Learning quantitative sequencefunction relationships from massively parallel experiments
Abstract
A fundamental aspect of biological information processing is the ubiquity of sequencefunction relationships – functions that map the sequence of DNA, RNA, or protein to a biochemically relevant activity. Most sequencefunction relationships in biology are quantitative, but only recently have experimental techniques for effectively measuring these relationships been developed. The advent of such “massively parallel” experiments presents an exciting opportunity for the concepts and methods of statistical physics to inform the study of biological systems. After reviewing these recent experimental advances, we focus on the problem of how to infer parametric models of sequencefunction relationships from the data produced by these experiments. Specifically, we retrace and extend recent theoretical work showing that inference based on mutual information, not the standard likelihoodbased approach, is often necessary for accurately learning the parameters of these models. Closely connected with this result is the emergence of “diffeomorphic modes” – directions in parameter space that are far less constrained by data than likelihoodbased inference would suggest. Analogous to Goldstone modes in physics, diffeomorphic modes arise from an arbitrarily broken symmetry of the inference problem. An analytically tractable model of a massively parallel experiment is then described, providing an explicit demonstration of these fundamental aspects of statistical inference. This paper concludes with an outlook on the theoretical and computational challenges currently facing studies of quantitative sequencefunction relationships.
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1 Introduction
A major longterm goal in biology is to understand how biological function is encoded within the sequences of DNA, RNA, and protein. The canonical success story in this effort is the genetic code: given an arbitrary sequence of messenger RNA, the genetic code allows us to predict with near certainty what peptide sequence will result. There are many other biological codes we would like to learn as well. How does the DNA sequence of a promoter or enhancer encode transcriptional regulatory programs? How does the sequence of premRNA govern which exons are kept and which are removed from the final spliced mRNA? How does the peptide sequence of an antibody govern how strongly it binds to target antigens?
A major difference between the genetic code and these other codes is that while the former is qualitative in nature, the latter are governed by sequencefunction relationships that are inherently quantitative. Quantitative sequencefunction relationships^{1}^{1}1These have also called quantitative sequenceactivity maps, or QSAMs Melnikov:2012dw (). describe any function that maps the sequence of a biological heteropolymer to a biologically relevant activity (Fig. 1a). Perhaps the simplest example of such a relationship is how the affinity of a transcription factor protein for its DNA binding site depends on the DNA sequence of that site (Fig. 1b). Such relationships are a key component of the more complicated relationship between the DNA sequence of a promoter or enhancer (which typically binds multiple proteins) and the resulting rate of mRNA transcription (Fig. 1c). In both of these cases, the activities of interest (affinity or transcription rate) can vary over orders of magnitude and yet still be finely tuned by adjusting the corresponding sequence (binding site or promoter/enhancer). Similarly other sequencefunction relationships, like the inclusion of exons during mRNA splicing or the affinity of a protein for its ligand, are fundamentally quantitative.
The study of quantitative sequencefunction relationships presents an exciting opportunity for the concepts and methods of statistical physics to shed light on biological systems. There is a natural analogy between biological sequences and the microstates of physical systems, as well as between biological activities and physical Hamiltonians. Yet we currently lack answers to basic questions a statistical physicist might ask, such as “what is the density of states?” or “is a relationship convex or glassy?” The answers to such questions may well have important consequences for diverse fields including biochemistry, systems biology, immunology, and evolution.
Experimental methods for measuring sequencefunction relationships have improved dramatically in recent years. In the mid 2000s, multiple “highthroughput” methods for measuring the DNA sequence specificity of transcription factors were developed; these methods include protein binding microarrays (PBMs) Mukherjee:2004um (); Berger:2006wp (), E. coli onehybrid technology (E1H) Meng:2005dp (), and microfluidic platforms Maerkl:2007tv (). The subsequent development and dissemination of ultrahighthroughput DNA sequencing technologies then led, starting in 2009, to the creation of a number of “massively parallel” experimental techniques for probing a wide range of sequencefunction relationships (Table 1). These massively parallel assays can readily measure the functional activity of to sequences in a single experiment by coupling standard benchtop techniques to ultrahighthroughput DNA sequencing.
