Bayesian analysis is a reliable framework that allows researchers to make probabilistic inference using prior knowledge and observed data. In practice, eliciting informative prior distributions can pose challenges, particularly when subjectivity comes into play. Elicitation methods of prior distributions then serve as an important tool for systematically integrating expert knowledge into the Bayesian framework.

In numerous real-world scenarios, such as medical research, quality control, social sciences and marketing, the parameter of interest is often a probability or proportion (Korkmaz 2020). In such situations, the binomial sampling arises in which independent identical trials are repeated, with success or failure outcomes. The probability of success is assumed to be fixed across trials. Accurately estimating this probability is crucial for effective decision-making across various domains.

Prior distributions for probabilities over the bounded unit interval [0, 1] have been limited to typical continuous distributions like the Beta, Triangular or Uniform distributions. Families of one-, two- and three-parameter distributions have appeared in the literature. Many univariate distributions over the unit interval are listed in Mazucheli et al. (2022).

The Topp-Leone distribution (Topp and Leone 1955) is a popular one-parameter distribution that was used in the late 90’s. The Alpha-Unit distribution (Concha-Aracena et al. 2022) has recently been introduced as a one-parameter distribution over [0, 1]. Among the two-parameter distributions, the Beta distribution has conventionally been the most popular choice due to its inherent confinement within the [0, 1] interval. Nonetheless, it does not necessarily capture all the true beliefs or uncertainties in every practical scenario.

To address the limitations of the Beta distribution, this study proposes elicitation methods of some non-conjugate priors from the one-parameter distributions: Continuous-Bernoulli, Alpha-Unit, and Topp-Leone; and from the two-parameter distributions: Kumaraswamy, Unit-Weibull, Beta Type 3, Unit-Chen, Logit-Normal, and Unit-Logistic. These priors exhibit similar properties to the Beta distribution, especially as elicited priors for the probability of success, p, in the binomial sampling. We also elicit and compare some three-parameter prior distributions that have been recently proposed: the Libby-Novick Beta, Libby-Novick Kumaraswamy, and Unit-Exponential Pareto.

Our choice of these priors to elicit is motivated by the advantages inherent in the availability of closed-forms of the CDF and quantile functions and the ease of obtaining the corresponding posterior distributions. Another key motivation for introducing these priors is their potential use in regression models where the response variable is constrained to the [0, 1] interval, which is common for modelling probabilities, proportions or rates. This will improve the traditional regression models, such as the Beta regression model, by providing better data fits across various applied domains, that gives a clearer understanding of complex relationships in areas where outcomes naturally fall between 0 and 1.

We also investigate the elicitation of Quantile-Parameterised Distributions (QPD) (Keelin and Powley 2011). These distributions were recently proposed as a flexible class of continuous probability distributions that use only quantiles to parameterise a distribution. These distributions are notable for their unique property that the quantiles obtained through the elicitation process directly and fully determine the prior distribution. No further adjustment nor reconciliation are needed. This feature ensures not only a strong fit but also enhances the interpretability of the parameters involved (Perepolkin et al. 2025). Among the QPDs, we have chosen the Johnson Quantile-Parameterised Distributions (J-QPD-B), which is defined on the bounded interval [0, 1] and can represent any symmetric percentile triplet (SPT) of quantile assessments (Hadlock and Bickel 2017).

In Bayesian analysis, prior elicitation often involves asking questions about distributions or probabilities (Galway 2007). Some elicitation methods focus on the quantiles of PDFs and CDFs (Winkler 1967), while others relate directly to probabilities and their relationship with these functions (O’hagan et al. 2006). For those new to Bayesian methods, it can be difficult to determine prior distributions without historical data, making expert opinions particularly valuable. However, this process can be complex, influenced by factors like cognitive biases and heuristics, as discussed in various studies (Gigerenzer and Hoffrage 1995; Koehler 1996; Daneshkhah 2004; Kynn 2008). Although we do not describe formal steps for bias-reduction in this paper, we make sure experts are well debriefed in advance about the elicitation tasks, then frequent graphical and numerical feedback are given throughout the process. Comprehensive reviews of expert elicitation methods, including cognitive bias-reduction techniques, can be found in O’Hagan (2019), Mikkola et al. (2024), Stefan et al. (2022) and Falconer et al. (2022).

In this paper, we adopt the approach of Elfadaly and Garthwaite (2013) and Elfadaly and Garthwaite (2017) by assessing a lower quartile, median, and upper quartile for the proposed univariate priors. This quartile method (also called variable interval method or bisection method (Spetzler and Stael von Holstein 1975)) has several advantages over other elicitation methods, as it provides a more structured approach to eliciting prior information (Kiefer 2016). The ease of the quartile method lies in its straightforward implementation and the minimal cognitive load it imposes on the expert. It does not require complex calculations or extensive knowledge of statistical concepts. Additionally, the method allows for easy adjustments and refinements based on the expert’s input, making it flexible in accommodating different levels of uncertainty or changing beliefs (Winkler 1967).

To improve the accuracy of assessments, we believe that the experts should be asked to provide a median as a location value along with the quartiles for scale. The median and quartiles are convenient for experts to assess as they can be obtained using the bisection method. Experts are asked to identify the median as the value at which the probability of success, p, is equally likely to be above or below. They then divide the interval above the median into two equally likely intervals to assess the upper quartile. The lower quartile is similarly assessed by dividing the interval below the median into two equally likely intervals.

It is argued, however, that relying on only three quartiles might lead to a poor fit for some highly-skewed or heavy-tailed distributions, or more importantly, the quartile approach might struggle to identify or distinguish among different distributions (Stefan et al. 2022). Obtaining more assessments from the expert is one way to overcome this. However, we prefer to elicit only three quartiles for the advantages mentioned before. We then ask the expert to choose the distribution that best represents his opinion based on both the graphical feedback and the optimisation error corresponding to each plausible distribution.

Bockting et al. (2024) use simulation-based and gradient-driven approaches to elicit priors from experts’ assessed moments, quantiles and histograms. Our proposed method is a simpler and more straightforward analogue of their quantile-based approach. The target quantity we directly assess here is the parameter p. So any set of hyperparameters for a prior distribution of p will simply give a specific set of three quartiles, given that the quartile function is available for all univariate priors we consider. Therefore, no iterative simulation is required, but we optimise a discrepancy function as in Bockting et al. (2024).

In Sect. 2 of this paper, we introduce an elicitation method for 14 prior distributions that are confined within the [0, 1] interval. Namely, the elicitation methods are given for the Continuous Bernoulli, Alpha-Unit, Topp-Leone, Beta, Kumaraswamy, Unit-Weibull, Beta Type 3, Unit-Chen, Logit-Normal, Unit-Logistic, Libby-Novick Beta, Libby-Novick Kumaraswamy, Unit-Exponential Pareto, and J-QPD-B distributions. These distributions provide a suitable framework for modelling various types of data and expressing expert opinions in a meaningful way. It offers a variety of options to capture different opinions, helping experts to customise their selections based on their insights and knowledge. Our goal is to improve the precision and effectiveness of Bayesian analyses by ensuring the elicited prior distributions accurately reflect the experts’ perspectives.

