An Investigation of Non-Informative Priors for Bayesian Dose-Response Modeling

Abstract

Toxicology analyses are built around dose-response modeling, and increasingly these methodologies utilize Bayesian estimation techniques. Bayesian estimation is unique because it includes prior distributional information in the analysis, which may impact the dose-response estimate meaningfully. As such analyses are often used for human health risk assessment, the practitioner must understand the impact of adding prior information into the dose-response study. One proposal in the literature is the use of the flat uniform prior distribution, which places a uniform prior probability over the dose-response model’s parameters for a chosen range of values. Though the motivation of such a prior distribution is laudable in that it is most like maximum likelihood estimation seeking unbiased estimates of the dose-response, one can show that such priors add information and may introduce unexpected biases into the analysis. This manuscript shows through numerous empirical examples why prior distributions that are non-informative across all endpoints of interest do not exist for dose-response models; that is, other quantities of interest will be informed by choosing one inferential quantity not informed.

1. Introduction

Recently, the World Health Organization (WHO) and the United Nations Food Agriculture Organization (FAO) recommended using Bayesian model averaging methodologies for dose-response modeling in their update of Chapter 5 of the Environmental Health Criteria 240 (WHO and FAO, 2020). The European Food Safety Authority also updated its recommendations for modeling dose-response to focus on Bayesian methodologies (John-More et al., 2022). This change is due to the fact that the Bayesian approach allows multi-model inferences using model averaging (Shao and Gift, 2014; Wheeler and Bailer, 2007; Wheeler et al., 2020), and it further allows the inclusion of prior information in data-poor situations. The latter ability is unique to Bayesian inference and may be controversial as it introduces subjective information into the analysis through the prior distribution (or prior for short). Because of the subjective flavor of Bayesian analysis, many will want objective, or noninformative, prior distributions that do not influence their analysis beyond the available data.

There are a variety of software platforms now available for Bayesian dose-response estimation (e.g., the Bayesian Benchmark dose system(BBMDS) (Shao and Shapiro, 2018), the US EPA’s Benchmark Dose Software 3.2 (EPA, 2012), the EFSA Benchmark Dose Software Suite (Interuniversity Institute for Biostatistics and Bioinformatics, 2022), and ToxicR (Wheeler et al., 2023) ), but only BBMDS provides uniform priors for non-informative distributions. Leading one to ask what the difference between the approaches is. In the statistical sciences, there is a vast literature on the use of prior information for Bayesian analysis, and this manuscript attempts to describe what may occur when uniform priors on the parameters are used (i.e., those proposed in Shao and Shapiro (2018)). However, it also raises questions about other priors and maximum likelihood inference.

A prior distribution may bias the analysis, and because of this, authors have developed strategies that attempt not to alter the analysis substantively beyond the information provided by the data. Such priors have existed since the beginning of Bayesian analysis (e.g., de Laplace (1820)). These priors are called non-informative. In principle, a non-informative prior should provide results like an analysis based on the maximum likelihood (ML) estimation principle, which finds the model parameters that are considered most likely given the observed data. Some (for example, Berger (2006)) argue for their use in regulatory settings.

Nevertheless, for an uninformative prior distribution to exist, certain conditions must be met that do not generally occur for dose-response analysis (for a complete treatment of uninformative priors, see Berger (2013)). Some (Bornkamp, 2012; Bornkamp, 2014), seeing this problem, develop a uniform prior over the space of functions considered by the dose-response model. Even here, they recognize their approach is informative for other quantities of interest in the analysis. For example, if one used their uninformative prior, they would be placing an informative prior on the dose associated with a given estimate of toxicological risk known as the benchmark dose (EPA, 2012). This work investigates a much simpler non-informative prior, which looks at the parameters of the dose-response function. It shows that these influence the risk assessment, but the arguments apply to any such prior.

