Prediction uncertainty assessment of a systems biology model requires a sample of the full probability distribution of its parameters

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- The initial submission of this article was received on March 10th, 2014 and was peer-reviewed by 2 reviewers and the Academic Editor.
- The Academic Editor made their initial decision on April 22nd, 2014.
- The first revision was submitted on May 22nd, 2014 and was reviewed by 1 reviewer and the Academic Editor.
- The article was Accepted by the Academic Editor on May 28th, 2014.

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Thank you for carefully revising your manuscript.

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The revised manuscript is improved substantially and the authors addressed all the questions satisfactorily.

One minor point:

- line 134: there is twice 'biologists are' in succession

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- submitted
May 22, 2014

Major Revisions

Both reviewers had specific comments. In particular the second reviewer raised several important points that you should address in your revision.

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The manuscript of Van Mourik et al. describes a methodology that allows to study the predictive quality of biological multi-parameter models by a full computational uncertainty analysis. They show that the predictive quality of sloppy biological models does not necessarily depends on the parameter uncertainty. Furthermore they show that the quality of predictions vary within one model and that this depends on the type of prediction. The methodology is very relevant since it gives systems biologist the opportunity to test their biological models for predictive quality for different predictions. However, I have some questions.

• For biologists it is not always essential to obtain quantitative predictions. The method described in the manuscript results in a quantifier, which is a degree of how accurate the predictions are quantitatively as far as I understood. However, in some cases a qualitative prediction can be enough to obtain new insights in how a biological system works. For instance it might be enough to know if a certain metabolite or flux is going up without an accurate prediction on how much the metabolite/flux goes up. Would it be possible with this method to obtain predictive values for qualitative predictions instead of quantitative predictions?

• The method shows that including data points can increase the prediction quality of the biological model. An experimentalist would like to know what kind of data/experiments are needed to increase the predictive quality. Would it be possible to use the method to determine what kind of data is needed to increase the predictive quality?

• In the last paragraph of the conclusion the authors state that a software package is made available. It would be useful to notify where we can find this software package (I could not find it, but maybe I overlooked it).

As far as I'm concerned the article addresses all areas that are associated to Basic Reporting's criteria. I would however like to suggest a few changes and point out some of the mistakes in the text that will boost up the readability of the text.

1) I do not find the choice of two text books that are cited with respect

to the Bayesian framework very suitable for this subject. A book from

Nate Silver (2012) is a popular science book, which is, apart from being

a good read, has no pedagogical value for learning statistics

and Bayesian inference. Another book, McGrayne (2011), is a historical

assessment on the importance of Bayes' theorem, which again is not meant

to be a text book for learning the subject. I suggest instead authors

refer to other valuable, and scientically sound, texts for this subject.

There are several excellent texts, such as a book from Edwin Jaynes titled

Probability Theory As Extended Logic, or David MacKay's "Information

theory, inference and learning algorithms", or "Data analysis: a Bayesian

tutorial\ by Sivia and Skilling.

2) At line 317, for the acronym GLUE, it is not dened that what is it exactly

standing for.

3) In line 21 there is one extra "for".

4) The title suggests that characterizing model's prediction quality needs

a "full" probability distribution. This is a valid suggestion, but this work does

not taking into account the entire probability distribution of parameters.

Whereas, a sample of parameters (1000 samples) drawn by a MCMC algorithm

is considered. It is of course not a single parameter fit, like maximum

likelihood estimation, but it is not also the entire distribution of parameter

space.

5) At line 57, for the frst time authors mention MCMC algorithms and

provide a definition to it. I find the definition, although correct, but rather

narrow. MCMC algorithms have broader applications such as estimating

posteriors, or numerically calculating an integral which is cumbersome to

calculate analytically. Hence I would suggest to put the MCMC approach

into a broader context and subsequently narrow it down to its particular

application for the current problem.

6) Just as a suggestion, although several texts refer to the uniform prior distribution

as "uninformative", I find it an incorrect usage of the term. The uniform probability still contains information, such as the boundaries of the parameter, or even saying that "every value of parameter within the boundary is equally likely" contains information, in my opinion. Therefore, maybe using other terms such as ignorant prior or even uniform prior is a better practice.

7) In line 303, authors mentioned that the prior distribution in Bayesian

analysis is a means to ensure that a parameter does not lose its physical

meaning at parameter fitting; I am not sure if I understand this statement,

so please clarify it further.

Authors suggested a quantity Q(theta) that is basically the log-ratio of the

model's prediction under parameter to the real value (or the model's prediction

under maximum likelihood parameter estimation (MLE), across time. A distribution

of Q(theta) is calculated, for a preferably large number of samples of parameters

drawn by a MCMC algorithm. With regard to this technique for characterizing

prediction quality I have several questions that would like to be addressed by

the authors.