sequence  activity  system  name  publication 

DNA binding sites  proteinDNA binding affinity  purified protein  BindnSeq  Zykovich et al., 2009 Zykovich:2009dx () 
HTSELEX  Zhao et al., 2009 Zhao:2009gs ()  
Jolma et al., 2010 Jolma:2010hz ()  
EMSASeq  Wong et al., 2011 Wong:2011gi ()  
SELEXSeq  Slattery et al., 2011 Slattery:2011hxa ()  
promoter/ enhancer DNA  transcription rate  purified protein  Patwardhan et al., 2009 Patwardhan:2009cw ()  
bacteria  SortSeq  Kinney et al., 2010 Kinney:2010tn ()  
cell culture  MPRA  Melnikov et al., 2012 Melnikov:2012dw ()  
mouse liver  Patwardhan et al., 2012 Patwardhan:2012hy ()  
yeast  Sharon et al., 2012 Sharon:2012io ()  
mouse retina  CRESeq  Kwasniesk et al., 2012 Kwasnieski:2012hu ()  
protein  ligand binding  phage display  DMS  Fowler et al., 2010 Fowler:2010gt () 
cellular growth rate  yeast  EMPIRIC  Hietpas et al., 2011 Hietpas:2011bp ()  
toxin activity  bacteria  Adkar et al., 2012 Adkar:2012ea ()  
H1N1 binding  yeast display  Whitehead et al., 2012 Whitehead:2012jt ()  
GPCR expression  bacteria  Schlinkmann et al., 2012 Schlinkmann:2012il ()  
RNA  mRNA translation  bacteria  Holmqvist et al., 2013 Holmqvist:2013hx ()  
sRNA targeting  bacteria  qSortSeq  Peterman et al., 2014 Peterman:2014cd ()  
mRNA translation  cell culture  Oikonomou et al., 2014 Oikonomou:2014goa ()  
mRNA translation  cell culture  FACSSeq  Noderer et al., 2014 Noderer:2014fj ()  
replication origins  DNA replication  yeast  ARSSeq  Liachko et al., 2013 Liachko:2013jc () 
endonuclease sites  DNA cutting  purified protein  Thyme et al., 2014 Thyme:2014hna () 
Massively parallel experiments are very unlike conventional experiments in physics: they are typically very noisy and rarely provide direct readouts of the quantities that one cares about. Moreover, the noise characteristics of these measurements are difficult to accurately model. Indeed, such noise generally exhibits substantial daytoday variability. Although standard inference methods require an explicit model of experimental noise, it is still possible to precisely learn quantitative sequencefunction relationships from massively parallel data even when noise characteristics are unknown Kinney:2007dh (); Kinney:2014ge ().
The ability to fit parametric models to these data reflects subtle but important distinctions between two objective functions used for statistical inference: (i) likelihood, which requires a priori knowledge of the experimental noise function and (ii) mutual information Cover:1991ti (), a quantity based on the concept of entropy, which does not require a noise function. In contrast to the conventional wisdom that more experimental measurements will improve the model inference task, the standard maximum likelihood approach will typically never learn the right model, even in the infinite data limit, if one uses an imperfect model of experimental noise. Model inference based on mutual information does not suffer from this ailment.
Mutualinformationbased inference is unable to pin down the values of model parameters along certain directions in parameter space known as “diffeomorphic modes” Kinney:2014ge (). This inability is not a shortcoming of mutual information, but rather reflects a fundamental distinction between how diffeomorphic and nondiffeomorphic directions in parameter space are constrained by data. Analogous to the emergence of Goldstone modes in particle physics due to a specific yet arbitrary choice of phase, diffeomorphic modes arise from a somewhat arbitrary choice one must make when defining the sequencedependent activity that one wishes to model. Likelihood, in contrast to mutual information, is oblivious to the distinction between diffeomorphic and nondiffeomorphic modes.
We begin this paper by briefly reviewing a variety of massively parallel assays for probing quantitative sequencefunction relationships. We then turn to the problem of learning parametric models of these relationships from the data that these experiments generate. After reviewing recent work on this problem Kinney:2014ge (), we extend this work in three ways. First, we show that “diffeomorphic modes” of the parametric activity model that one wishes to learn are “dual” to certain transformations of the corresponding model of experimental noise (the “noise function”). This duality reveals a symmetry of the inference problem, thereby establishing a close analogy with Goldstone modes. Next we compute and compare the Hessians of likelihood and mutual information. This comparisons suggests an additional analogy between this inference problem and concepts in fluid mechanics. Finally, we work through an analytically tractable model of a massively parallel experiment of proteinDNA binding. This example explicitly illustrates the differences between likelihood and mutualinformationbased approaches to inference, as well as the emergence of diffeomorphic modes.