We show that the non-informative priors on parameters, such as those used in Shao and Shapiro (2018) and Shao et al. (2021), add information to the analysis, which can cause unintuitive results that vary depending on the choice of the prior. This result is shown herein with experimental datasets, and the mathematical rigor, explaining why it is valid for any non-informative prior, is given in the supplement. In two of these examples, it is notable that the datasets were used in regulatory settings, and thus they cannot be considered nonstandard data for regulatory toxicology.

In discussing this issue, we also consider maximum-likelihood (ML) inference, which has problems in the same situations but for different reasons. In ML estimation, the problem lies with the theoretical underpinnings of the central limit theorem (see Van der Vaart (2000)), whose assumptions might not be met for the model/data combination. Such a violation of assumptions, in turn, results in the construction of confidence intervals whose performance is not known because the standard asymptotic approximation used is not applicable.

2. Bayesian Dose-Response Analysis

2.1. Likelihood-Based Dose-Response Estimation

Since Finney introduced the log-Probit model in his work “Probit Analysis: A Statistical Treatment of the Sigmoid Response Curve,” likelihood-based estimation has been a staple in analyzing dose-response data (Finney, 1952). In this work, the dose-response model was formalized. It has since been used to specify a mathematical relationship to estimate the expected adverse response (for continuous data) or the probability of adverse response (for dichotomous data). The dose response is the response in the absence of experimental scatter. Experimental scatter is used generically to describe unexplained-for noise in the experiment.

For our purposes, $f(dose\mid \text{Θ})$ is a parametric dose-response function defined by the parameters $\text{Θ}$. In what follows, $f(dose\mid \text{Θ})$ takes many forms depending on the context, but the dose-response function will be clear in context. In Finney’s context, one has the log-probit model given by

$$f\left(dose\mid \text{Θ}={\left[\text{g},\text{a},\text{b}\right]}^{\prime }\right)=g+\left(1-g\right)\times \text{Φ}\left(a+b\log \left[dose\right]\right),$$

where $\text{Θ}=\left[\text{a},\text{b},\text{g}\right]$ and $\text{Φ}(\cdot )$ is the cumulative distribution function of a standard normal random variable. As the data are observed with error, the vector $\text{Θ}$ is estimated with $\widehat{\text{Θ}}$, which is the most probable value, given statistical assumptions. For example, we often assume a binomial distribution for dichotomous data and a log-normal or normal distribution for continuous data. Combined with the dose-response function, $f(dose\mid \text{Θ})$, and observed data, Y, this distribution defines a likelihood function, $L\left(Y|f\left[dose\mid \text{Θ}\right]\right)$, which is maximized over all plausible values of $\text{Θ}$. The value that maximizes this function is the maximum IML estimate, represented as $\widehat{\text{Θ}}=\left[\widehat{a},\widehat{b},\widehat{g}\right]$.

Typically, a dose-response analysis relies on the theory of ML estimation, which is vast (for a comprehensive treatment of asymptotic maximum likelihood theory, see chapter 5 of Van der Vaart (2000) ). For ML, statistical inference of dose-response curves is based upon the asymptotic normality of the maximum likelihood estimator, i.e., the central limit theorem, which impacts the construction of confidence bounds. This fact is also valid when constructing profile likelihood confidence intervals (Molenberghs and Verbeke, 2007). To apply this theorem, one cannot have a parameter estimated on a boundary, which frequently occurs in dose-response estimation, and when it is on a boundary, the asymptotic behavior is known to diverge from normality; again see Molenbergs and Verbeke. Additionally, there are times when the ML solution is not unique, which can occur when a parameter’s value goes to infinity or when the likelihood approaches some asymptote. When any of these conditions occur, the correctness of statistical inference using traditional approaches cannot be guaranteed. Further, bootstrapping one’s data cannot escape this problem because similar conditions exist when using bootstrap estimators (Andrews, 2000). See chapter 23 of Van der Vaart (2000) for an exposition of the theory of bootstrapping.