1) How the equation for Q(theta) is derived? Under what assumptions such a

formula can be obtained?

2) I suppose this concept relies on a choice of "correct" prediction, from

which the dispersion of model's predictions is quantfied when the model's

parameters are perturbed. Here it is suggested that the predictions of the

model under maximum likelihood parameter MLE represent the correct

prediction. How much this choice is justified under different models? Under

what conditions, and assumptions, the model that is trained by

MLE provides the correct prediction?

3) Looking at the figure 2C and 2D, they resemble the figure 4B from the

reference (Gutenkunst 2007b). This raises a question that how novel is the

proposed quantity Q. If they are based on different assumptions, it can

be useful to clarify this explicitly in the paper. Plus, I think a comparison

between these two approaches is necessary to pin down differences clearly.

4) It is described that the LCA method is also considered for characterizing

prediction uncertainty, I find it useful to mention and consider MCA

method as well, which can be found in (Gutenkunst 2007a). This again

boils down to comparison of the proposed quantity Q and Gutenkunst et

al.

5) In the text, I found it several times that authors used the term "dimensionless"

for Q (e.g., lines 237 and 128), as it is an extra advantage for Q, when

compared to other approaches. If it is important to highlight the so-called

dimensionless quality of Q, it can be useful to discuss why it is important

and whether other methods are lacking this quality.

6) In section "Estimation of the posterior distribution of the parameters",

the general parameter estimation under Bayesian framework is discussed.

Bayesian framework enjoys incorporating prior knowledge on the acceptable parameter's value, and the current observed data, into a unified

framework for parameter estimation. In the text, authors used a uniform

distribution for the prior distribution which is for situations that given

our information we are ignorant about the parameter's value. Firstly, it

might be constructive if authors explicitly discuss the different choices of

the prior distribution.

7) If an ignorant (uniform) prior is replaced by another prior such as Gaussian

prior or other members of exponential family, how this will be translated

into the sloppy parameter sensitivity. And consequently, how much this

will affect the prediction quality Q of the model. Does this boost the

model's prediction quality or does not change the outcome?

1) It is mentioned in discussion (line 346) that the proposed approach is

applied on noiseless data, and where finding MLE is computationally

easy. However, once finding MLE is difficult, authors suggested to take

the average predictions. First, average predictions under what choice of theta. Second, how this point is formally justified?

2) I suggest to have an extra section in the main text to discuss the application

of this approach on noisy data. And discuss different scenarios that

might be faced when dealing with a data that is contaminated with various

noise components. Therefore, the authors can justify different choices

for when MLE is difficult to determine.

3) At line 364, authors concluded that the publication of models should include

a number of parameter samples from the parameter space posterior,

which is an acceptable point. However, this has been suggested several

times before in other publications, such as reference (Brown 2003). If there is

any novelty in the suggestion, authors might consider elaborate this point

further, otherwise mention previous works that have pointed this out.

The proposed quantity Q in fact quantities the dispersion of predictions

around a reference point, that is proposed to be fixed at the predictions from the

maximum likelihood estimation (MLE) parameters. Therefore, the quality of

predictions is implicitly defined as how much predictions diverge from a reference

point. In Bayesian framework, on the other hand, a credible interval is defined as

an interval that comprises a given amount of posterior distribution. The credible

(and confidence) intervals are associated to the "certainty" of predictions, and

the stability of predictions in proximity of a reference point is not of concern. In

the text, terms "certainty" and "quality" (as it is characterized by Q) are used

interchangeably. However, quality of prediction is not a well-defined concept,

but the certainty of predictions are rigorously defined in relevant literature. All

in all, I did not find the argument that Q quantfies the certainty in prediction

convincing, it is merely a measure of dispersion around a single reference point.

Therefore, I would like to ask authors to clarify this point, otherwise it would

lead to confusion and misconception for the readers.

Based on the definition of sloppy parameter sensitivity, it is logically easy

to see that under variation of parameters along the sloppy axis the model's

prediction does not change dramatically. Therefore, some would mistake

this as the robustness in model. Model's with a high-degree of sloppiness

would exhibit a more robust character. However, there is another concept

where the certainty in predictions that when a parameter estimation is

less certain, the prediction is less trustworthy. The proposed quantity

Q actually characterizes the former, the variation of model's prediction,

or loosely defined robustness. I believe it is more useful to quantify the

model's prediction certainty. Or it must be explicitly discussed in the text.

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