We developed a Shiny R application, PEBSI (Prior Elicitation for Binomial Sampling models on the unit Interval) [https://unnipillai.shinyapps.io/PEBSI/], that implements the proposed elicitation methods for all the 14 suggested prior distributions. The Shiny R application is described in Sect. 3. It features an interactive interface that facilitates the elicitation of the proposed priors and provides visualisation of the elicited prior distributions with their corresponding posterior distributions and a triplot, facilitating a full Bayesian analysis framework. Section 4 gives three examples from ecology, genetics and veterinary medicine that demonstrates the usability and usefulness of our developed tool. Finally, some concluding comments are given in Sect. 5.

Uncertainty about bounded phenomena is a prevalent aspect in applied statistics. Many fields involve variables that fall within the unit interval [0, 1], such as proportions, test scores or rates. These variables have been extensively explored in diverse applications (Cribari-Neto and Souza 2013). To effectively model these phenomena probabilistically, it is crucial to use continuous probability distributions on the unit interval [0, 1]. Such distributions play a significant role in capturing the inherent nature of these variables and in facilitating insightful analysis in various domains.

While the Beta distribution is the most popular choice to model variables in the [0, 1] interval, it has some limitations. The lack of analytical expressions for its distribution function, which is given in terms of an incomplete beta function, can sometimes complicate calculations and limit its use in certain analytical contexts. Moreover, it lacks flexibility in capturing multimodal or complex shapes and has limited tail behaviour. Therefore, in order to find a more flexible prior distribution for the success probability parameter, p, in binomial sampling models, we have explored various continuous probability distributions over the unit interval [0, 1]. From these distributions, we have selected specific distributions that exhibit close connections to the Beta prior and demonstrate better performance compared to the Beta distribution in certain domains. The subsequent subsections delve into the significance and prominence of these selected distributions.

A unified elicitation framework is given for all priors with one-, two-, and three-parameters for modelling binomial sampling with a probability parameter p. We start by asking the expert to assess a lower quartile \((Q_1)\), median \((Q_2)\), and an upper quartile \((Q_3)\). We then minimise the least-squares function

where \(Q_D(r)\) is the quantile function of distribution D evaluated at r. We used the Rsolnp package in R (Ghalanos et al. 2012) to numerically minimise \(W_D\) to determine the parameters of various prior distributions. This optimisation function was used in Oakley and Hagan (2010) and Elfadaly and Garthwaite (2013, 2017).

The Rsolnp package in R implements a Sequential Quadratic Programming (SQP) algorithm for solving constrained nonlinear optimisation problems. We apply it to minimise the least-squares error function \(W_D\) that quantifies the deviation between expert-given quartiles and the quartiles of chosen prior distributions. To improve convergence and avoid local minima, particularly in multi-parameter distributions, a coarse grid search is first employed by the package to identify plausible initial values, which are then refined by the optimisation routine.

This approach is reliable and efficient across all 14 prior distributions. Each optimisation typically completes in under one or two seconds on a standard machine, making it suitable for interactive use and rapid prior comparison in the Shiny R. The use of \(Q_1\), \(Q_2\), and \(Q_3\) as the basis for optimisation is a practical choice, as these quartiles are intuitive for many experts and provide a simple yet informative summary of location and spread. However, the method is not limited to these specific values. In principle, any three informative quantiles can be used, and the approach can be adapted to alternative summaries commonly reported in applied fields (Cook 2010). For example, in clinical or epidemiological research, it is common to report a median and a \(95\%\) confidence interval.

While this approach successfully captures central tendency and spread, it may be less sensitive to differences in tail behavior between distributions with similar inter-quartile ranges. This limitation is particularly relevant when comparing skewed or heavy-tailed distributions, where tail properties are often scientifically meaningful. To mitigate this, users are encouraged to compare the shape and fit of chosen priors both visually and numerically using the Shiny R application.

In the following Sects. 2.1–2.3, we demonstrate how the optimisation function in (1) is used to elicit prior distributions in each category of one-, two- and three-parameter priors, respectively. We have also conducted a comparison of priors in each category to show their relative importance and significance, offering key insights into their respective strengths and weaknesses.

Although two-parameter distributions, such as the Beta distribution, offer flexibility and depth in conveying prior knowledge about the probability parameter p, there are some situations in which one-parameter distributions might be preferable. These situations arise when computational simplicity is required or when there is little prior knowledge about the parameter. Experts in these situations might find it easier to give only one accurate assessment, such as the median, say, for the parameter of interest, reducing the cognitive load and the chance of bias in specifying more assessments. This usually involves simpler questions or fewer steps, simplifying the process and making it quicker and more efficient. Eliciting a one-parameter distribution requires only one assessment, although more assessments can also be obtained for over-fitting.

In what follows, we give a method to elicit three one-parameter priors, namely the Continuous-Bernoulli, Alpha-Unit, and Topp-Leone distributions that are defined over the interval [0, 1] and can serve as prior distributions for the binomial models. These priors differ in several ways from their multi-parameter counterparts, but they can still capture the key features of prior beliefs about the probability of success p in some specific situations, as detailed below.

The Continuous-Bernoulli distribution (Loaiza-Ganem and Cunningham 2019) relaxes the discreteness constraints of the Bernoulli distribution. It is defined on the continuous range [0, 1] and has only one shape parameter, say \(\lambda \). Its PDF, CDF and quantile function are given in Table 5.

This distribution is frequently encountered in the field of artificial intelligence, particularly in modelling pixel intensities, hence facilitating various applications in image processing, computer vision, and related fields (Korkmaz et al. 2023). This distribution is therefore relevant in these fields and can be widely used as a reasonable prior in the context of elicitation. Also, it is worth noting that there exists a multivariate extension of this distribution, namely the Continuous-Categorical distribution (Gordon-Rodriguez et al. 2020).

To elicit a Continuous-Bernoulli distribution, we minimise the least-square optimisation function \(W_D\) in (1) using three assessed quartiles \(Q_1\), \(Q_2\) and \(Q_3\) where \(Q_D(r)\) is taken as the quantile function of the Continuous-Bernoulli distribution (see Table 5). This elicits the hyperparameter \(\lambda \) and thus determines the Continuous-Bernoulli distribution.

For a one-parameter prior, it suffices to use only one assessed quartile to obtain the parameter \(\lambda \). However, we use the principle of over-fitting here and make use of the three-quartile assessments. Over-fitting, in this context, is beneficial for reducing any imprecision in the assessed summaries. This flexibility enables the fitted compromise, through the optimisation function \(W_D\), before potentially providing a more accurate depiction of the expert’s knowledge, as noted in O’hagan et al. (2006).

