \right)P\left(\theta \right)$$

(3)

1.3. Liklihood Function

In many hydrological applications, the likelihood function is chosen based on the statistical characteristics of the data. For example, if the observed runoff y is assumed to be normally distributed with mean μ(θ) (which depends on the parameters θ) and known variance σ², then the likelihood is given by:

$$P\left(y|\theta \right)={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}\exp \left(-{\displaystyle \frac{{\left(y-\mu \left(\theta \right)\right)}^2}{2{\mathsf{\sigma }}^2}}\right)$$

(4)

When multiple independent observations y = {y₁, y₂, …, y_n} are available, the joint likelihood becomes:

$$P\left(\mathit{y}|\theta \right)={\displaystyle \prod }_{i=1}^n{\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}\exp \left(-{\displaystyle \frac{{\left(y_i-\mu \left(\theta \right)\right)}^2}{2{\mathsf{\sigma }}^2}}\right)$$

(5)

1.4. Prior Distribution

The prior P(θ) encapsulates our knowledge about the parameter θ before observing the data. Priors can be informative (based on previous studies or expert judgement) or non-informative (designed to have minimal influence). For example, a common non-informative prior for a parameter that can take any real value is the uniform prior:

$$P\left(\theta \right)=constant$$

(6)

Alternatively, if previous information suggests that θ is normally distributed with mean μ₀ and variance τ², the prior might be:

$$P\left(\theta \right)={\displaystyle \frac{1}{\sqrt{2\pi {\tau }^2}}}exp\left(-{\displaystyle \frac{{\left(\theta -{\mu }_0\right)}^2}{2{\tau }^2}}\right)$$

(7)

1.5. Posterior Distribution and Bayesian Updating

Combining the likelihood and the prior via Bayes’ theorem yields the posterior distribution:

$$P\left(\theta |\mathit{y}\right)={\displaystyle \frac{P\left(\mathit{y}|\theta \right)P\left(\theta \right)}{\int P\left(\mathit{y}|\theta \right)P\left(\theta \right)d\theta }}$$

(8)

In practice, the denominator (the evidence) is often difficult to compute directly, particularly for complex models or high-dimensional parameter spaces. As a result, numerical methods such as Markov Chain Monte Carlo (MCMC) are used to sample from the posterior distribution without having to calculate the normalising constant explicitly.

1.6. Bayesian Estimation

A common point of estimate from the posterior distribution is the posterior mean:

$$E[\theta |\mathit{y}]=\int \theta P\left(\theta |\mathit{y}\right)d\theta$$

(9)

This estimate can be used as a input in predictive models, though it is often more informative to consider the full posterior distribution when quantifying uncertainty.

1.7. Predictive Distribution

To predict a new observation y_new given the data y, the predictive distribution is computed by integrating over the uncertainty in θ:

$$P\left(y_{new}|\mathit{y}\right)=\int P\left(y_{new}|\theta \right)P\left(\theta |\mathit{y}\right)d\theta$$

(10)

This equation underscores one of the key advantages of the Bayesian framework—uncertainty in parameter estimates is naturally propagated to the predictions.

1.8. Practical Application in Hydrology

In hydrological modelling, such as in rainfall–runoff prediction or flood-frequency analysis, Bayesian methods allow for:

• Integration of Diverse Data Sources: Prior information from regional studies can be combined with local observations.
• Uncertainty Quantification: The full posterior distribution gives insights into the range of possible outcomes rather than a single deterministic prediction.
• Adaptive Learning: As new data become available (e.g., during a flood event), the posterior can be updated, leading to improved and dynamic forecasting.

For example, when estimating flood quantiles, the use of a Bayesian framework might involve specifying a likelihood function for the observed peak flows, choosing an appropriate prior based on historical records, and using MCMC techniques to obtain a posterior distribution for flood quantiles. The resulting credible intervals provide decision-makers with a clear measure of the uncertainty associated with flood predictions.

The next section provides an outline of the methods employed in the literature review. This includes the theoretical foundations of Bayesian inference as well as the systematic approach used to identify, evaluate, and synthesise relevant studies in the field of hydrological modelling.

2. Methodology and Literature Review

2.1. Databases and Timeframes

Searches were conducted in Scopus, Web of Science, and Google Scholar to cover both multidisciplinary and domain-specific journals. Publications from 1975 to the present were considered. This period captures the formative years of Bayesian applications in hydrology (notably in flood-frequency analysis and parameter estimation) as well as more recent innovations (e.g., hybrid Bayesian–machine learning models).

2.2. Search Strategy and Keywords

Keywords: The following terms were used in various combinations:

(i)

Core Bayesian Terms: “Bayesian”, “Bayes”, “Markov Chain Monte Carlo”, “hierarchical Bayesian”, “Bayesian network”, “approximate Bayesian computation”.

(ii)

Hydrology Keywords: “hydrology”, “flood forecasting”, “flood frequency”, “drought prediction”, “rainfall modelling”, “water quality”, “streamflow”, “runoff”, “groundwater”, “hydraulic”. Advanced Topics: “machine learning”, “data assimilation”, “uncertainty quantification”, “dynamic Bayesian networks”, “spatio-temporal modelling”.

(iii)

Boolean Operators: Queries combined these keywords with AND, OR, and NOT to refine results. For example:

(“Bayesian” OR “Bayes”) AND (“hydrology” OR “flood forecasting” OR “rainfall modelling”)

(“Bayesian network” OR “hierarchical Bayesian”) AND (“uncertainty” OR “uncertainty quantification”)

2.3. Initial Screening and Eligibility Criteria

(i)

Title and Abstract Screening:

• Articles were initially filtered based on relevance to Bayesian methods in hydrology. Titles and abstracts mentioning Bayesian approaches, hydrological modelling, or related topics were retained.

(ii)

Language and Publication Type:

• Only English-language peer-reviewed articles, conference proceedings, and book chapters were considered to ensure the quality of the materials and feasibility of analysis.

(iii)

Inclusion Criteria:

• Explicit use or discussion of Bayesian methods in hydrological contexts.
• Sufficient methodological detail, allowing for an assessment of how Bayesian inference was applied (e.g., parameter estimation, flood-frequency analysis, model calibration).

(iv)

Exclusion Criteria:

• Studies focusing purely on statistical theory without hydrological application.
• Non-academic literature, short news articles, or publications lacking methodological details.

2.4. Full- and Partial-Text Review and Quality Appraisal

(i)

Full-Text Retrieval:

• Potentially relevant articles were downloaded and reviewed, some partially and some in full.

(ii)

Quality Assessment:

• Studies were assessed on the clarity of Bayesian methodology, data availability, and rigour of results.
• Articles that did not provide enough detail on Bayesian model construction or uncertainty analysis were excluded.

(iii)

Reference Checking and the Snowball Approach:

• Reference lists of key articles were scanned to identify additional relevant publications. This iterative process helped capture important works that might not have surfaced in the initial database queries.