For example, the log-probit model is often estimated so that that $b\ge 1$. When one has $\widehat{b}=1$, the central limit theorem does not apply. For another example, suppose the likelihood reaches a plateau as b goes to infinity. Due to numerical issues, estimation is bounded at a pre-specified value, such as 18, so $\widehat{b}=18$ in the case of the plateau. For a quantity that utilizes $\widehat{b}$ the statistical inference will also be incorrect. For example, the benchmark dose (BMD) (Crump, 1984) will have the same theoretical issues. When this happens, much of classical dose-response inference using ML is not founded upon solid underpinnings, and this impacts Bayesian analysis because it relies on the likelihood.

2.2. Bayesian Dose-Response Analysis

A Bayesian analysis uses the likelihood function but incorporates additional information by defining a prior distribution over $\text{Θ}$. This prior distribution is denoted as $\mathrm{Pr}\left(\text{Θ}\right)$. With the prior, Bayesian inference proceeds by computing the distribution

$$\mathrm{Pr}\left(\Theta |Y\right)\left(\text{Θ}|Y\right)=\frac{L\left(Y|f\left[dose,\text{Θ}\right]\right)\mathrm{Pr}\left(\text{Θ}\right)}{\int L\left(Y|f\left[dose,\text{Θ}\right]\right)\mathrm{Pr}\left(\text{Θ}\right)d\text{Θ}},$$ (1)

which is called the posterior distribution or ‘posterior.’ For the posterior distribution defined in (1), one can see that the prior is multiplied by the likelihood and divided by a constant. Values weighted at zero a priori (i.e., $\mathrm{Pr}\left(\text{Θ}=\text{θ}\right)=0$) have a zero probability a posteriori; similarly, values downweighed in the prior distribution may be impacted in the posterior. Consequently, the choice of the prior is essential in any Bayesian analysis.

Inference over (1) is based upon integration. This integral is usually analytically intractable and estimated using simulation methods like Monte Carlo Markov Chain (MCMC) integration. These methods avoid the need to calculate the denominator of (1) and integrals in general; however, if the denominator goes to infinity as ϴ becomes unbounded because there is some part of $L\left(Y|f\left[dose,\text{Θ}\right]\right)\mathrm{Pr}\left(\text{Θ}\right)$ that is flat, problems occur. In such a situation, MCMC algorithms will explore this flat surface in a way where the convergence of the Markov chain will be difficult to ascertain. Numerical integration may be more accurate because it considers the entire domain of Θ (though it requires significantly more computational time). Consequently, numerical integration is used below as assessing convergence using MCMC methods for the datasets is challenging.

2.3. Benchmark Dose Estimation

In dose-response estimation, toxicity is frequently defined using the BMD. The BMD is the dose corresponding to a particular definition of risk, where risk is a function of some pre-specified change from background based upon the dose response. This quantity is a function of the estimated dose-response curve, and inference about the BMD depends upon the model parameters. Likelihood inference for the BMD depends on normal approximation theory. In contrast, Bayesian inference requires only the integration to be correct. The BMD depends on the type of data (e.g., continuous or dichotomous) and dose-response function, so in what follows, we define the BMD in the context of the given risk analysis. The benchmark response (BMR) is a central quantity in the BMD estimation, and in what follows, it is a modeler-defined quantity specifying the risk level a priori.

3. Prior Distributions and Information

Prior selection is a huge topic within the Bayesian literature. For a comprehensive discussion, the reader may be interested in Gelman et al. (2013), or for an extensive theoretical perspective, Berger (2013). The following discussion focuses only on the idea of non-informative priors.

A non-informative prior distribution is a distribution that does not favor any value of $\text{Θ}$. In the most straightforward situation, where $\text{Θ}$ is a single parameter taking n discrete values, one would specify a probability of $\frac{1}{n}$ for each value. For example, suppose one is rolling a die and is interested in determining whether it is fair. A priori, one does not want to assume that it is not, and it is reasonable that each number (1,2,3,4,5,6) has a $\frac{1}{6}$ probability of occurring. When used as a prior distribution, an equal-weighted prior favors no value of $\text{Θ}$ before the experiment, letting the data fully inform the answer.