The Alpha-Unit distribution is a recently proposed distribution generated using the moment of order two of the standard normal distribution (Concha-Aracena et al. 2022). It is a unit distribution defined on the interval [0, 1] with a single positive parameter, say a.

If Y is distributed as a bimodal half-normal distribution with order one, then \(Y^2 \sim \chi ^2_3,\) a chi-square distribution with 3 degrees of freedom. The Alpha-Unit model is generated by the transformation \(X = \exp \{-a |Y|\}\). Its PDF, CDF and quantile function are given in Table 5.

The Alpha-Unit distribution is then fully elicited once a is determined, based on an expert’s three assessed quartiles, using the optimisation function \(W_D\) in Eq. (1).

The Topp-Leone distribution (Topp and Leone 1955) is a valuable distribution for modelling lifetime data. It is distinguished by a J-shaped density function and a bathtub-shaped hazard function. The PDF, CDF and quantile functions of the Topp-Leone distribution are given in Table 5 (see also Nadarajah and Kotz 2003).

Recent studies have concentrated on the Bayesian estimation techniques for the shape parameter of the distribution, utilising approaches like normal approximation and the Tierney-Kadane method, alongside various informative and non-informative priors (Sindhu et al. 2013; Sultan and Ahmad 2015). The Topp-Leone distribution has also been compared to the Beta distribution as a prior in Bayesian analysis for binomial sampling (Lin and Balakrishnan 2023).

The PDF curves of all the three one-parameter distributions are plotted in Fig. S1 in Supplementary Material.

Table 1 provides a comparative error analysis of the three one-parameter priors - Continuous-Bernoulli (CB), Alpha-Unit (AU) and Topp-Leone (TL) for different sets of assessed quartiles. The errors, calculated using the least-squares optimisation function in (1), indicate how well the expert-assessed quartiles align with the actual quartiles of each distribution. The lower the error, the better the fit with the expert assessments.

For the quartile sets (0.1, 0.2, 0.3), (0.3, 0.4, 0.5) and (0.4, 0.5, 0.6), the AU prior has the lowest error. This suggests that it captures expert opinions most effectively in this range. The AU prior exhibits a strong positive skewness for (0.1, 0.2, 0.3) and slightly positive skewness for the other sets of quartile combinations. Such skewness is commonly observed in scenarios like low-risk returns and early-stage adoption rates, where most outcomes are concentrated at the lower end, but there is potential for higher returns.

The CB prior performs best for the set (0.1, 0.2, 0.9) and exhibits a decreasing PDF. Scenarios where a decreasing PDF is relevant include assessing risks for investments with limited returns, where the likelihood of surpassing a specific threshold decreases (Tankov 2003). This pattern can also happen in quality control, where improved processes lead to a lower probability of higher defect rates (Montgomery 2019).

For the quartile set (0.7, 0.8, 0.9), the TL prior has the lowest error compared to others. The TL has a negatively skewed PDF. These can be seen in stock market returns, where investors experience small gains on most trading days but may face occasional significant losses, resulting in a distribution that skews to the left.

Two-parameter distributions provide greater flexibility than the one-parameter distributions and can represent a variety of PDF shapes. They can model increasing, decreasing, unimodal or bathtub shapes.

In the Bayesian analysis for binomial sampling models, the choice of a prior distribution for p significantly impacts the results. While the Beta distribution is the most popular choice due to its conjugacy to the binomial likelihood, it may not always reflect the true prior beliefs about p, depending on the specific domain of applications.

In the following sections, we briefly discuss elicitation methods for a Beta distribution and for six other two-parameter distributions that can serve as priors for p in Binomial sampling models. These priors were chosen because they all have closed-form CDFs and quantile functions which simplifies the elicitation method as well as the calculation of the posterior distribution. We do a comparative study with these priors and the most popularly used Beta prior, highlighting the benefits of each prior and emphasising the importance of using new flexible two-parameter priors.

Different methods exist in the literature for eliciting expert opinions about the parameters of a Beta distribution. Some methods ask an expert for a specific location measure such as the mean, median or mode (Gross 1971), or dispersion measure such as the mean deviation about median (Pham-Gia et al. 1992), while others ask for a confidence interval first and then derive the mean from the center of the interval (Cuevas 2012). The methods also differ in how the elicited values are used to obtain the hyperparameters of the Beta distribution. The bisection method was used to ask an expert to assess the three quartiles and to then use them to elicit the hyperparameters of the Beta distribution in Elfadaly and Garthwaite (2013) and Elfadaly and Garthwaite (2017). Their method is based on minimising the \(W_D\) function in Eq. (1) and is used in this current work and the accompanying software for eliciting the Beta distribution.

The Kumaraswamy distribution (Kumaraswamy 1980) is a continuous probability distribution with two shape parameters, say a and b, over the interval [0, 1]. Its PDF can exhibit various forms, such as the unimodal, monotonic, symmetric (when \(a=b\)), asymmetric, or uniform (when \(a=b=1\)) (Fletcher and Ponnambalam 1996). This distribution is sometimes referred to as the double-bounded distribution and is similar in shape to the Beta distribution (Jones 2009). The PDF, CDF and quantile function of the Kumaraswamy distribution are given in Table 5.

The CDF and quantile function of the Kumaraswamy distribution have closed forms, making it more practical for various applications, simulation studies and specifically useful in the elicitation context. An intriguing feature of utilising this distribution lies in the fact that obtaining the parameters from the assessed quartiles becomes considerably easier due to the closed-form expression of the quantile function.

The Kumaraswamy distribution also offers some other notable benefits over the Beta distribution, such as the explicit formulas for the moments of order statistics and a straightforward method for generating random variables, as highlighted by Ishaq et al. (2019). The Kumaraswamy distribution finds broad applications in hydrology (Kumaraswamy 1980) and other earth sciences (Fletcher and Ponnambalam 1996), where Beta struggles to fit in these domains. It has also been used in various engineering subfields, such as electrical, civil, and financial engineering, to describe various phenomena (Michalowicz et al. 2013).

Mazucheli et al. (2018) proposed the Unit-Weibull distribution on the unit interval \(\left[ 0, 1 \right] \) as an alternative to the Kumaraswamy distribution for modelling quantiles. The authors established various structural properties of the distribution and demonstrated its flexibility and competitiveness compared to distributions like the Beta and Kumaraswamy or modelling quantiles conditional on covariates. Unlike the Beta distribution, the Unit-Weibull distribution possesses a closed-form expression for its CDF and quantile function. The distribution is widely used for reliability and lifetime data. In several applied areas, such as estimating maximum flood levels, characterising petroleum reservoirs, assessing cost-effectiveness in risk management, and modelling CD34+ cell recovery rates, this distribution has been found to perform better than alternatives like the Beta and Kumaraswamy (Mazucheli et al. 2020).