It should be noted that the inference of receptive fields in sensory neuroscience is another area of biology in which mutual information has proved useful as an objective function, and that work in this area has also provided important insights into basic aspects of machine learning Paninski:2003eq (); Sharpee:2004ty (); Sharpee:2006vb (); Kouh:2009ik (); Rajan:2011vi (). Indeed, the problem of learning quantitative sequencefunction relationships in molecular biology is very similar to the problem of learning receptive fields in neuroscience Kinney:2014ge (). The discussion of this problem in the neuroscience context, however, has largely avoided indepth analyses of how mutual information relates to likelihood, as well as of how diffeomorphic modes emerge.
2 Massively parallel experiments probing sequencefunction relationships
All of the massively parallel experiments in Table 1 share a common structure (Fig. 2a). The first step in each experiment is to generate a large set of (roughly to ) different sequences to measure. This set of sequences is called the “library.” Multiple different types of libraries can be used depending on the application. One then performs an experiment that takes this library as input, and as output provides a set of one or more “bins” of sequences. Each output bin contains sequences selected from the library with a weight that depends on the measured activity of that sequence. Finally, a sample of sequences from each of the output bins, as well as from the input library, are determined using ultrahighthroughput DNA sequencing. The resulting data thus consists of a long list of (typically nonunique) DNA sequences, each assigned to a corresponding bin (Fig. 2b). It is from these data that we wish to learn quantitative models of sequencefunction relationships.
Some of the earliest massively parallel experiments were designed to measure the specificity of purified transcription factors for their DNA binding sites Zykovich:2009dx (); Zhao:2009gs (); Jolma:2010hz (); Wong:2011gi (); Slattery:2011hxa () (Fig. 2c). The library used in such studies consists of a fixedlength region of random DNA flanked by constant sequences used for PCR amplification. This library is mixed with the transcription factor of interest, after which proteinbound DNA is separated from unbound DNA, e.g., by running the proteinDNA mixture on a gel. Proteinbound DNA is then sequenced, along with the input library.
Using a library of random DNA to assay proteinDNA binding has the advantage that the same library can be used to study each protein. This is particularly useful when performing assays on many different proteins at once (e.g., Jolma:2010hz (); Jolma:2013fh ()). On the other hand, only a very small fraction of library sequences will be specifically bound by the protein of interest. Moreover, because proteins typically bind DNA in a nonspecific manner, such experiments are often performed serially in order to achieve substantial enrichment.^{2}^{2}2This serial enrichment approach is known as SELEX and is much older than ultrahighthroughput DNA sequencing; see Oliphant:1989vb (); Tuerk:1990ux (); Ellington:1990ct (); Blackwell:1990tq (); Wright:1991uw ().
The first massively parallel experiment to probe how multiproteinDNA complexes regulate transcription in living cells was SortSeq Kinney:2010tn () (Fig. 2d). The sequence library used in this experiment was generated by introducing randomly scattered mutations into a “wildtype” sequence of interest, specifically, the 75 bp region of the promoter of the lac gene in E. coli depicted in Fig. 3a. A few million of these mutant promoters were cloned upstream of the green fluorescent protein (GFP) gene. Cells carrying these expression constructs were grown under conditions favorable to promoter activity and were then sorted into a small number of bins according to each cell’s measured fluorescence. This partitioning of cells was accomplished using fluorescenceactivated cell sorting (FACS) Herzenberg:1976th (), a method that can readily sort cells per second. The mutant promoters within each sorted bin as well as within the input library were then sequenced, yielding measurements for variant promoter sequences. We note that advances in DNA sequencing have since made it possible to accumulate much more data, and it is no long difficult to measure the activities of different sequences in this manner.