(iv)

Data Extraction and Categorising Information:

• For each article, the following information was recorded in a spreadsheet.
• Study objective and hydrological domain (flood, drought, water quality, rainfall–runoff, etc.).
• Type of Bayesian method used (MCMC, hierarchical Bayesian, Bayesian network, etc.).
• Key findings, especially regarding uncertainty quantification and parameter estimation.
• Articles were organised into histograms reflecting historical development, current applications, advancements and innovations, and challenges in Bayesian hydrology.
• Overlapping themes (e.g., flood forecasting plus machine learning) were cross-referenced to avoid duplication and to highlight interdisciplinary links.

(v)

Final Selection and Synthesis

• Final Dataset: After removing duplicates and non-relevant items, the final list comprised the core references used in this review.
• Synthesis Approach: Literature findings were summarised in narrative form, highlighting the evolution of Bayesian methods in hydrology, their current state-of-the-art applications, and the outstanding challenges.
• Tables and figures (e.g.,

  Table 1. (a) Historical development of Bayesian methods in hydrological modelling. (b) Current applications of Bayesian methods in hydrological modelling. (c) Advancements, innovations, and challenges in Bayesian hydrology.

  (a)
  Subcategory	Key Contributions	Selected References
  Early Foundations	Introduction of Bayesian theory into hydrology; early work on parameter estimation and uncertainty quantification	Box & Tiao [1]; Vicens et al. [2]; Sorooshian & Dracup [3]; Kuczera [4,5,6]; Pericchi & Rodriguez-Iturbe [7]; Kitanidis [8]; Kuczera [9]
  Flood-Frequency Analysis	Application of Bayesian methods to combine regional and site-specific data for robust flood-quantile estimation; reduction in bias under short records	Kuczera [5,6]; Stedinger & Cohn [10]; Merz & Thieken [11]; Reis et al. [12]; Seidou et al. [13]; Micevsk & Kuczera [14]; Gaume et al. [15]; Haddad & Rahman [16]; Haddad et al. [17]; Parkes & Demeritt [18]; Guame [19]; Mehmood et al. [20]; Qu et al. [21]; Reis et al. [22]; Shang et al. [23]; Jarajapu et al. [24]; Barna et al. [25]; Lei et al. [26]; Alexandre et al. [27]; Lucas et al. [28]
  Drought Forecasting	Development of probabilistic frameworks for seasonal drought prediction and severity assessment	Madadgar & Moradkhani [29,30]; Avilés et al. [31]; Kim et al. [32] Ali et al. [33]; Raza et al. [34]; Wu et al. [35,36]
  Integration with Geostatistics	Use of Bayesian kriging and spatial analysis to interpolate sparse hydrological data; incorporation of spatial dependencies	Diggle et al. [37]; Renard et al. [38,39]; Nowak et al. [40]; Verdin et al. [41]; Gupta et al. [42]; Yang & Ng [43]; Lima et al. [44]; Senoro et al. [45]; Zaresefat [46]
  (b)
  Subcategory	Key Contributions	Selected References
  Flood Risk Assessment	Combination of historical and observed data to probabilistically estimate extreme flood events and derive credible intervals for risk management	Fill & Stedinger [47]; Marshall et al. [48]; Reis & Stedinger [49]; Ribatet et al. [50]; Micevski & Kuczera [14]; Viglione et al. [51]; M. Šraj et al. [52]; Liu et al. [53]; Naseri & Hummel [54]; Rampinelli et al. [55]
  Rainfall Modelling	Improvement of spatiotemporal rainfall predictions and rainfall–runoff model calibration by incorporating prior knowledge and uncertainty quantification	Kuczera [56]; Khan & Coulibaly [57]; Martina et al. [58]; Smith & Marshall [59]; Lima & Lall [60]; Haddad et al. [61]; Molina et al. [62]; Little et al. [63]; Ombadi et al. [64]; Ossandón et al. [65]; Nguyen et al. [66]; Zorzetto et al. [67]
  Water Quality Assessment	Integration of Bayesian inference with machine learning to model pollutant dynamics and assess uncertainty in water quality predictions	Gronewold et al. [68]; Jin et al. [69]; Haddad et al. [70]; Egodawatta et al. [71]; Zhao et al. [72]; Liang et al. [73]; Peng et al. [74]; Perera et al. [75]; Jackson-Blake et al. [76]; Chowdhury & Egodawatta [77]; Spezia et al. [78]; Zhang et al. [79]
  Hydraulic Systems and Modelling	Quantification of uncertainty in hydraulic grade line models and simulation of water networks; application of Bayesian methods to calibrate and validate models	Nadiri et al. [80]; Camacho et al. [81]; Shrestha & Kozlowski [82]; Han & Coulibaly [83]; Liu & Merwade [84]; Garcia et al. [85]; Yin et al. [86]; Chowdhury & Egodawatta [77]; Badjadi et al. [87]; Pourahari et al. [88]
  (c)
  Category and Subcategory	Key Contributions	Selected References
  Advancements and Innovations Hybrid Methods	Combination of Bayesian inference with machine learning for improved predictive accuracy and enhanced model robustness	Nadiri et al. [80]; Humphrey et al. [89]; Han & Coulibaly [83]; Alexander et al. [90]; Haddad & Rahman [91]; Cheng & Hsu [92]; Ossandón et al. [65]; Senoro et al. [45]; Jarajapu et al. [2]; Nguyen et al. [66]; Xu et al. [93]; Sheikh & Coulibaly [94]; Sattari et al. [95]; George & Athira [96]
  Big Data Integration	Assimilation of remote sensing data and large-scale datasets into Bayesian frameworks to support real-time updating and model calibration	Ponce Romero et al. [97]; Fer et al. [98]; Monrat et al. [99]; Marcot & Penman [100]; Wu et al. [101]; Donratanapat et al. [102]; Gaffoor et al. [103]; Yuan et al. [104]; Tanguy et al. [105]
  Dynamic Bayesian Networks	Modelling of complex temporal and spatial interactions via network-based approaches, allowing dynamic updating and improved forecasting	Molina et al. [62]; Leu & Bui [106]; Humphrey et al. [89]; Xue et al. [107]; Jäger et al. [108]; Chen et al. [109]; Mihunov & Lam [110]; Das & Chanda [111]; Ma et al. [112]; Jackson-Blake et al. [76]; Xie et al. [113]; Albert et al. [114]; Pourahari et al. [88]
  Bayesian Decision Frameworks	Application of decision theory to balance competing objectives in water-resource management through probabilistic assessment	Martina et al. [58,115]; Pagano et al. [116]; Bertone et al. [117]; Madadgar et al. [118]; Jäger et al. [108]; Roozbahani et al. [119]; Phan et al. [120]; Kostyuchenko et al. [121]; Garzon et al. [122]; Huang & Merwade [123]; Lu et al. [124]; Tanim et al. [125]; Xue et al. [126]
  Challenges Data Quality	Addressing issues related to incomplete, noisy, or sparse datasets which affect the reliability of Bayesian inferences in hydrological modelling	Renard et al. [127]; Rode et al. [128]; Renard et al. [39]; Liu et al. [129]; Hernández & Liang [130]; DeChant & Moradkhani [131]; Qamar et al. [132]; Wu et al. [133]; Yoon et al. [134]; Zhou et al. [135]; Alexandre et al. [27]
  Computational Intensity	Mitigating the high computational demands of advanced Bayesian methods, especially for high-dimensional parameter estimation	Liu et al. [136]; Sadegh & Vrugt [137]; Kavetski et al. [138]; Moges et al. [139]; Gupta & Govindaraju [140]; Saha & Bradley [141]; Ulzega & Albert [142]; Andraos [143]; Nizeyimana [144]
  Parameter Uncertainty	Managing the subjectivity in prior selection and the propagation of uncertainty in parameter estimates, which is critical in influencing posterior outcomes	Kuczera [5]; Kitanidis [8]; Kuczera [145]; Kavetski et al. [146,147]; Kuczera et al. [148]; Thyer et al. [149]; Zhang et al. [150]; Ouarda & El-Adlouni [151]; Haddad et al. [17]; Haddad & Rahman [16,91]; Yoon et al. [134]; Ulzega & Albert [142]; Gui et al. [152]; Andraos [143]
  Interdisciplinary Integration	Bridging statistical modelling with physical hydrology and engaging stakeholders to ensure that probabilistic predictions are actionable and comprehensive	Kuczera [9]; van Dam et al. [153]; Liu et al. [53]; Senoro et al. [45]; Kumari & Pandey [154]; Kneier et al. [155]; Dubbert et al. [156]; Huang & Merwade [123]; Rampinelli et al. [157]; Lu et al. [124]