Suppose a random variable Y follows the Weibull distribution, then \(X=\exp (-Y)\) is said to follow the Unit-Weibull distribution with PDF, CDF and quantile function as given in Table 5.

The Beta Type 3 distribution (Cardeno et al. 2005) is obtained by transforming a standard Beta distribution as follows. Let Y follow a Beta distribution with parameters a and b, and define \(X=Y/(2-Y)\). Then X is said to follow the Beta Type 3 distribution with PDF, CDF and quantile function as given in Table 5.

Like the standard Beta distribution, the shape of the Beta Type 3 distribution varies with its parameters a and b. This flexibility allows it to capture different prior beliefs about a binomial parameter p.

A multivariate version of the Beta Type 3 distribution has been suggested as the Dirichlet Type 3 distribution (Mecene and Ghorbel 2023). Beta type 3 distributions have applications in various areas, including the study of matrix variate distributions and modelling data on the unit disk (Gupta and Nagar 2009). The choice between a Beta Type 3 and a standard Beta distribution as a prior depends on the specific needs of the model and which distribution best reflects the prior beliefs about the binomial parameter p.

Korkmaz et al. (2022) introduced the Unit-Chen distribution, which was found to outperform other models including Beta and Kumaraswamy distributions in some specific applications. The distribution has wide applications on stress-strength models and lifetime data (Kayal et al. 2020; Tarvirdizade and Ahmadpour 2021). This distribution also provided a better fit compared to Beta and Kumaraswamy in quantile regression models (Sarhan 2025).

Let Y be a non-negative random variable with the Chen distribution, then \(X=\exp (-Y)\) is said to have the Unit-Chen distribution with PDF, CDF and quantile function as given in Table 5. Depending on its parameters, the PDF can be left-skewed, right-skewed, or bathtub-shaped, and display unimodal, reverse J-shaped, or U-shaped characteristics.

In many statistical applications, the logit-normal distribution has drawn interest. It is obtained by transforming a normal distribution using the logistic transformation. In comparison to the normal distribution, it provides benefits for modelling data that is bounded between 0 and 1, such as probabilistic and compositional data (Atchison and Shen 1980a).

Cohen et al. (2008) utilised the logit-normal distribution in a Bayesian model for probabilistic grammars, achieving notable improvements in unsupervised grammar induction. Lenk (1988) expanded its use to model the common density of an exchangeable sequence of observations, offering a robust framework for predictive density computation. Atchison and Shen (1980b) showed its superior fit for pollen count proportions in environmental studies, where Beta tails decayed too quickly. Thomson et al. (2019) introduced the logit-normal as a continuous analog of the spike-and-slab prior, demonstrating comparable performance to machine learning methods in biological data analysis. These studies highlight the logit-normal distribution’s utility in diverse fields, from genomics to linguistics, offering a reliable prior for binomial sampling models.

Let Y be normally distributed as \(Y \sim N(\mu , \sigma )\), and let \(X= \exp (Y)/[1+\exp (Y)].\) Then X is said to follow the logit-normal distribution with PDF as given in Table 5.

The CDF and quantile function do not have analytical forms but can be easily obtained in R using the greybox package (Mead 1965).

The unit-logistic distribution is versatile and useful for modelling data on the unit interval [0, 1] (Menezes et al. 2018). The flat probability scale of the unit-logistic prior ensures that it does not introduce strong biases, making it suitable for a wide range of applications (Bickis 2009).

Let Y follow a standard logistic distribution and, for any \(\mu >0\) and \(b>0\), take \(X=\exp [(Y-\mu )/b]/\{1+\exp [(Y-\mu )/b]\}\). Then X is said to follow the unit-logistic distribution with PDF, CDF and quantile function as given in Table 5.

The PDF of the Unit-Logistic distribution exhibits remarkable versatility in its shapes, ranging from unimodal and bell-shaped to increasing, decreasing, or constant, depending on its parameters \(\mu \) and b. It can be symmetric or asymmetric, and even U-shaped for certain parameter combinations.

The unit-logistic distribution, though versatile, hasn’t been widely studied in the literature. However, recent research has started to explore its properties and potential applications, particularly in regression analysis and modelling proportion data (Iliev et al. 2019) and hence we propose eliciting the distribution as a promising alternative prior for binomial sampling models.

PDF curves of all the seven two-parameter distributions are plotted in Figs. S2 and S3.

Table 2 presents a comparison of optimisation errors for various sets of assessed quartiles for different prior distributions: Beta (B), Kumaraswamy (K), Unit-Weibull (UW), Beta Type 3 (B3), Unit-Chen (UC), Logit-Normal (LN), and Unit-Logistic (UL). These errors are the minimum values obtained by minimising the least-square optimisation function in Eq. (1). This error measures how closely the actual quartiles of each elicited distribution align with the quartiles assessed by the expert. The distribution with the lowest error is considered to align best with the expert’s assessments.

The Kumaraswamy distribution is a preferable choice when quartiles are closely clustered together. For example, for quartile combinations like (0.1, 0.2, 0.3) and (0.3, 0.4, 0.5), the elicited Kumaraswamy distribution shows the smallest error, indicating it aligns closely with the quartiles assessed by experts. In the case of (0.1, 0.2, 0.3), the distribution shows a high positive skewness. Meanwhile, (0.3, 0.4, 0.5) indicates a distribution that is evenly spread across the middle half, giving a fairly balanced or slightly skewed shape. These findings show how well Kumaraswamy captures the essence of expert opinions, particularly in accurately portraying central quartiles with a touch of variability, making it a viable choice for modelling scenarios like hydrology (Kumaraswamy 1980) needing precise central tendency and moderate spread.

The Beta, Logit-Normal, and Unit-Logistic distributions all have an error of 0, for the quartile combination (0.4, 0.5, 0.6), showing perfect alignment with the experts’ quartile assessments. In this case, they exhibit symmetric shapes, providing valuable practical characteristics in various scenarios. Symmetric distributions make it easier to interpret data and improve the accuracy of statistical tests since the mean, median, and mode are the same. In finance, they help with evaluating risk and managing portfolios by giving equal chances to extreme outcomes (Hürlimann 2001).

The Unit-Weibull prior distribution emerged as a favorable choice for the quartile combinations, such as (0.1, 0.6, 0.7). One particularly interesting aspect of the Unit-Weibull distribution is its strong performance when the PDF curve exhibits a bathtub shape. This shape corresponds to situations where the failure rate of a system is initially high, then decreases, and rises again towards the end of its lifespan. The Unit-Weibull distribution captures this pattern exceptionally well, resulting in minimal error from the optimisation function. The close alignment between the assessed quartile values and the calculated quartile values indicates that the Unit-Weibull distribution accurately represents the underlying assessments.