Massively parallel experiments using partially mutagenized sequences provide data about sequencefunction relationships within a localized region of sequence space centered on the wild type sequence of interest. Measuring these local relationships can provide a wealth of information about the functional mechanisms of the wild type sequence. For instance, the SortSeq data of Kinney:2010tn () allowed the inference of an explicit biophysical model for how CRP and RNAP work together to regulate transcription at the lac promoter (Fig. 3b). In particular, the authors used their data to learn quantitative models for the in vivo sequence specificity of both CRP and RNAP. Model fitting also enabled measurement of the proteinprotein interaction by which CRP is able to recruit RNAP and upregulate transcription.
Partially mutagenized sequences have also been used extensively for “deep mutational scanning” experiments on proteins, starting with Fowler:2010gt (); Hietpas:2011bp (). In this context, selection experiments on partially mutagenized proteins allow one to identify protein domains critical for folding and function. A variety of deep mutational scanning experiments are described in Fowler:2014gq ().
3 Inference using likelihood
The inference of quantitative sequencefunction relationships from massively parallel experiments can be phrased as follows. Data consists of a large number of sequences , each sequence having a corresponding measurement . Due to experimental noise, repeated measurements of the same sequence can yield different values for . Our experiment therefore has the following probabilistic form form: {diagram}[LaTeXeqno] sequenceS & \rTo^p(M—S)experiment & measurementM. If we assume that the measurements for each sequence are independent, and if we have an explicit parametric form for , then we can learn the values of the parameters by maximizing the perdatum log likelihood,
(1) 
In what follows we will refer to the quantity simply as the “likelihood.”
In regression problems such as this, one introduces an additional layer of structure. Specifically, we assume the measurement of each sequence is a noisy readout of some underlying activity that is a deterministic function of that sequence. We call the function relating to the “activity model” and denote it using . This activity model is ultimately what we want to understand. The specific way the activity is read out by measurements is then specified by a conditional probability distribution, , which we call the “noise function.”^{3}^{3}3We use the term “noise function” in order to be consistent with the terminology of Kinney:2014ge () and to avoid deviating too much from the more standard terms “noise model” and “error model” used in the statistics and machine learning literature. We emphasize, however, that defines much more than just the characteristics of experimental noise; entirely specifies the relationship between measurements and the underlying activity . Were it not for prior terminology, the term “measurement function” might be preferable to “noise function.” Our experiment is thus represented by the Markov chain {diagram}[LaTeXeqno] sequenceS & \rTo^θ(S)model & activityR & \rTo^π(M—R)noise function & measurementM. The corresponding likelihood is
(2) 
The model we adopt for our experiment therefore has two components: , which describes the sequencefunction relationship of interest, and , which we do not really care about.
Standard statistical regression requires that the noise function be specified upfront. can be learned either by performing separate calibration experiments, or by assuming a functional form based on an educated guess. This can be problematic, however. Consider inference in the large data limit, , which is illustrated in Fig. 4. Likelihood is determined by both the model and the noise function (Fig. 4a). If we know the correct noise function exactly, then maximizing over is guaranteed to recover the correct model . However, if we assume an incorrect noise function , maximizing likelihood will typically recover an incorrect model (Fig. 4b).
4 Inference using mutual information
Information theory provides an alternative inference approach. Suppose we hypothesize a specific model , which gives predictions . Denote the true model and the corresponding true activity . The dependence between , , , and will then form a Markov chain, {diagram}[LaTeXeqno] R & \lTo^θ & S & \rTo^θ^* & R^* & \rTo^π & M. From the simple fact that depends on only through the value of , any dependence measure that satisfies the Data Processing Inequality (DPI) Cover:1991ti () must satisfy
(3) 
Therefore, in the set of possible models , the true model is guaranteed to globally maximize the objective function .
One particularly relevant dependence measure that satisfies DPI is mutual information, a quantity that plays a fundamental role in information theory Cover:1991ti ().^{4}^{4}4See Kinney:2014vn () for an extended discussion of mutual information as a measure of statistical association. For the massively parallel experiments such as those in Fig. 2, is continuous and is discrete. In these cases, mutual information is given by
(4) 
where is the joint distribution of activity predictions and measurements resulting from the model . If one is able to estimate from a finite sample of data, mutual information can be used as an objective function for determining without assuming any noise function .