  (a), (b), (c), timeline diagrams) were developed to provide a structured overview of major contributions, authorship networks, and emerging trends.

(vi)

Limitations in the Overall Search

• While multiple databases and a broad range of keywords were used, some relevant studies may not have been retrieved (e.g., non-English publications or grey literature).
• The chosen timeframe (1975–present) aimed to capture seminal works and current research but may exclude earlier formative studies not widely digitised.

This multi-step, systematic search strategy helped ensure a comprehensive and unbiased coverage of Bayesian hydrology research. By integrating keyword-based database queries, careful eligibility screening, quality assessment, and iterative reference checking, it helped identify the most relevant and methodologically robust studies. These selected works form the basis of the review and inform the discussions on historical development, current applications, and future directions in Bayesian hydrological modelling.

3. Evolution of Bayesian Methods in Hydrology

The following section traces the evolution of Bayesian methods in hydrology, highlighting key developments from their early theoretical foundations to modern applications in flood-frequency analysis and forecasting, drought prediction, and hydrological modelling.

Table 1 (a), (b), and (c) present a comprehensive overview of the historical development, current applications, advancements, and challenges related to Bayesian methods in hydrology, while Figure 1 succinctly summarises the Bayesian hydrology evolution timeline. The historical development section traces the evolution of Bayesian approaches in hydrology, from the early contributions by [1,2,3,4,5,6,7,8,9] to the adoption of geostatistical methods e.g., [40]. Current applications highlight how Bayesian frameworks are used in flood risk assessment [51], rainfall modelling [111] and drought forecasting [158]. Advancements include the integration of machine learning with Bayesian methods [159] and dynamic Bayesian networks [110], which address complex hydrological dynamics. The challenges section identifies issues such as data quality [33], computational intensity [105], and parameter uncertainty [142,146,147,148,160,161].

The flowchart in Figure 1 provides an overview of the evolution of Bayesian methods in hydrology, showcasing its development and increasing complexity over time. Beginning with foundational applications in uncertainty quantification for basic hydrological processes, these methods evolved to address spatial dependencies using hierarchical models and geostatistical approaches. The integration of temporal dynamics enabled the improved modelling of seasonal and inter-annual hydrological variability, particularly under the influence of climate phenomena such as ENSO.

Recent advancements have focused on coupling Bayesian frameworks with machine learning techniques, such as Bayesian neural networks, to enhance predictive accuracy and account for non-linear relationships. Additionally, modern Bayesian methods now facilitate multi-hazard risk assessment and decision making under uncertainty, offering probabilistic insights critical for sustainable water-resource management. This evolution underscores the transformative impact of Bayesian approaches on hydrology, supporting more robust and informed modelling across diverse contexts.

Figure 2 depicts the key contributions used in Bayesian hydrology from 1975 to 2024. To develop this figure a combination of keywords, Boolean operators, and truncations were used to capture the breadth of relevant studies in Scopus, Web of Science, and Google Scholar. Detailed steps regarding this approach are discussed in Section 2. For Bayesian methods in hydrology, the following was considered in summary:

Core keywords:

General: Bayesian; Bayes; hydrology; hydrological

Applications: flood; drought; water quality; groundwater; streamflow; hydraulic

Advancements: Bayesian network; hierarchical Bayesian; Markov Chain Monte Carlo (MCMC); copula Bayesian; Bayesian geostatistics

Challenges: uncertainty; bias; model validation; interdisciplinary

Since the 1990s, the number of publications based on the different categories as per Table 1 and Figure 2 has gradually increased, with the highest number of publications starting in 2009 and 2010. As shown in Figure 2, Bayesian methods in the current application subcategory has been the most popular, followed by the advancements and innovations subcategory. However, the popularity of challenges in Bayesian methods in hydrology has been increasing in recent years, particularly from 2008 onwards.

Figure 3 shows a network diagram map of authorship of papers on Bayesian methods in hydrology and the network of co-authorships. Here, the bigger the node (in blue), the greater the number of papers published by the country. It can be seen that the US, UK, Canada, and Australia have published the highest number of articles in this research field, which is followed by China, Germany, and France. It can also be seen that US has the strongest network of co-authorship (linked with UK, Canada, and Australia), followed by UK (linked with USA, Germany, France, and Greece).