For quartile combinations such as (0.1, 0.2, 0.9), the Logit-Normal and Unit-Logistic distributions have the smallest error. These distributions are notable for their shapes, which can show a high concentration of values near zero, indicating that very small probabilities are most likely. As the value of p increases, the density rapidly decreases, creating a long tail towards 1. This shape is applicable in areas such as econometrics for modelling proportion data, medicine for estimating treatment success rates, and environmental science for assessing the risk of rare events (Muse et al. 2021). Essentially, it helps in scenarios where probabilities are expected to be closer to 0 or 1 rather than evenly spread out.

The Beta Type 3 distribution has the least error for the quartile combinations (0.4, 0.6, 0.8) and (0.5, 0.6, 0.7). This indicates that this distribution performs better than the Beta distribution when there is a high negative skewness.

While one- and two-parameter distributions offer reasonable flexibility as elicited priors, three-parameter distributions can provide even greater versatility by accommodating complex prior beliefs and intricate shapes like skewness and peakedness. Here, we introduce elicitation methods for some three-parameter priors suitable for binomial sampling models, outlining their forms and potential advantages, disadvantages, and domains of applications.

The Libby-Novick Beta (LNB) distribution is a generalised family of continuous probability distributions that extends the standard Beta distribution. It was proposed as a new class of distributions with an additional shape parameter, say c, offering more flexibility for modelling real-world data compared to the standard Beta distribution (Iqbal et al. 2021).

Let Y follow a standard Beta distribution with parameters a and b. If we set \(X=Y/[c+(1-c)Y]\), for \(c>0\), then it is said that X has the LNB distribution with PDF, CDF and quantile function as given in Table 5.

This distribution provides greater flexibility in modelling skewness, kurtosis and tail weights of data compared to the Beta distribution and other related distributions like the Kumaraswamy distribution (Ghosh 2023). The additional shape parameter in the LNB distribution allows for better control over the distribution’s shape, potentially providing improved fits to real data in various fields (Cordeiro et al. 2014). The LNB distribution has recently gained attention for its flexibility and has found many applications in various fields (Ghosh 2023).

The additional parameter, c, offers more flexibility and allows for a more accurate representation of expert knowledge, especially when dealing with complex prior beliefs that simpler distributions cannot adequately capture.

Mitnik and Baek (2013) highlighted the practical advantages of the Kumaraswamy distribution, such as its closed-form CDF and quantile function, making it suitable for simulation studies. Saboor et al. (2021) built on this work by introducing the Libby-Novick Kumaraswamy (LNK) distribution, which combines the flexibility of the LNB family with the practical advantages of the Kumaraswamy distribution. This was independently introduced as the New form Libby-Novick distribution (NLN) by Ali Ahmed (2021) as an extension of LNB distribution and has been compared to Beta and Kumaraswamy distributions by fitting it to lifetime and engineering data (Saboor et al. 2021), demonstrating its practical applicability.

Unlike the LNB, the LNK distribution has a closed-form CDF and quantile function and its moments are analytically tractable, making it simpler for eliciting priors from the assessed quartiles. The properties of the LNK distribution are very promising and the distribution can be used as a flexible prior in Bayesian analysis. The additional flexibility provided by its three parameters makes it a suitable choice for modelling prior beliefs in certain scenarios, particularly those involving data with potential skewness.

The PDF, CDF, and quantile function of the LNK distribution are given in Table 5.

Haj Ahmad et al. (2023) proposed the Unit-Exponential Pareto (UEP) as a three-parameter distribution on [0, 1] to model COVID-19 recovery rates, demonstrating its flexibility in handling decreasing, symmetric, and asymmetric data with monotone failure rates.

If Y follows the Exponential Pareto distribution then \(X = Y/(1 + Y)\) has the Unit-Exponential Pareto (UEP) distribution with PDF, CDF and quantile function as given in Table 5.

This distribution is particularly useful in modelling scenarios where extreme values are of interest, such as the recovery rates in epidemiological studies. Its ability to capture heavy-tailed behaviour makes it a suitable distribution for understanding the distribution of rare events or outliers in data (Haj Ahmad et al. 2023) and hence it can be a flexible choice for use as a prior in our elicitation context.

The Johnson Quantile-Parameterised Distribution (J-QPD) is a system of distributions that extends the standard Johnson distribution by allowing distributions to be directly parameterised using symmetric triplets of quantiles (Hadlock and Bickel 2019). In the context of elicitation, this system of distributions avoids the need for curve fitting and ensures the distribution fully aligns with the assessed quantiles. Moreover, J-QPDs are very flexible, able to mimic the shapes of many standard distributions like Beta and Kumaraswamy (Hadlock and Bickel 2019). The concept builds on the earlier work with quantile-parameterised distributions (QPD), developed to overcome limitations in traditional probability elicitation methods used in decision analysis (Keelin and Powley 2011).

J-QPDs can effectively approximate a wide variety of commonly used distributions, including beta, gamma, lognormal, and Weibull distributions. This versatility makes them ideal for applications where traditional distributions might not be a good fit. Furthermore, an optimisation step is not necessary for determining the parameters of the distribution (Perepolkin et al. 2025). The J-QPD system is divided into two subfamilies: J-QPD-B (bounded) and J-QPD-S (semi-bounded), enabling the modelling of distributions with known support bounds, which is often required in practical scenarios. Our interest here is in using J-QPD-B as a potential prior for binomial sampling models over the bounded interval [0, 1].

The J-QPD-B distribution is entirely defined by the three (assessed) quartiles \(Q_1\), \(Q_2\) and \(Q_3\). Its quantile function is given in Table 5.

Some PDF curves of all the four three-parameter distributions are illustrated in Fig. S4 for different sets of parameters.

Table 3 compares the performance of LNB, LNK, and UEP across different quartile combinations in terms of the least-squares error, which measures how closely the quartiles from the expert assessments align with the quartiles derived from the respective distributions. Note that J-QPD-B is a distribution that is parametrised using the assessed quartiles, so no optimisation is needed, and the distribution quartiles are always equal to the assessed quartiles. The error will therefore always be 0. Hence, this distribution is not compared to the other three distributions in the error analysis.

For the quartile combination (0.1, 0.2, 0.3), UEP has the lowest error compared to the other priors. The PDF is positively skewed, which can be practically applicable in relevant scenarios, including risk assessment, early-stage forecasting, and quality control (Hamedani et al. 2023).

LNB is more suitable for quartile combinations such as (0.4, 0.5, 0.6), showing a lower error than that of the LNK and UEP. This quartile combination is one of these scenarios that typically reflect data concentrated in the lower to middle ranges. The PDF of the distribution is characterised by a single peak skewed towards lower values. These align well with applications such as risk assessments and initial market analyses, where understanding outcomes at lower percentiles is crucial (Campisi et al. 2023).