The previous example assumed discrete possibilities for the parameter of interest. For continuous distributions, one standard approach is to let $\mathrm{Pr}\left(\Theta \right)=c$, where c is a constant for all values Θ. In nonlinear models, such as many of those used in the dose-response analysis, one cannot do this because the resultant posterior integration may not have a finite solution (i.e., it may not be a probability distribution). As a fix, some set $\mathrm{Pr}\left(\text{Θ}\right)=c$ for a range of values, which guarantees the posterior will be a probability distribution. This approach defines a uniform prior distribution, which is the approach used by Shao and Shapiro (2018). As shown in the supplement, a uniform prior of this kind is not uninformative for dose response and risk estimation. The supplement’s arguments can be extended to any uniform prior distribution, including those of (Bornkamp, 2014). They show that one can specify an uninformative prior distribution for a parameter of interest, but this will inform all other quantities of interest. The empirical results below are specific examples where the uniform prior of Shao and Shapiro produces unintuitive results due to problems with the likelihood surface. For an additional example, section two of the supplement describes this problem for a simulated dataset and the Hill model.

3. Case Studies

Experimental datasets are analyzed using uniform priors on the model parameters in what follows. In each case, the suggested priors of Shao and Shapiro (2018) and variants of these priors are used. This analysis shows how the abovementioned likelihood problems translate to inferential problems using uniform priors. In the supplement, we analyze the same datasets using diffuse but informative priors over all parameters of interest, alleviating these problems.

3.1. Formaldehyde

Kerns et al. (1983) investigated respiratory squamous cell carcinoma in mice exposed to formaldehyde. Mice were exposed to 0, 2, 5.6, and 14.3 ppm of formaldehyde in the air and observed to have 0/156, 0/159, 2/153, and 94/140 squamous cell carcinomas in their respiratory tract. These data were fit to the 3-parameter Weibull model, which defines the probability of carcinoma as

$$f\left(dose,\text{Θ}=\left[a,b,g\right]\right)=g+\left(1-g\right)\times \left(1-\exp \left[-b\times dos{e}^a\right].\right).$$

For a given benchmark response or BMR based on extra risk, the benchmark dose is

$$\text{BMD}={\left[\raisebox{1ex}{$-\log \left(1-\text{BMR}\right)$}\!\left/ \!\raisebox{-1ex}{$b$}\right.\right]}^{1/a}.$$

In the extra risk definition, the BMD is independent of ‘g.’

Parameter ‘a’ is the problem parameter even though the likelihood has a well-defined maximum. Figure 2 shows the profile likelihood for different values of ‘a.’ The likelihood attains a maximum at approximately a=4.75 and then drops about 2.2 units before plateauing. At the plateau, all values of ‘a’ give the same likelihood value. Given that the profile likelihood typically closely approximates the marginal posterior distribution of a parameter (see chapter 5 of Leonard and Hsu (2001)), it is reasonable to ask, what is the impact of using flat uniform priors on the posterior when the likelihood plateaus for this dataset?

Figure 2:

Profile likelihood over different values of ‘b’ for the log-probit model fit to experimental data from a two-year inhalation study (NTP 2011) of mice exposed to 1-Bromopropane. The maximum likelihood estimate of ‘b’ is 0.272, but the likelihood is not statistically different than the maximum at the 95% level at the dose of 0.025.

To answer this question, we place a Uniform(0,1) on ‘g’ and a Uniform(0,50) prior on b, which are the same priors used by Shao and Shapiro (2018). A Uniform prior is placed on parameter ‘a’ with one as the minimum and the maximum set to 7.5, 15, 50, 100, or 300, respectively. When using a Uniform(1,15) prior distribution, one arrives at the Weibull model defined by Shao and Shapiro. All integrals were estimated using numerical quadrature methods. With one million quadrature points, the answers given are close to exact.