4. Historical Development of Bayesian Methods in Hydrology

The use of Bayesian methods in hydrology began with early attempts to quantify uncertainty in the estimation of hydrological parameters. In the 1980s and 1990s, the development of Bayesian models in hydrology was initially focused on flood-frequency analysis, where they were used to estimate the probability of extreme flood events based on limited historical data. One of the major advantages of Bayesian methods is their ability to incorporate prior knowledge, such as expert judgement or physical principles, into the analysis. This was particularly valuable in flood modelling, where data scarcity often limits the reliability of traditional methods. A number of influential papers using Bayesian empirical methods for flood-frequency analysis were by [4,5,6]. Kuczera [4] evaluates the performance of flood-frequency estimators under varying conditions, focusing on the robustness and efficiency of empirical Bayes estimators. He introduces hybrid models that combine regional and site-specific data, demonstrating their superiority for short record lengths. The models are designed to balance resistance to bias with efficiency, ensuring reliable flood-quantile estimates even under adverse data conditions. This work emphasised the importance of robust modelling techniques to enhance flood risk assessments. Kuczera [5] addresses the challenge of integrating site-specific and regional data for flood analysis. The incorporation of spatial correlation into an empirical Bayes framework, showed improved estimates under sampling uncertainty. This method reduced bias by leveraging shared regional characteristics, advancing reliability in flood predictions. Kuczera [6] also introduced a Bayesian approach to model regional skew in flood-frequency analysis. By treating regional skew as a prior distribution, he was able to create a surrogate that reduces the sensitivity of flood-quantile estimates to localised data, providing robust and consistent results.

Gelman et al. [162] introduced Markov Chain Monte Carlo (MCMC) methods, which would later become the cornerstone of modern Bayesian analysis in hydrology. This breakthrough allowed for more efficient sampling from posterior distributions and laid the groundwork for Bayesian methods to be applied to more complex hydrological models.

Another key milestone in Bayesian hydrology was the development of the Bayesian methodology for model calibration by [163]. Their study examined the relationship between data characteristics and the precision of parameter estimates in hydrologic models. It highlighted how the quality and quantity of data influence model parameter uncertainty. The study used sensitivity analysis to show that data with greater variability and better alignment with model predictions lead to more precise parameter estimates. The work presented by [163] here complemented and extended the work of [5,6] who presented a Bayesian approach for the evaluation of flood-frequency models.

One of the most significant contributions to Bayesian hydrology came from [164], who extended Bayesian methods to address uncertainty in hydrological model predictions. Beven’s work on the Topmodel framework demonstrated how Bayesian inference could be used to quantify uncertainty in runoff predictions, particularly in the context of spatial variability and limited observational data. As Bayesian methods gained popularity, researchers began to apply it to other areas of hydrology, such as parameter estimation for rainfall–runoff models. Early work in this area involved using Bayesian inference to estimate parameters in lumped models, which treat the catchment as a single entity—[165]. However, this approach was soon expanded to distributed models, which account for spatial variations in rainfall and runoff across a catchment. With the advent of more powerful computing resources in the late 1990s and early 2000s, Bayesian methods became increasingly sophisticated. The development of MCMC techniques allowed for more efficient sampling from complex posterior distributions, enabling Bayesian inference in more computationally demanding models [146,147]. Hierarchical Bayesian models, which allow for the incorporation of multi-level data (e.g., temporal and spatial data), climate-change impacts on extremes, also became more common [166,167,168,169]. These advancements allowed Bayesian methods to be applied to more complex hydrological systems, such as river basins and groundwater flow models.

5. Current Applications of Bayesian Methods in Hydrology

5.1. Flood Risk Assessment

Flood risk assessment is one of the most prominent areas where Bayesian methods are applied in hydrology. Flood models typically rely on historical data to estimate the frequency and magnitude of extreme flood events. However, the uncertainty inherent in these estimates can be significant due to the limited length of the historical record and the complex nature of flood processes. Bayesian methods help address these challenges by providing a probabilistic framework that accounts for uncertainty in both the data and model parameters.

For example, a Bayesian framework can be used to estimate flood hazard maps that predict the probability of flood events at various locations. This is particularly useful in flood-prone regions where flood mitigation measures need to be prioritised. Bayesian models can also incorporate spatially and temporally varying information, such as climate data, land-use patterns, and river flow characteristics, to provide more accurate flood predictions.

Little et al. [63] explored Bayesian approaches to classify extreme rainfall events across the UK for flood risk management. By using a Bayesian objective classification system, the study evaluated the probability of extreme rainfall events and their associated risks. This methodology allows for dynamic updates as new data become available, enhancing the precision and adaptability of flood risk predictions. The Bayesian framework in this application provided a structured way to incorporate uncertainties and offers probabilistic outputs that are particularly valuable for decision-making in flood risk management.

Parkes & Demeritt [18] developed a Bayesian statistical model for flood-frequency analysis that integrates historic flood data with annual maximum flow data. By applying Bayesian methods, the model effectively combines disparate data sources to reduce uncertainties in estimating rare flood events, such as the 100-year flood. The authors demonstrate that including historical data significantly narrows confidence intervals by as much as 50% for certain return periods as compared to traditional frequency analysis methods. They applied the model to the River Eden in the United Kingdom (UK) and highlighted the importance of addressing uncertainties related to historical channel changes and discharge estimates. This Bayesian framework provided a robust tool for flood risk assessment by leveraging both systematic and historical information. Supplementary Example S1 illustrates a typical Bayesian model setup in R and JAGS (Just Another Gibbs Sampler), implementing a Bayesian MCMC framework to estimate the parameters of the Log Normal distribution (location and scale). It also shows the setting up of priors based on expert knowledge and historical data with the graphing of flood-frequency curves with credible limits and a boxplot comparison of observed vs. posterior flood-quantile estimation.

Cai & Wang [170] examined the uncertainty in precipitation forecasts using the TIGGE (THORPEX Interactive Grand Global Ensemble) database. It combines fuzzy probability with Bayesian theory to improve the modelling and quantification of forecast uncertainty. The Bayesian component of the study focused on probabilistically estimating forecast distributions and updating these estimates as new observational data become available. This hybrid approach improved the reliability of precipitation forecasts, which is crucial for managing flood risks and other hydrological applications.

Overall, it can be seen how Bayesian methods provide a versatile and robust framework for addressing uncertainties inherent in hydrological and meteorological data. These studies also highlighted the value of integrating historical data, probabilistic reasoning, and adaptive learning in enhancing predictive models for water-resource and flood risk management.

5.2. Drought Prediction

Drought prediction is another key area where Bayesian methods are becoming increasingly popular. Droughts, defined by prolonged periods of abnormally low rainfall, are difficult to predict due to their dependence on complex climate patterns. Bayesian models are well-suited for this task because they can integrate information from different sources, such as past drought events, climate indices (e.g., ENSO), and atmospheric conditions. Bayesian hierarchical models have been used to predict drought severity based on regional and global climatic factors. These models allow for the estimation of uncertainty in both short-term and long-term drought predictions, providing decision-makers with more reliable forecasts.

Madadgar & Moradkhani [29,30] introduced a comprehensive Bayesian framework for seasonal drought forecasting. Bayesian inference was utilised to integrate historical hydrologic data with dynamic climate model outputs. The methodology incorporated prior knowledge of drought characteristics and updated it with new information, enabling probabilistic forecasts of drought indices. The Bayesian framework provided a mechanism to quantify uncertainty in seasonal drought forecasts, offering a robust tool for water-resource management and planning.