LNK demonstrates strong performance in quartile combinations like (0.1, 0.6, 0.7), (0.3, 0.4, 0.5) and (0.7, 0.8, 0.9) surpassing LNB and UEP with lower errors. These quartiles typically represent shapes with slightly symmetric, broader ranges, extremes, and high negative skewness, respectively. For most quartile combinations, we found that LNK has lower errors compared to the other two prior. The PDF of LNK exhibits a wider peak and extended tails, making it well-suited for modelling financial variables, environmental data, and high-performance metrics where variability and outliers are prevalent (Riva et al. 2015).

To further investigate the effect of the assessed skewness on the optimisation error of each potential prior distribution, we investigated all possible combinations of quartile triplets on the range \(0.01\le Q_1<Q_2<Q_3\le 0.99\), with all possible values of \(0.01, 0.02, \ldots , 0.99\). For each quartile triplet we calculated the quartile skewness \(S = (Q_1 + Q_3 - 2Q_2)/(Q_3 - Q_1)\). This skewness captures the degree of asymmetry in the distribution’s shape using only the assessed quartiles with no need for the parameters of any chosen distribution or its moments.

For each prior distribution, we calculated the average of the optimisation error (AOE) across those negatively skewed, symmetric and positively skewed quartile triplets, respectively. We also calculated the overall AOE across all triplets for each prior distribution. The results of this simulation study are given in Table 4.

As expected, the three-parameter priors have AOE that is approximately zero (because three assessments were used to elicit three parameters). The two parameter priors have less AOE than the one-parameter prior distributions. Three-parameter priors achieve \(W_D \simeq 0\) by construction when elicited using three assessed quartiles, since the optimization has zero remaining degrees of freedom. This is a property of interpolation, not evidence that a three-parameter prior captures the expert’s belief better than a one- or two-parameter prior. Selection among prior families should be made on the basis of distributional shape and domain appropriateness, not error magnitude.

All priors are doing better for symmetric quartile triplets than skewed ones. If a one-parameter prior is to be chosen, results recommend the Continuous-Bernoulli prior over the other two one-parameter distributions, especially for skewed quartiles.

All two-parameter priors are more or less equally performing, with an overall AOE of about 0.008. However, the Kumaraswamy prior seems to do better than all other two-parameter distributions for symmetric quartiles. If a two-parameter distribution is to be chosen for a negatively (positively) skewed quartile triplet, the Unit-Weibull (Unit-Chen) is recommended based on the AOE. The Libby-Novick Kumaraswamy is the best-performing three-parameter prior, especially for symmetric cases.

It is worth mentioning that the AOE is a diagnostic of optimization performance under the \(W_D\) objective within a given prior family, and should not be used as a model-selection criterion across families of differing parameter counts.

Table 5 summarises the prior distributions, outlining their advantages, disadvantages, domains of application, key characteristics, and practical guidelines for selection.

Software tools are essential for the elicitation of expert opinion as they make the elicitation process easily accessible and user-friendly for users in decision-making scenarios. While established tools like the Predictive Modal (PM) method (Chaloner and Duncan 1983), \(E^{3}\) language (Jaccheri et al. 1998), and PROBES (Lau and Leong 1999) mostly based on JAVA, have been beneficial for Bayesian inference and graphical visualisation, they have limitations like complexity, verbose code, and performance issues. Our Shiny R application, PEBSI [https://unnipillai.shinyapps.io/PEBSI/], aims to tackle these issues by being more user-friendly and efficient for the elicitation framework.

Other tools, like those developed by Chaloner et al. (1993), Craig et al. (1998), and Van Lenthe’s ELI method (Van Lenthe 1993a, b, 1994), are often too specialised, with limited flexibility in specific contexts. More recent tools, such as the R package makemyprior (Hem et al. 2021), |tPRiors| (Pateras and Kostoulas 2022), and BayesESS (Song et al. 2023), have made improvements in Bayesian modelling and prior elicitation, but they also have their challenges, like high computational demands and limitations in handling complex models. While the Sheffield Elicitation Framework (SHELF) (Gosling 2018) effectively gathers expert knowledge to support decision-making under uncertainty by using probability or quantile judgments to construct cumulative distribution functions, it can be resource-intensive (Stringer et al. 2023) and it doesn’t have a mechanism to provide the posterior distribution based on the elicited priors. Moreover, SHELF just elicits the Beta distribution as the only available prior over the unit interval. Our software elicits a set of 14 priors.

An overview of the developed Shiny R application, PEBSI, is briefly presented in Sect. 3.1.

We developed a user-friendly web-based Shiny R application, PEBSI, crafted to facilitate the elicitation of univariate distributions over the bounded interval [0, 1]. PEBSI is a handy tool that features the proposed univariate priors, calculates their elicited hyperparameters, and plots the prior and posterior distributions, including a triplot (prior, posterior, and likelihood) for binomial sampling models. The developed software thus helps users gain a comprehensive understanding of the full Bayesian framework in an easily accessible and user-friendly way. We have also produced an R package that can be quickly used by Bayesian researchers. The package is called ’UPElicit’ and is provided as Supplementary Material to this paper.

Our software user is asked to assess three quartile values \((Q_1, Q_2, Q_3)\), representing their initial beliefs about the success probability parameter p. The software calculates the elicited hyperparameters of the selected prior distribution and plots them. The user can choose one prior from a list of 3 one-parameter, 7 two-parameter, and 4 three-parameter prior distributions discussed in Sects. 2.1, 2.2 and 2.3, respectively. This helps the users understand how their inputs affect the distribution’s appearance in terms of its location and shape.

Another distinctive feature of our software is that users can adjust the quartiles by directly dragging them on the PDF plot and, upon pressing the calculate button, the users immediately see the updated elicited hyperparameters and the updated PDF plot. Although the prior distributions discussed here are different from the widely-used Beta prior, they exhibit some similar characteristics and can be used in various domain areas.

Users can also input the number of trials, n, and successes, x, from their binomial experiment. This data is then used together with the elicited prior to obtain and plot the posterior distribution using numerical integration.

PEBSI simplifies the elicitation process, makes it more user-friendly, and enhances the overall quality of Bayesian analysis. As output, it displays the summary statistics: mean, median, standard error, and a 95% credible interval of the posterior distribution, showing how initial beliefs are adjusted with available data. Users can compare priors, likelihoods, and posterior distributions using a triplot (Fig. S5). By understanding the similarities between priors, experts can confidently choose among them for their assessed quartile combinations, knowing that similar priors will likely produce comparable results in terms of shape, central tendency, and spread.

Additionally, the software includes another tab (Fig. S6) that lets users choose a set of different prior distributions and compare them visually. In this comparison tab, users can explore three main features:

• Prior Plots: The determined prior distributions are plotted based on the user-input quartiles \((Q_1, Q_2, Q_3)\), showing how each selected distribution behaves before considering any data and allowing users to compare the elicited prior distributions. It also identifies which prior has the minimum error obtained from minimising Eq. (1), indicating the best fit to the input quartiles.