Table 1 gives the resultant expected value of ‘a, the BMD, and the probability that the BMD is less than 70% of the MDT. The table shows that the expected values of ‘a’ and BMD increase as the upper bound on the prior increases, i.e., the Bayesian estimates change by merely changing the upper bound on the prior. This change is large; the BMD increases from 0.64 to 0.95 by increasing the upper bound. More concerningly, the tails of the distribution, which affects the BMDL, also change. The table shows the probability that the BMD is less than 70% of the MDT goes from 84.7% to 2.6% just by changing the upper bound on the prior for parameter ‘a’. Increasing the upper bound will push the BMD toward the MDT. One can make the lower bound on the BMD as close to the MDT as one would like, which is different from the ML estimate. This effect occurs assuming one is not adding information to the analysis.

Table 1:

Parameters from the Bayesian posterior distribution of the Weibull model. The posterior is formed using various uniform priors and the same Formaldehyde data set from Kerns et al. (1983). Doses were rescaled so that 1 was the maximum dose tested (MDT). All estimates, including the BMD estimate, are noticeably impacted by the range of the uniform prior. Further, the BMD distribution increases simply by increasing the size the prior’s range.

Uniform Prior	E[a]	E[BMD]	Pr(BMD < 0.7 × MDT)
0.5 ≤ d ≤ 7.5	5.1	0.64	0.847
0.5 ≤ d ≤ 15	8.1	0.72	0.457
0.5 ≤ d ≤ 50	24.5	0.85	0.151
0.5 ≤ d ≤ 100	52.6	0.98	0.083
0.5 ≤ d ≤ 300	146.7	0.95	0.026

3.2. 1-Bromopropane

The National Toxicology Program (NTP) studied the effects of 1-Bromopropane on B6C3F1 mice in a two-year inhalation study (NTP, 2011). This study investigated the excess probability of pulmonary adenomas or carcinomas for mice exposed to 0, 65.5, 125, and 250 ppm of 1-Bromopropane in the air. Here, 1/50, 9/50, 8/50, and 14/50 animals were observed with pulmonary adenoma or carcinoma, showing a clear link between exposure and cancer. In the dose-response analysis, the probability of pulmonary adenomas or carcinomas is modeled as a function of dose with the log-probit model:

$$f\left(dose|\text{Θ}=\left[g,a,b\right]\right)=g+\left(1-g\right)\times \text{Φ}\left[a+b\ast \log \left(dose\right)\right],$$

where $\text{Φ}[\cdot ]$ is the cumulative distribution function (CDF) for the standard normal distribution¹.

In this case, the unconstrained log-probit model, using the method of ML, BMR=0.1, and extra risk BMD definition estimates a BMD using 11% of the maximum dose tested with a confidence interval of (0%,41%) of the MDT. The confidence interval is not usable in a risk assessment because the lower bound of the BMD includes zero, and figure 2 shows why. The profile likelihood of parameter ‘b’, which is the parameter that causes issues, shows a very weakly defined maximum for a value of ‘b’ less than 1. We see the same issue as with the Weibull model above. There is a maximum, and then the likelihood flattens. For b>3, any value of ‘b’ will model the data similarly. However, unlike the Weibull case, there is no information in the low-dose region either. Overall, one has little information to estimate ‘b.’

As above, four uniform distribution widths for ‘b’ are used to analyze these data. Due to numerical issues, the lower bound of the uniform distribution is set to 0.1 for the first two conditions. For the last two, it is set to 0.25 and 1.0. For the upper bound of the distribution, the first condition uses 2, and the last three conditions use 15. Table 2 shows these results, and the estimates are counterintuitive. The probability that the BMD is less than 50% of the MDT is inversely related to the expected BMD. In the first case, the expected value is greater than 14 billion times the MDT, but the probability of it being less than 50% of the MDT is 0.74. The model returns nonsense because of the strange behavior of the likelihood. As Bayesian methods integrate the flat region seen in figure 2, there is an increasing mass on values of ‘b’ when the uniform prior width increases. The greater the prior width, the more counterintuitive the result. When parameter ‘b’ is constrained to be greater than 1, the BMD is equal to the MDT, and the probability that the BMD is less than 50% of the MDT is 0.006. Again, the choice of bounds impacts the estimate.