Madadgar et al. [171] developed a hybrid approach combining statistical and dynamical models for drought prediction in the southwestern United States (U.S). Bayesian methods were employed to estimate probabilistic drought predictions by assimilating diverse sources of uncertainty from the climate models and observational data. Bayesian inference helped calibrate and validate the statistical–dynamical model ensemble, enabling robust seasonal drought forecasting. The framework highlighted the importance of Bayesian approaches for decision support by addressing the variability and uncertainty in climate predictors and improving the prediction of drought indices such as the Standardised Precipitation Index (SPI).

Ali et al. [33] proposed a Bayesian network approach to regional drought monitoring. The methodology involved integrating diverse drought indicators (e.g., precipitation deficits and soil moisture) within a probabilistic framework. Bayesian inference was used to evaluate the likelihood of drought conditions based on observed data and model predictions. The approach was particularly effective in handling data sparsity and provided a robust regional drought index that combined multiple seasonal factors, enhancing the reliability of drought-monitoring efforts. These studies highlight the versatility of Bayesian methods in handling uncertainties, integrating multi-source data, and providing probabilistic predictions for drought management and decision making. In Supplementary Example S2, a simple Bayesian linear regression model to predict future drought indices based on historical data is illustrated, incorporating uncertainty in the predictions based on the Palmer Drought Severity Index. The developed model can be used as a basis to predict future drought occurrences and quantify the risk of extreme drought events and a probabilistic assessment of drought severity under different scenarios.

5.3. Rainfall Modelling

Rainfall modelling is a core application of Bayesian methods in hydrology. Rainfall data are often sparse, particularly in regions with limited monitoring stations or unreliable data sources [61]. Bayesian methods can help integrate various sources of information to improve rainfall predictions [172,173]. By using priors based on historical rainfall patterns and incorporating data from remote sensing technologies (e.g., satellite-based rainfall estimates), Bayesian models can offer more accurate rainfall forecasts. For instance, Bayesian networks have been used to model the spatiotemporal distribution of rainfall, taking into account spatial correlations between rainfall stations and temporal dependencies [174,175]. This approach has been applied in various parts of the world, including tropical regions where rainfall patterns exhibit significant variability. Bayesian methods have been extensively applied to rainfall–runoff modelling, especially in the context of model calibration [143,176,177]. By using Bayesian inference, hydrologists can estimate parameters such as soil moisture retention, evapotranspiration, and runoff coefficients. In Australia, a Bayesian hierarchical model was presented by [178] to improve rainfall prediction by integrating data from ground-based stations and satellite observations. The results showed that the Bayesian model outperformed traditional models in terms of both accuracy and uncertainty quantification. Previous studies have also showed similar results e.g., [61,179].

Supplementary Examples S3 and S4 illustrate comprehensive examples of a Bayesian rainfall–runoff model using R and JAGS. The Bayesian rainfall–runoff model estimates the relationship between rainfalls, evaporation, and temperature inputs (while taking into account the dependences between the covariates) and runoff output, and the Bayesian framework in both examples quantifies the uncertainty in the parameters.

6. Advancements and Innovations in Bayesian Methods for Hydrology

6.1. Machine Learning and Bayesian Methods

The integration of machine learning (ML) with Bayesian methods is a growing trend in hydrology. Traditional Bayesian models rely on prior distributions and likelihood functions, but machine learning algorithms can learn patterns directly from data [180,181,182]. Combining these two approaches allows for the creation of more flexible models that can handle large and complex datasets.

For instance, Bayesian neural networks (BNNs) combine the flexibility of neural networks with the probabilistic framework of Bayesian inference. BNNs can be used to model non-linear relationships between hydrological variables, such as the impact of climate factors on river flow or rainfall. By estimating the uncertainty in model predictions, BNNs provide a more reliable tool for flood and drought prediction. In a study by [183], they show that hybrid modelling that combines hydrological models, BNNs, and uncertainty analysis improves streamflow prediction in alpine regions influenced by rainfall, snow, and glacier melt and also achieves higher precision and more reliable uncertainty intervals than traditional approaches. Tested in the Yarkant River basin, this hybrid approach outperforms alternative methods, making it a robust tool for global alpine streamflow prediction.

Vasheghani et al. [184] used the BNN model integrated with atmospheric teleconnection patterns to predict monthly dam inflows in Iran, showing superior performance compared to Artificial Neural Networks (ANN). By leveraging teleconnection indices and local variables, the BNN model achieved lower normalised root mean square errors across three major dams, demonstrating enhanced predictive accuracy and probabilistic reliability. While both models struggled with peak flow predictions, BNN’s ability to incorporate large-scale climate patterns into hydrological forecasting represents a significant advancement in water-resource management. Supplementary Example S5 illustrates the combination of Bayesian methods and ML in hydrology. This example focuses on a scenario where we model streamflow (runoff) based on multiple covariates (e.g., rainfall, temperature, and evaporation), using a Bayesian approach for parameter estimation and a machine learning technique, such as Random Forest, to improve predictions.

6.2. Integration of Remote Sensing and Big Data

The availability of big data and remote sensing technologies has revolutionised hydrological modelling. Satellite-based measurements of rainfall, temperature, and soil moisture are now widely used to supplement ground-based observations. These data sources can be incorporated into Bayesian models to improve predictions of hydrological extremes. Bayesian models can handle the vast amount of data generated by remote sensing technologies and use it to update model parameters in real time. For example, satellite-based rainfall estimates can be used to update flood risk models during a storm event, providing more accurate and timely predictions. Weather forecasting plays a crucial role in various sectors like agriculture, transportation, and emergency planning, all of which can be affected by flooding. A study by [185] introduced a Bayesian model that improves weather predictions by integrating remote sensing data from sources such as satellites, radar, and ground observations. The model employs data pre-processing, Bayesian probability frameworks, and ensemble predictions to provide probabilistic forecasts, enhancing reliability and capturing uncertainties. Preliminary results demonstrated improved accuracy for both short- and long-term forecasts, particularly for extreme weather events. By offering probabilistic insights, this approach supports smarter decision-making across industries and highlights the benefits of incorporating remote sensing data into weather prediction models.

6.3. Hybrid Models

Hybrid models, which combine physical models with data-driven approaches, are gaining increasing attention in hydrology due to their ability to integrate the strengths of both methodologies. These models typically use physical laws (e.g., conservation of mass and energy) to represent hydrological processes and combine them with data-driven methods (e.g., machine learning or Bayesian networks) to estimate unknown parameters or improve prediction accuracy. Example: Bayesian inference with hydrological simulation models: One of the leading examples of hybrid models in hydrology is the integration of Bayesian inference with hydrological simulation models like SWAT (Soil and Water Assessment Tool) or HEC-HMS (Hydrologic Engineering Center’s Hydrologic Modelling System). In these models, Bayesian techniques are applied to optimise parameters that cannot be directly measured, such as soil permeability or vegetation parameters. These parameters are often inferred by Bayesian methods based on available observational data, which improves the model’s predictive power and provides a rigorous measure of uncertainty.