• Posterior Plots: Here, users can see how these distributions change after incorporating their data (n and x), giving the posterior distribution.

• Distances: The software also outputs a table (Fig. S7) that compares how similar or different each prior distribution is from the other priors. Lower values indicate that the distributions are quite similar, while higher numbers suggest they are more different. The given values in the table are Bhattacharrya distance discussed in Sect. 3.2.

To evaluate the introduced priors, we use a distance metric to gauge their similarity and measure the distance between each pair of them. For instance, Fig. S7 illustrates the distances between each pair of all the elicited two-parameter priors for a particular set of quartiles. These distances offer a quantitative measure of the similarity or differences between the priors, given specific quartile inputs.

The software uses the Bhattacharyya distance (Bhattacharyya 1946) that measures the overlap between distributions, focusing on the similarity in their shape and spread. Mathematically, the Bhattacharyya distance between two probability distributions P(x) and Q(x) is calculated as \(D(P,Q)=-\log \left( \int \sqrt{P(x)Q(x)}\, dx\right) \).

For example, from Fig. S7, for quartile combination (0.4, 0.5, 0.6), the Beta and Logit-Normal prior distributions have an extremely small Bhattacharyya distance (0.0002), suggesting they are nearly identical in quantifying the expert’s belief. On the other hand, distributions like Kumaraswamy and Unit-Chen have a larger distance (0.0434), indicating a more noticeable difference in how they represent the same set of beliefs.

Fritillaria meleagris, known as the snake’s head fritillary, is a significant plant in the British and Irish countryside. It is especially important because it grows in the floodplain hay meadows, which are rare and valuable habitats (Tatarenko et al. 2022). The species is seen as a key indicator of traditional, undisturbed grasslands and has been the focus of conservation work (Walker 2021).

In a study of plant competition in wet meadows, the presence/absence of Fritillaria meleagris in specific small areas of land, called quadrats, within a meadow ecosystem was studied based on data available from 2003. However, it is believed that the year 2003 was the driest year on that meadow, and hence expert opinion on the presence/absence of plants would help reach conclusions that are not highly impacted by the exceptionally dry season when the data was collected.

In this example, a plant ecologist used our elicitation method and software to quantify his opinion about the presence/absence of Fritillaria meleagris. It can be assumed that the probability, p, of the plant being present in a single quadrat is fixed across quadrats and that the presence or absence in one quadrat is independent of other quadrats. We model this scenario using the Binomial distribution; \(X\sim \text {Binomial}(n, p)\), with X is the number of quadrats where Fritillaria meleagris is present, n is the total number of quadrats sampled.

The expert used our Shiny R application and gave the following quartiles for the probability p: \(Q_1=0.4\), \(Q_2=0.65\) and \(Q_3=0.8\). These quartiles represent the expert’s uncertainty about the true value of p. The expert was then shown the PDF curves produced by the software for different elicited prior distributions of p. The expert was asked to choose the distribution that best represents his knowledge about Fritillaria meleagris presence. Among the one-parameter distributions, the expert selected the Topp-Leone distribution as the one that most accurately reflected his belief. From the two-parameter distributions, the expert chose the Unit-Logistic distribution, and from the three-parameter distributions, the LNK distribution was chosen as the best fit.

The three selected distributions were shown to the expert for comparison in a panel similar to that in Fig. S8, and after considering them, he chose the LNK distribution as the most suitable prior distribution for p based on the distribution’s shape that adequately reflected his beliefs. The elicited hyperparameters of this distribution are \(a\simeq 0.54\), \(b\simeq 15.37\) and \(c\simeq 0.01\) (see Fig. S5).

Based on a subset of 10 quadrats \((n=10)\) from the available data, Fritillaria meleagris were present in 3 of them \((x=3)\). Hence, the classical estimate of p is \(\widehat{p}=0.3\). However, the prior-to-posterior analysis based on the elicited LNK prior gives a point estimate of p as 0.36 with the \(95\%\) credible interval (0.12, 0.65). Quantifying expert opinion in the elicited prior has increased the estimate of p from 0.3 to 0.36. The expert was shown the obtained posterior plot and a triplot as in Fig. S5.

The expert commented on the usefulness of the visualisation that the software provides and stressed that the given feedback, in terms of the PDF of different prior distributions, was also very useful.

The increase in the estimate of p based on the elicited prior was not very big since the prior was rather weak with a high variance. However, the Bayesian approach has helped obtain more reliable estimates about the presence of Fritillaria meleagris than that depending only on the data from the driest year. The Bayesian approach with the elicited LNK will better support the conservation efforts and facilitate more efficient management of the ecosystem.

Drosophila melanogaster, commonly known as the fruit fly, plays a central role in the studies of genetics and developmental biology (Tolwinski 2017; Mirzoyan et al. 2019). Of particular interest are mutations that affect the structure of the endoplasmic reticulum (ER) in neurons, especially mutations in genes comparable to those implicated in Hereditary Spastic Paraplegia (HSP) in humans (Sonda et al. 2021; Byrne et al. 2022). These mutations are known to disrupt endoplasmic reticulum morphology in neurons, potentially impairing early development and reducing survival rates, including the likelihood of successful hatching from pupae. It turned out that the proportion of flies hatched under one of these mutations is around 10%.

In a further study to investigate the effect of the ER in Drosophila melanogaster, a new diet has been used to feed the flies throughout the experiment. However, this diet was found to be nutritionally richer than the usual diet. It is hypothesised that altering the diet might improve hatching in the mutated flies pupae. Diets rich in nutrients, particularly those high in protein and carbohydrates, may help improve cellular function. This is especially important in early development and may improve the cellular function resulting from abnormal ER structure in fly larvae carrying mutations (Dinh et al. 2022).

In this example, an expert in genetics used our elicitation software to quantify her belief about the probability of pupal hatching in Drosophila melanogaster under the richer diet. The expert believes that the highly nutritional diet of the mutated parents should slightly improve the hatching probability of their laid eggs compared to those under the usual diet.

The number, X, of successfully hatched pupae out of a fixed number, n, of laid eggs is assumed to follow a binomial distribution, \(X\sim \text {Binomial}(n, p)\), where p is the probability of hatching. Using our PEBSI application, the expert assessed the following quartiles for the probability p: \(Q_1=0.2\), \(Q_2=0.25\) and \(Q_3=0.3\). She was then shown the PDF curves for different elicited prior distributions of p and was asked to choose the distribution that best represented her belief. Among the one-parameter distributions, the expert selected the Alpha-Unit distribution; from the two-parameter distributions, she chose the Unit-Chen distribution; and from the three-parameter distributions, the Unit-Exponential Pareto distribution was chosen as her best fit.