Table 2:

Parameters from the Bayesian posterior distribution of the log-probit model. The posterior is formed using various uniform priors and the 1-Bromopropane data from (NTP 2011). Doses were rescaled so that one was the maximum dose tested (MDT). All estimates, including the BMD estimate, are noticeably impacted by the range of the uniform prior.

Uniform Prior	E[b]	E[BMD]	Pr(BMD < 0.5 MTD)
0.1 ≤ b ≤ 2	0.56	14e9	0.74
0.1 ≤ b ≤ 15	2.7	8e9	0.44
0.25 ≤ b ≤ 15	5.2	30.40	0.29
1 ≤ b ≤ 15	7.7	1.1	0.006

3.3. PFOA toxicogenomic data on GLIPR1 receptor

In this data example, which comes from analyzing data from the GLIPR1 receptor, we investigate problems with the Hill model that occur when there is insufficient in the region where response changes the most, i.e., one misses the dose-response curve in the experiment. The Hill model is

$$f\left(dose|\text{Θ}=\left[\text{a,b,c,d}\right]\right)=a+\frac{b\times dos{e}^d}{c^d\times dos{e}^d},$$

and the problem occurs with the shape parameter ‘d’.

Figure 3 shows the data for this example based on PFOA exposure data from Gwinn et al. (2020). This figure also plots a Hill model fit using ML. The maximum change, parameter ‘b,’ and the background rate, parameter ‘a’, are well defined from the data, but there is little information about where the dose-response occurs. The lack of information causes the parameter ‘d’ to be ill-defined. Figure 4 shows this by plotting the profile likelihood for this parameter. The maximum is essentially the same past value of 10, which causes the parameter estimate to be on the upper bound of the parameter space, i.e., d = 18.

Figure 3:

Plot of toxicogenomic dose-response data receptor ‘GLIPR1’ and the fitted Hill model using maximum likelihood methods and data from the study of Gwinn et al. (2020). Here the estimated dose-response curve suggests a sharp change around 10% of the MDT.

Figure 4:

Plot of the maximized log-likelihood as a function of Hill parameter ‘d’ using toxicogenomic dose-response data on the ‘GLIPR1’ receptor for study of Gwinn et al. (2020).

To understand this behavior, we investigate a Bayesian model with the following uniform prior bounds. For ‘a,’ one can use a uniform prior distribution over the range (0,10), and for ‘b,’ a range of (−15,0.). Further, one can place prior bounds between 0 and 2 times the maximum tested dose for the value to cover all possible values of ‘c.’ For ’d,’ the parameter is given a lower bound of 0.5 with upper bounds of 3.25, 7.5, 15, 30, 60, and 120. For this posterior, deterministic quadrature methods fail. Here, 60 million function evaluations of the distribution were taken using an adaptive Monte Carlo integration algorithm that utilized importance weighting to guarantee correctness in the results.

Table 3 gives the results of the analysis. here, it is seen that as the uniform’s prior mass becomes more diffuse on ‘d’, the parameter’s posterior mean increases. As the likelihood is essentially flat past values of ‘d’ greater than 7, as shown in Figure 4, increasing the limits of integration merely increases the expected value of the shape parameter proportional to the amount of density added. After a point, doubling the integration region doubles the estimate of ‘d,’ which centers the BMD estimate at 0.1, the first observed dose exhibiting a statistically significant change in the experiment.

Table 3:

Parameters from the Bayesian posterior distribution of the Hill model. Doses were rescaled so that 1 was the maximum dose tested (MDT). All estimates, including the BMD estimate, are noticeably impacted by the range of the uniform prior.