It should be noted that there are multiple dependence measures that satisfy DPI. One might wonder whether maximizing multiple different dependence measures would improve on the optimization of mutual information alone. The answer is not so simple. In Kinney:2014ge () it was shown that if the correct model is within the space of models under consideration, then, in the large data limit, maximizing mutual information is equivalent to simultaneously maximizing every dependence measure that satisfies DPI. On the other hand, one rarely has any assurance that the correct model is within the space of parameterized models one is considering. In this case, considering different DPIsatisfying measures might provide a test for whether is noticeably outside the space of parameterized models. To our knowledge, this potential approach to the model selection problem has yet to be demonstrated.
5 Relationship between likelihood and mutual information
A third inference approach is to admit that we do not know the noise function a priori, and to fit both and simultaneously by maximizing over this pair. It is easy to see why this makes sense: the division of the inference problem into first measuring , then learning using that inferred , is somewhat artificial. The process that maps to is determined by both and and thus, from a probabilistic point of view, it makes sense to maximize likelihood over both of these quantities simultaneously.
We now show that, in the large limit, maximizing likelihood over both and is equivalent to maximizing the mutual information between model predictions and measurements. Here we follow the argument given in Kinney:2014ge (). In the large limit, likelihood can be written
(5)  
(6) 
where
(7) 
is the KullbackLeibler divergence between the assumed noise function and the observed noise function , and is the entropy of the measurements, which does not depend on . To maximize it therefore suffices to maximize over alone, then to set the noise function equal to the empirical noise function , which causes to vanish.
Thus, when we are uncertain about the noise function , we need not despair. We can, if we like, simply learn at the same time that we learn . We need not explicitly model in order to do this; it suffices instead to maximize the mutual information over alone.
The connection between mutual information and likelihood can further be seen in a quantity called the “noiseaveraged” likelihood. This quantity was first described for the analysis of microarray data Kinney:2007dh (); see also Kinney:2014ge (). The central idea is to put an explicit prior on the space of possible noise functions, then compute likelihood after marginalizing over these noise functions. Explicitly, the perdatum log noiseaveraged likelihood is related to via
(8) 
We will refer to simply as “noiseaveraged likelihood” in what follows.
Under fairly general conditions, one finds that noiseaveraged likelihood is related to mutual information via
(9) 
Here, the effect of the noise function prior is absorbed entirely by the term . Under very weak assumptions, vanishes in the limit and thus becomes irrelevant for the inference problem Kinney:2007dh (); Kinney:2014ge ().
6 Diffeomorphic modes
Mutual information has a mathematical property that is important to account for when using it as an objective function: the mutual information between any two variables is unchanged by an invertible transformation of either variable. So if a change in model parameters, , results in changes in model predictions that preserves the rank order of these predictions, then
(10) 
and and are judged to be equally valid.
By using mutual information as an objective function, we are therefore unable to constrain any parameters of that, if changed, produce invertible transformations of model predictions. Such parameters are called “diffeomorphic parameters” or “diffeomorphic modes” Kinney:2014ge (). The distinction between diffeomorphic modes and nondiffeomorphic modes is illustrated in Fig. 5.
6.1 Criterion for diffeomorphic modes
Following Kinney:2014ge (), we now derive a criterion that can be used to identify all of the diffeomorphic modes of a model .^{5}^{5}5Here, as throughout this paper, we restrict our attention to situations in which is a scalar. The case of vectorvalued model predictions is worked out in Kinney:2014ge (). Consider an infinitesimal change in model parameters , where the components of are specified by
(11) 
for small epsilon and for some vector in space. This change in will produce a corresponding change in model predictions , where
(12) 
In general, the derivative can have arbitrary dependence on the underlying sequence . This transformation will preserve the rank order of values only if is the same for all sequences having the same value of . The change must therefore be a function of and have no other dependence on . A diffeomorphic mode is a vector field that has this property at all points in parameter space. Specifically, a vector field is a diffeomorphic mode if and only if there is a function such that
(13) 
6.2 Diffeomorphic modes of linear models
As a simple example, consider a situation in which each sequence is a dimensional vector and is an affine function of , i.e.
(14) 
for model parameters . The criterion in Eq. (13) then gives
(15) 
Because the left hand side is linear in and is linear in , the function must be linear in . Thus, must have the form
(16) 
for some functions and . The corresponding diffeomorphic mode is
(17) 
which has two degrees of freedom. Specifically, the component of corresponds to adding a constant to while the component corresponds to multiplying by a constant.