From those three distributions, the expert then identified the Unit-Exponential Pareto distribution as her most adequate prior distribution for p, based on the distribution shape. The elicited hyperparameters of this distribution are \(a\simeq 2.91\), \(b\simeq 0.197\) and \(c\simeq 0.145\), with an error of approximately \(4\times 10^{-6}\).

Empirical data were then collected from a single vial containing \(n = 10\) pupae from mutated parents of Drosophila melanogaster, from which only \(x = 1\) pupa hatched successfully, giving a classical estimate of \(\widehat{p} = 0.10\). Since this sample was too small, it did not reflect the expected probability under the richer diet. However, when combined with the expert-elicited Unit-Exponential Pareto prior, the posterior mean of p was found to be 0.21, showing the impact of expert opinion in updating the too low classical estimate of p under the richer diet.

Adequate passive transfer of maternal immunoglobulins is essential for neonatal foal health. Foals are born with minimal circulating antibodies and depend entirely on colostrum intake during the first 24 h of life for immune protection. Failure of passive transfer (FPT) predisposes foals to septicemia and other infectious diseases with potentially fatal consequences (Francesca et al. 2017).

A veterinarian specializing in equine neonatal care with 14 years of experience managing foaling operations at multiple large breeding farms was consulted to quantify her belief about the probability of successful passive transfer (serum IgG concentration > 800 mg/dL at \(18-24\) hours post-birth) in Thoroughbred foals born on well-managed farms where: mares receive appropriate peripartum vaccination, foaling is directly observed, colostrum intake is supervised within the first 2 h, and IgG testing is performed routinely.

The probability, p, of achieving successful passive transfer can be assumed fixed across foals under these optimal management conditions. We use the Binomial distribution; \(X\sim \text {Binomial}(n, p)\), where X represents the number of foals achieving adequate IgG levels, and n is the total number of foals tested. Using PEBSI, the expert assessed: \(Q_1 = 0.90\), \(Q_2 = 0.92\), \(Q_3 = 0.95\). These quartiles reflect her belief that intensive management achieves very high passive transfer success rates, while acknowledging occasional failures due to factors like premature lactation, poor colostrum quality, or foal weakness affecting nursing ability.

The expert examined the obtained prior distributions. Among the one-parameter priors, the Continuous-Bernoulli showed limited flexibility for this high-probability scenario. From two-parameter distributions, the Unit-Logistic provided a reasonable fit. But the Libby-Novick Kumaraswamy (LNK) was identified as the prior with the most adequate distribution shape, with parameters \(a \simeq 26.21\), \(b \simeq 77.71\), \(c \simeq 0.059\), with an error of \(5.2 \times 10^{-5}\).

Observational data from one breeding season at a large Thoroughbred operation included \(n = 68\) foals that were born alive and tested for IgG at 18–24 h. Of these, \(x = 62\) foals achieved successful passive transfer (IgG > 800 mg/dL), yielding \(\widehat{p} \simeq 0.91\). Bayesian analysis with the elicited LNK prior produced a posterior mean \(p = 0.92\) with a 95% credible interval (0.86, 0.96). The posterior shows strong agreement between expert assessment and observational outcomes, reflecting that intensive management protocols effectively prevent FPT.

This example demonstrates prior elicitation in a preventive medicine context where proactive management aims for very high success rates. The posterior estimates can inform decisions about resource allocation for foaling supervision, economic analysis of colostrum supplementation programs, and risk assessment for insurance purposes.

We aimed to introduce elicitation methods to quantify expert knowledge for different prior distributions over the unit interval beyond the usual choice of the Beta prior. We suggested another 13 priors, with different numbers of hyperparameters, to represent prior beliefs about the probability of success, p, for binomial sampling. With this wider selection, we improve how experts and researchers elicit subjective priors that better represent the range of beliefs and uncertainties involved across different fields. This gives more accurate results and better decision-making based on expert opinion.

The Shiny R application proposed here for eliciting expert opinions, PEBSI [https://unnipillai.shinyapps.io/PEBSI/], serves as an important tool in the elicitation process. It provides an interactive, user-friendly graphical interface for experts to contribute their insights and preferences. The interactive features and visualisations of the app empower experts to explore and compare different prior distributions with ease. Although the expert’s assessments only capture 50% of the elicited prior, the given visual feedback helps the expert check the prior shape, including its tail behaviour. This functionality supports informed decision-making based on the expert’s specific domain of knowledge and preferences.

Furthermore, the Shiny R application supports the comparison of different priors, providing plots of the prior and posterior distributions and a triplot. These comparisons allow experts to evaluate how different choices of prior distributions and associated parameters influence the resulting posterior distributions. By directly observing the changes in shape, spread, and location of the distributions, experts can make well-informed judgments regarding the suitability of each prior in capturing their domain-specific knowledge. Moreover, the software also calculates the distances between different priors. These take into consideration all the points of the distribution shape and can also be useful in informing the expert’s decision.

The strength of the proposed method is that it is structured in a way that it can be used to elicit any univariate distribution over the unit interval once its PDF and quantile function are available either in a closed form or as a transformation from another known distribution. Several univariate distributions over the unit interval are being frequently proposed in the literature. Although our current version of the software elicits 14 prior distributions, a forthcoming version of the software will be easily extended to allow for eliciting user-defined prior distributions. An option will be added for the users to input their formulae of the PDF and quantile function for any distribution, and the software will elicit its hyperparameter(s) based on a set of three assessed quartiles.

A limitation of the current approach is that the least-squares optimisation function, \(W_D\), treats all quartile differences equally on the probability scale, even though their practical importance may differ across the unit interval. In particular, discrepancies near 0 or 1 can have a much larger impact than similar discrepancies near the centre. While the current formulation provides a simple and stable approach, it does not explicitly account for this.

A natural extension would be to use a weighted version of \(W_D\), where different quartile differences are given different importance in terms of weights. For example, an expert may be more confident in their median assessment than in the lower and upper quartiles. So a higher weight can be assigned to the median. This would allow the elicited prior to better reflect the most reliable aspects of the expert’s beliefs, particularly in applications where behaviour near the boundaries is important.

The work reported here was motivated by two important extensions we are currently working on. First, the availability of a unified structured elicitation method and software to elicit many different univariate priors on the unit interval will facilitate similar extensions to the multivariate case of eliciting more flexible multivariate distributions over the simplex beyond the Dirichlet distribution. The wide set of univariate distributions is a very flexible start to using more flexible copula functions that model the multivariate distributions using different reasonable marginal distributions.

The second important usage of the methods developed here is to help elicit priors for the parameters of regression models in which the dependent variable is defined over the unit interval. Namely, we are working on elicitation methods for Beta regression models and their proposed alternatives, such as the Kumaraswamy regression models and those based on some other univariate distributions over the unit interval.