Uniform Prior	E[d]	E[BMD]	Pr(BMD < 0.07 × MTD)
1 ≤ d ≤ 3.25	1.81	0.14	0.21
1 ≤ d ≤ 7.5	3.5	0.13	0.21
1 ≤ d ≤ 15	7.09	0.11	0.19
1 ≤ d ≤ 30	14.68	0.11	0.17
1 ≤ d ≤ 60	29.93	0.10	0.15
1 ≤ d ≤ 120	60.31	0.10	0.14

The centering is evident in the width of the credible interval around this point (0.07 was a dose chosen below 0.1 for this illustration). Initially, there is a 21% probability that the BMD is less than 0.07 times the MDT when d < 3.25, but when d < 120, this probability decreases to 0.14%. This result is counterintuitive, i.e., by making the uniform prior more diffuse, we get increased certainty on the BMD. The reason for this behavior is based on the functional form of the BMD, which is a function of the dose raised to the inverse power of ‘d;’ as the shape parameter increases, the BMD will become more centered around one value.

4. Discussion

The above examines the concept of uninformative priors in dose-response modeling applied to BMD estimation. We showed that, for dose-response estimation, uniform priors, proposed by Shao and Shapiro (2018), give information in the analysis and should not be considered as allowing for completely objective analyses. The supplement shows that any uniform prior will result in an informative distribution for other parameters. One can have specific parameters as uninformative, but this leads to other parameters being informed by the analysis. Seemingly unimportant changes in the priors may result in critical inferential differences; thus, these prior distributions should not be considered in a regulatory setting unless sufficient evidence exists that the likelihood is not ill-posed for a given dataset.

This behavior occurs when there is not enough information to inform the parameter, but as shown, this occurs in various situations, even when the likelihood has a single mode. Such a prior may lead to non-intuitive results. As an example of this possible behavior, Shao et al. (2021) used a Uniform(0,50) prior for parameter ‘d’ in the Hill model, which is more diffuse than the prior for the same parameter described in Shao and Shapiro (2018). In this work, Shao, Zhou et al. describe an arsenic meta-analysis where ‘d’ was given a prior distribution for each study’s dose response. However, in the supplement of that work, a very different result is reported when a hierarchical prior was placed over the dose-response curves, allowing for information sharing between studies. This difference in results may indicate that the priors used in the main manuscript have an undue influence on the result.

One may question if Bayesian methods are appropriate for dose-response analysis in a regulatory setting. Any Bayesian analysis provides subjective information that may unduly influence the analysis, which requires objectivity. Yet, the problem is common to all Bayesian analyses because the issues result from an ill-posed likelihood where the maximum is not defined or is weakly specified. We see these issues for maximum likelihood estimation too, and they occur for the same data. The only other option is to provide priors that give specific information on the parameters of interest, which was the opinion of the WHO/FAO committee when they recommended the use of informative priors (WHO and FAO, 2020) and the European Food Safety Authority (Committee et al., 2022).

Additional research on priors is necessary to guarantee correct inference without undue prior influence. In this endeavor, one could focus inference on a region of interest as described in section 5 of Geweke (1999). Simulation experiments can then be used to make optimal priors. Other approaches like those outlined in (Wheeler et al., 2019) or (Shao et al., 2022) may also be appropriate in certain situations. Regardless of the approach, it must be transparent and show that the above problems do not occur, or if they do occur, the prior distributions chosen to ameliorate the issue are scientifically reasonable to the research question.

Figure 1:

Profile likelihood over different values of ‘a’ for the Weibull model fit to data in Kerns et al. (1983). The maximum likelihood estimate of ‘a’ is 4.75, but the likelihood does not change numerically past 15, which causes problems for Bayesian inference when using uniform priors.

Highlights:

• Bayesian dose-response modeling requires careful examination on the prior distribution.

• Uniform parameter priors cause unexpected biases in risk assessment.

• Universally uninformative dose-response priors do not exist.

• Research is needed to better understand the impact in regulatory toxicology.