Note that if we had instead chosen , i.e. left out the constant component , then there would be only one diffeomorphic mode, corresponding to multiplication of by a constant. This fact will be used when we analyze the Gaussian selection model in Section 8.
6.3 Diffeomorphic modes of a biophysical model of transcriptional regulation
Diffeomorphic modes can become less trivial in more complicated situations. Consider the biophysical model of transcriptional regulation by the E. coli lac promoter (Fig. 3). This model was fit to SortSeq data in Kinney:2010tn (). The form of this model is as follows. Let denote a matrix representing a DNA sequence of length and having elements
(18) 
where and . The binding energy of CRP to DNA was modeled in Kinney:2010tn () as an “energy matrix”: each position in the DNA sequence was assumed to contribute additively to the overall energy. Specifically,
(19) 
where are the parameters of this energy matrix. Similarly, the binding energy of RNAP to DNA was modeled as
(20) 
Both energies were taken to be in thermal units (). The rate of transcription resulting from these binding energies was assumed to be proportional to the occupancy of RNAP at its binding site. This is given by
(21) 
where is the interaction energy between CRP and RNAP (again in units of ).
Because the binding sites for CRP and RNAP do not overlap, one can learn the parameters and from data separately by independently maximizing and . Doing this, however, leaves undetermined the overall scale of each energy matrix as well as the chemical potentials and . The reason is that the energy scale and chemical potential are diffeomorphic modes of energy matrix models and therefore cannot be inferred by maximizing mutual information.
However, if and are inferred together by maximizing instead, one is now able to learn both energy matrices with a physically meaningful energy scale. The chemical potential of CRP, , is also determined. The only parameters left unspecified are the chemical potential of RNA polymerase, , and the maximal transcription rate . The reason for this is that in the formula for in Eq. (21) the energies and combine in a nonlinear way. This nonlinearity eliminates three of the four diffeomorphic modes of and .^{6}^{6}6The one additional diffeomorphic mode is created by the introduction of the parameter . See Kinney:2014ge () for the derivation of this result.
6.4 Dual modes of the noise function
Diffeomorphic transformations of model parameters can be thought of as being equivalent to certain transformations of the noise function . Consider the transformation of model parameters
(22) 
where is an infinitesimal number and is a vector in space.^{7}^{7}7For the sake of clarity we suppress the dependence of , , and in what follows. For any sequence , this transformation induces a transformation of the model prediction
(23) 
To see the effect this transformation has on likelihood, we rewrite Eq. (2) as,
(24) 
where indicates an average taken over the measurements and predictions for all of the sequences in the data set. The change in likelihood resulting from Eq. (23) is therefore given by
(25) 
Now suppose that there is a noise function that has an equivalent effect on likelihood, i.e.,
(26) 
for all possible data sets . We say that this transformation of the noise function is “dual” to the transformation of model parameters. The transformed noise function will necessarily have the form
(27) 
for some function . To determine we consider the transformation of likelihood induced by :
(28) 
Comparing Eq. (25) and Eq. (28), we see that will be dual to for all possible data sets if and only if
(29) 
for all sequences .
For general choice of vector , no function will exist that satisfies Eq. (29). The reason is that will typically depend on the sequence independently of the value of . In other words, for a fixed value of and , the left hand side of Eq. (29) will retain a dependence on . The right hand side, however, cannot have such a dependence. The converse is also true: for general choice of the function , no vector will exist such that Eq. (29) is satisfied for all sequences. This is evident from the simple fact that is a finite dimensional vector while is a function of the continuous quantity and therefore has an infinite number of degrees of freedom.
In fact, Eq. (29) will have a solution if and only if
(30) 
for some function . Here we have added the superscript “dif” because this is precisely the definition of a diffeomorphic mode given in Eq. (13). In this case, the function dual to this diffeomorphic mode is seen to be
(31) 
These findings are summarized by the Venn diagram in Fig. 6. Arbitrary transformations of the model parameters will alter likelihood in a way that cannot be imitated by any change to the noise function . The reverse is also true: most changes to cannot be imitated by a corresponding change in . However, a subset of transformations of are equivalent to corresponding dual transformations of . These transformations are precisely the diffeomorphic transformations of . This partial duality between and has a simple interpretation: the choice of how we parse an experiment into an activity model and a noise function is not unique. The ambiguity in this choice is parameterized by the diffeomorphic modes of and the dual modes of .