A recent example of hybrid modelling for flood forecasting is the combination of Bayesian methods with ensemble machine learning techniques can be read in [186]. This study introduced an advanced ensemble learning model, combining a Bayesian Belief Network with an extreme learning machine and a back-propagation structure optimised by a genetic algorithm. Using a dataset of 194 flood locations and ten conditioning factors, the model outperformed benchmark established methods. The findings demonstrate the hybrid model that combines genetic algorithms and Bayesian Belief Networks has potential as a robust tool for managing flood risks globally. Han & Coulibaly [83,182] also provide good examples of Bayesian flood-forecasting hybrid methods.

6.4. Data Assimilation

Data assimilation, the process of combining model predictions with observational data, is another area where Bayesian methods have led to significant advancements. By incorporating real-time observations into hydrological models, data assimilation methods improve model forecasts and reduce uncertainty. The Kalman filter, a recursive Bayesian estimation technique, has been widely used in hydrological modelling for data assimilation [187,188,189]. It is particularly effective in real-time forecasting, where observations become available at different time steps and need to be incorporated into ongoing model predictions. The Kalman filter has been applied in flood-forecasting models to assimilate streamflow observations and adjust the model’s parameters accordingly, improving flood prediction accuracy [53,190]. For non-linear and non-Gaussian systems, particle filters offer a more flexible alternative to Kalman filters. Particle filters approximate the posterior distribution by using a set of random samples or “particles” to represent the distribution. This method has been successfully applied in hydrological models to assimilate data from multiple sources, such as rainfall and streamflow measurements, leading to improved forecasts in flood and drought prediction [191,192,193].

6.5. Spatial and Temporal Modelling

The growing complexity of hydrological systems and their interactions with climatic and anthropogenic factors have underscored the critical importance of spatial–temporal modelling in hydrology. These models provide a framework for capturing the variability and interdependence of hydrological processes across space and time [194], which is vital for informed water-resource management. Spatial–temporal modelling allows researchers to better understand and predict phenomena such as rainfall distribution, runoff patterns, and flood events by incorporating location-specific features such as topography, land use, and soil characteristics [65]. Bayesian models have emerged as a powerful tool for modelling spatial correlations in hydrology, offering a probabilistic approach that quantifies uncertainty while accounting for inter-site dependencies [195]. For instance, Gaussian Process models and Bayesian hierarchical models are commonly used to analyse spatial rainfall patterns, estimate runoff in ungauged basins, and predict flood risks by leveraging spatially distributed data and covariates such as elevation and land cover [44].

Bayesian methods are particularly well-suited for capturing spatial correlations in hydrological data due to their flexibility and ability to incorporate prior information. Bayesian spatial models, such as those using Gaussian–Markov Random Fields (GMRFs) or spatial autoregressive priors, enable the explicit modelling of dependencies between measurements taken at different locations [196,197]. These models can integrate diverse data sources, such as satellite precipitation estimates and ground-based measurements, to provide more robust and comprehensive insights into spatial hydrological processes. In the context of rainfall, Bayesian kriging has been used to interpolate rainfall data over large areas [44], while Bayesian copulas have been applied to model joint distributions of flood peaks across multiple sites [198]. Such spatially aware Bayesian frameworks allow hydrologists to make predictions in data-scarce regions, assess risks more comprehensively, and optimise infrastructure design in the face of spatially variable hydrological inputs.

In addition to spatial correlations, Bayesian methods are invaluable for modelling temporal dependencies in hydrological processes, particularly under the influence of climate variability and periodic patterns, [62,199,200]. These models enable the incorporation of temporal covariates such as El Niño–Southern Oscillation (ENSO) indices, seasonal cycles, and long-term climate trends, allowing researchers to understand how hydrological processes evolve over time. For example, Bayesian state-space models are often used to analyse streamflow time series, while dynamic linear models capture shifting relationships between rainfall and runoff in response to ENSO events. By integrating historical and real-time data, Bayesian frameworks help assess the impacts of changing climate regimes on hydrological processes, providing probabilistic forecasts of droughts, floods, and water availability [201,202,203]. This ability to quantify uncertainty and infer temporal dynamics makes Bayesian methods an essential tool for developing adaptive strategies to manage water resources in the face of climate change.

7. Challenges in Bayesian Hydrology

7.1. Current Trends

Bayesian methods are now widely used in hydrology, particularly for assessing uncertainty in model predictions, model calibration, and data assimilation. One of the major developments has been the application of Markov Chain Monte Carlo (MCMC) methods, which have made Bayesian methods more feasible for complex hydrological systems. The availability of open-source software packages like R version 4.4.0, Python version 3.12.2, JAGS version 4.31, WinBUGS (Windows Bayesian Inference Using Gibbs Sampling) version 1.4.3, and Stan version 2.35.0 has significantly lowered the barriers to applying Bayesian methods in hydrology.

7.2. Advancements in Computational Tools and Software

The development of more efficient MCMC algorithms and parallel computing has made Bayesian analysis more accessible for hydrologists. Integrated Nested Laplace Approximation (INLA), a method that provides approximate Bayesian inference, has also gained traction due to its speed and efficiency, especially in spatial modelling contexts [204]. Additionally, the rise in high-performance computing and cloud-based solutions has further accelerated the adoption of Bayesian methods.

Despite the considerable advancements in Bayesian methods for hydrology, there remain several challenges that limit their widespread adoption in practice. These challenges range from issues related to data and model complexity to computational demands and the interpretation of results [205].

The advancements in Bayesian hydrology have been significantly influenced by computational tools like R version 4.4.0, Python version 3.12.2, JAGS version 4.31, WinBUGS version 1.4.3, and Stan version 2.35.0 These tools have revolutionised how researchers approach uncertainty, spatial–temporal modelling, and predictive analysis in hydrology, offering both flexibility and efficiency in implementing Bayesian methods. Stan has been transformative in Bayesian hydrology due to its high-performance Hamiltonian Monte Carlo (HMC) algorithm, which allows researchers to tackle complex, high-dimensional models with faster convergence and improved accuracy. Its gradient-based approach makes it particularly suitable for handling spatial-temporal data in hydrology, such as rainfall–runoff modelling or groundwater flow simulations [206]. Stan’s integration with R and Python through packages like rstan and pystan has made it a cornerstone for hydrologists seeking advanced hierarchical modelling capabilities [207].

JAGS and WinBUGS remain popular choices for Bayesian hydrology due to their simplicity and effectiveness in modelling smaller datasets and moderate complexity problems. These tools leverage Gibbs Sampling, which, while slower than HMC, provides an intuitive framework for parameter estimation and hierarchical modelling [208,209]. JAGS, in particular, have advanced hydrology research through its seamless integration with R (via rjags), enabling flexible scripting and visualisation [210]. WinBUGS, with its graphical interface, has been instrumental in the early adoption of Bayesian methods for flood-frequency analysis and parameter uncertainty studies [127]. R and Python, as programming environments, have further propelled Bayesian hydrology by offering diverse libraries for Bayesian computation, data manipulation, and visualisation. R packages like brms (which interfaces with Stan), rjags, and coda simplify Bayesian analysis and provide tools for model diagnostics and posterior summaries [211]. Python’s libraries such as PyMC3 [212] mirror these functionalities, offering dynamic options for building Bayesian models with custom workflows.

Collectively, these tools have allowed hydrologists to develop more realistic models that capture the uncertainty inherent in hydrological processes. They support advanced applications like spatial correlation modelling, probabilistic flood forecasting, and climate impact assessment, cementing Bayesian analysis as a fundamental approach in modern hydrology.

Table 2. Comparison of computational tools for Bayesian hydrology.

Feature/Tool	Stan	WinBUGS	JAGS
Core Algorithm	Hamiltonian Monte Carlo (HMC)	Gibbs Sampling	Gibbs Sampling
Ease of Use	Medium (requires coding)	High (graphical interface)	Medium (integrates with R easily)
Speed/Performance	High	Low	Medium
Scalability	Excellent (handles complex models)	Limited	Moderate
Integration	R, Python, CmdStan	Standalone	R (rjags), Python
Applications	Large-scale and complex models	Basic models	Moderate to complex models

, below, compares these computational tools on specific features from ease of use to speed and performance, while also comparing how well they integrate with other software. Overall, Table 2 illustrates that each tool has its strengths, and the choice software depends on the complexity of the model, computational needs, and user familiarity.

7.3. Prior Selection and Subjectivity

One of the main challenges in applying Bayesian methods is the selection of appropriate priors. In Bayesian inference, the choice of prior distribution reflects prior knowledge or beliefs about the model parameters. However, the selection of priors can be subjective, and different priors can lead to significantly different posterior distributions, especially when data are sparse.

Commonly used priors in hydrological applications include non-informative priors (e.g., uniform or Jeffreys’ prior) when minimal prior knowledge exists, or informative priors derived from historical data and expert judgement. For example, [213] provides tools to investigate prior distributions in Bayesian hydrology, while [214] demonstrate the application of empirical Bayes estimation in peak over threshold modelling. The selection of a prior can influence the posterior results, particularly when data are scarce, and thus must be justified based on physical reasoning or historical evidence.

In at-site and regional flood-frequency analysis, the choice of prior distribution for the parameters of a flood-frequency distribution (e.g., Generalised Extreme Value and Log-Pearson type 3 distributions) can heavily influence the resulting risk estimates [12,22,49,51,215,216]. If the prior is chosen poorly or without sufficient justification, it may lead to biased flood risk assessments, which could have significant implications for flood management decisions. To address this challenge, empirical Bayes methods have been proposed as a way to estimate priors directly from the data [4,49,61,160,161,217,218,219]. However, even these methods require careful consideration, as they may still introduce biases, particularly in the case of complex models with many parameters.

The practical implementation of Bayesian methods in this study demonstrates that the choice of prior distributions can influence the posterior results and subsequent uncertainty quantification. In the examples provided in the Supplementary Materials, the selection of priors has been tailored to the specific requirements of each hydrological application, as detailed below.

Example 1. 

Bayesian Flood Risk Assessment

Let Y denote the annual maximum streamflow. We assume that log(Y) ∼ N (μ, σ²). The priors for the parameters are chosen as follows:

For the location parameter (μ): μ ∼ N (0,τ_μ⁻¹), with τ_u = 0.0001, meaning that μ has a very wide Gaussian prior. For the scale parameter (σ):

σ ∼ U (0,10). In this formulation, the precision is defined as τ = 1/σ²

Example 2. 

Bayesian Drought Prediction

Let D denote the drought index (e.g., Palmer Drought Severity Index). We assume that D ∼ N (μ, σ²).

The priors are specified as: For μ: μ ∼ N (0, 10⁶) (using a precision of 1.0 × 10⁻⁶), providing a very uninformative prior. For σ: σ ∼ Gamma (0.1, 0.1), where the Gamma distribution is used to model the precision indirectly.

Example 3. 

Bayesian Rainfall–Runoff Modelling

Consider the linear regression model: Q = β₀ + β₁R + β₂E + β₃T + ε, ε ∼ N (0, σ²), where Q is the runoff, and R, E, and T represent rainfall, evaporation, and temperature, respectively. The priors are defined as: For the regression coefficients (β₀, β₁, β₂, β₃): β_i ∼ N (0, 10³) for i = 0, 1, 2, 3 (using a precision of 0.001). For the precision (τ): τ ∼ Gamma (0.1, 0.1), and the standard deviation is recovered as σ = ${\displaystyle \frac{1}{\sqrt{\tau }}}$.

Example 4. 

Parameter Uncertainty Via GLUE Framework

For a simple rainfall–runoff model defined as Q = aP^b, where Q is runoff and P is precipitation, the parameters are assumed to have uniform priors: For a: a ∼ U (0.1, 1).

For b: b ∼ U (0.5, 1.5). This approach reflects uncertainty by allowing a broad range of plausible values for both parameters.

Example 5. 

Combining Bayesian and machine learning for rainfall–runoff modelling: The Bayesian component of this example employs the same linear regression model as in Example 3: Q = β₀ + β₁R + β₂E + β₃T + ε, ε ∼ N (0, σ²), with priors: for each βᵢ: β_i ∼ N (0, 10³) for i = 0, 1, 2, 3. For τ: τ ∼ Gamma (0.1, 0.1), and σ = ${\displaystyle \frac{1}{\sqrt{\tau }}}$. The machine learning (Random Forest) component does not involve an explicit prior.

Example 6. 

Bayesian hydrology with an interdisciplinary objective: in the interdisciplinary example, where agricultural yield (Y) is modelled as a function of rainfall (X₁) and streamflow (X₂), we consider the linear model

Y = β₀ + β₁X₁ + β₂X₂ + ε, ε ∼ N (0, σ²)

The priors in this example are slightly more informative: for the coefficients (β₀, β₁, β₂): β_i ∼ N (0, 10²) for i = 0, 1, 2 (using a precision of 0.01).

For σ: σ ∼ U (0, 50), with the precision defined as τ = 1/σ².

Therefore, in summarising mathematically:

Normal (Gaussian) priors are used for regression coefficients in Examples 3, 5, and 6, with varying degrees of informativeness (precision of 0.001 vs. 0.01).

Uniform priors are applied for σ in Example 1 and Example 6, as well as for parameters a and b in Example 4.

Gamma priors are chosen for τ in Examples 2, 3, and 5. These choices reflect the need to accommodate data sparsity, capture the skewed nature of hydrological variables, and provide flexibility in uncertainty quantification.

It is important to acknowledge that some degree of subjectivity is inevitable in the selection of priors.

7.4. High-Dimensional Parameter Estimation

Many hydrological models, particularly those used for flood forecasting or climate-change impact assessment, involve high-dimensional parameter spaces. Estimating the posterior distributions of these parameters using Bayesian inference can be computationally expensive, especially when using methods like MCMC. For large-scale models such as those used in global flood or drought forecasting, the parameter spaces may involve hundreds or even thousands of parameters. The computational burden of estimating these parameters, particularly with MCMC or other sampling methods can make Bayesian inference impractical for many real-world applications [220,221,222,223]. Recent advances in variational inference and approximate Bayesian computation (ABC) offer promising alternatives to MCMC. These techniques allow for faster computation by approximating the posterior distribution, though they may sacrifice some of the accuracy of the results [224,225,226].

7.5. Data Limitations and Uncertainty and Computational Demands

Another significant challenge in applying Bayesian methods in hydrology is the inherent uncertainty in the available data. Hydrological data are often sparse, particularly in remote regions or in the context of extreme events like floods and droughts. Additionally, data may be noisy, biased, or incomplete, which can further complicate the modelling process. In regions where streamflow data are sparse, Bayesian methods are often used to estimate parameters by integrating prior knowledge and expert judgement [227,228]. However, the limited number of observations can lead to high uncertainty in model predictions, which can affect decision making in water-resource management. Bayesian methods, particularly MCMC and particle filters, can be computationally expensive, especially when applied to large-scale models or complex systems with many parameters. This is particularly problematic in operational settings, where real-time predictions are needed [229,230,231,232]. In operational flood forecasting, where predictions need to be updated in real time based on incoming data, the computational cost of Bayesian methods may limit their feasibility. Hybrid approaches that combine Bayesian methods with faster machine learning algorithms are increasingly being explored to address this challenge [232,233].

7.6. Interdisciplinary Integration in Bayesian Hydrology

Bayesian hydrology has increasingly embraced interdisciplinary approaches to address the multifaceted challenges posed by climate change, water scarcity, and environmental management. By integrating insights from disciplines such as climatology, ecology, social sciences, and data science, Bayesian methods provide a holistic framework for understanding complex hydrological systems. For instance, climatological inputs, including sea surface temperatures, atmospheric pressure indices, and global circulation models, are commonly incorporated into Bayesian hydrological models to improve predictions of rainfall and streamflow patterns under varying climate scenarios [234,235]. Additionally, coupling Bayesian frameworks with ecological models helps evaluate the impacts of hydrological variability on aquatic ecosystems, such as fish habitats and wetland dynamics [236,237]. This interdisciplinary integration enables the simultaneous consideration of natural variability and anthropogenic influences, improving decision making in water-resource management. Supplementary Example S6 illustrates an interdisciplinary objective where Bayesian hydrology is integrated with agriculture to assess flood risks and their impact on agricultural yield, focusing on the joint modelling of extreme rainfall and streamflow using copulas and Bayesian inference.

Advances in computational science and machine learning have further expanded the interdisciplinary applications of Bayesian hydrology. Techniques like Bayesian neural networks and hierarchical Bayesian models are increasingly used to merge hydrological data with socioeconomic factors, enabling the assessment of water availability, agricultural productivity, and flood risk in a socio-environmental context [238,239,240]. Moreover, Bayesian decision analysis frameworks integrate stakeholder inputs to balance competing objectives in water-resource planning, such as flood protection and ecosystem preservation [241]. These approaches underscore the strength of Bayesian methods in synthesising diverse datasets and perspectives, providing probabilistic insights that align with interdisciplinary goals. By bridging domains, Bayesian hydrology is poised to address urgent global water challenges with robust, data-driven, and integrative solutions.

7.7. Limitations of the Current Study and the Way Forward

Despite the strengths of Bayesian methods, several limitations remain. The computational demands of advanced Bayesian models, especially when using MCMC techniques for high-dimensional problems, can be significant. Additionally, the subjectivity involved in selecting prior distributions may influence the results, particularly in data-sparse situations. This review focuses on the methodological aspects and literature trends, yet future work should aim at developing more efficient algorithms and objective criteria for prior selection to enhance reproducibility.

8. Conclusions

In this review, we have traced the historical development of Bayesian methods in hydrology, highlighting key contributions, advancements, and the challenges that have shaped their application. From their origins in parameter estimation and uncertainty quantification to their current use in advanced predictive modelling, Bayesian methods have proven to be invaluable tools in hydrological research and practice. The ability to incorporate prior knowledge, update predictions with new data, and quantify uncertainty allows hydrologists to make more informed decisions, particularly in contexts where data are sparse or uncertain.

However, as has been discussed, there are still several challenges that must be addressed for Bayesian methods to be more widely adopted in operational hydrology. These include the subjective nature of prior selection, high-dimensional parameter estimation, data limitations, and the significant computational resources required for real-time forecasting. Overcoming these challenges will require continued research into alternative methods for prior selection, as well as advancements in computational techniques such as variational inference, parallel computing, and hybrid models that integrate machine learning with traditional hydrological models. Moreover, R has emerged as a powerful tool for implementing Bayesian methods in hydrology, with applications spanning parameter estimation, flood-frequency analysis, drought prediction, rainfall–runoff modelling, sediment transport analysis, and uncertainty quantification. The six examples presented in the Supplementary Materials demonstrate how Bayesian frameworks can address a wide range of hydrological challenges effectively.

Looking to the future, the integration of big data, machine learning, and real-time data assimilation with Bayesian methods is set to be a major area of growth. With the increasing availability of high-resolution data from remote sensing, sensor networks, and other sources, there is a growing opportunity to combine data-driven insights with physical models, improving the accuracy of predictions while maintaining the rigorous uncertainty quantification provided by Bayesian inference. Hybrid models, which leverage the strengths of both physical and data-driven approaches, will likely become more prevalent, providing a powerful framework for addressing complex hydrological challenges. Furthermore, as climate change continues to affect water resources and hydrological extremes, the need for robust uncertainty quantification in climate-change modelling will only grow. Bayesian methods offer a powerful approach to capturing the uncertainty associated with future climate projections, making them indispensable in water-resource management and flood risk assessment.

In conclusion, the future of Bayesian hydrology is bright, with many exciting opportunities for innovation and advancement. As computational capabilities continue to improve and as new data sources become available, Bayesian methods will remain at the forefront of hydrological research, helping scientists and engineers tackle some of the most pressing water-related challenges of our time. Whether in flood forecasting, flood-frequency analysis, drought prediction, water quality monitoring, or climate-change impact assessment, Bayesian methods will continue to offer critical insights that improve our understanding of hydrological processes and inform better decision making.