7 Error bars from likelihood, mutual information, and noiseaveraged likelihood
We now consider the consequences of performing inference using various objective functions at large but finite . Specifically, we discuss the optimal parameters and corresponding error bars that are found by sampling from posterior distributions of the form
(32) 
for the following choices of the objective function :

is likelihood computed using the correct noise function .

where differs from by a small but arbitrary error.

where differs from by a small amount along a dual mode.

is the mutual information between measurements and model predictions.

is the noiseaveraged likelihood.
To streamline notation, we will use to denote averages computed in multiple different contexts. In each case, the appropriate context will be specified by a subscript. As above will denote averaging over a specific data set . will indicate averaging over an infinite number of data set realizations. , , , and will respectively denote averages over the distributions , , , and , the empirical distributions obtained in the infinite data limit. will indicate an average computed over parameter values sampled from the posterior distribution . Subscripts on or should be interpreted analogously.
7.1 Likelihood
Consider Eq. (32) with at large but finite . The posterior distribution will, in general, be maximized at some choice of parameters that deviates randomly from the correct parameters . At large , will become sharply peaked about with a peak width governed by the Hessian of likelihood; specifically
(33) 
where
(34) 
is the Hessian of the likelihood. It is also readily shown (see Appendix A) that this peak width is consistent with the correct parameters , in the sense that
(35) 
In Appendix A we show that the Hessian of likelihood, Eq. (122), is given by
(36) 
where
(37) 
is the Fisher information of the noise function . This Fisher information is a nonnegative measure of how sensitive our experiment is in the vicinity of .^{8}^{8}8In what follows we assume that almost everywhere. This just reflects the assumption that our experiment actually does convey information about through the measurements it provides. We thus see that, as long as the set of vectors spans all directions in parameter space, the Hessian matrix will be nonsingular. Using will therefore put constraints on all directions in parameters space, and these constraints will shrink with increasing data as . This situation is illustrated in Fig. 7a.
Now consider what happens if instead we use a noise function that deviates from in a small but arbitrary way. Specifically, let
(38) 
for some function and small parameter . It is readily shown (see Appendix A) that the maximum likelihood parameters will deviate from by an amount
(39) 
This expected deviation does not depend on and will therefore not shrink to zero in the large limit. Indeed, for any choice of , there will always be an large enough such that this bias in dominates over the uncertainty due to finite sampling.
Is there any restriction on the types of biases in that can be produced by the choice of incorrect noise function ? In general, no. Because the Hessian matrix is nonsingular, one can always find a vector such that the deviation of from in Eq. (39) points in any chosen direction of space. As long as the functions
(40) 
are linearly independent for different indices , a function can always be found that generates the vector .
We therefore see that arbitrary errors in the noise function will bias the inference of model parameters in arbitrary directions. This fact presents a major concern for standard likelihoodbased inference: if you assume an incorrect noise function , the parameters that you then infer will, in general, be biased in an unpredictable way. Moreover, the magnitude of this bias will be directly proportional to the magnitude of the error in the log of your assumed noise function. This problem is illustrated in Fig. 7b.
There is a case that deserves some additional consideration. Suppose we use a noise function that differs from only along a dual mode , i.e.,
(41) 
The maximum likelihood parameters of will still deviate from by an amount that does not shrink to zero in the limit. However, this bias in parameter values will be restricted to the diffeomorphic mode to which is dual, i.e.,
(42) 
This state of affairs ain’t so bad since the incorrect noise function will lead to model parameters that are inaccurate only along modes that we already know we cannot learn from the data. This situation is illustrated in Fig. 7c; see Appendix A for the derivation of Eq. (42).
7.2 Mutual information
The constraints on parameters imposed by using mutual information as the objective function in Eq. (32) are determined by the Hessian
(43) 
Appendix B provides a detailed derivation of this Hessian, which after some computation is found to be given by
(44) 
Comparing Eq. (44) and Eq. (36), we see that for any vector in parameter space,
(45) 
Likelihood is thus seen to constrain parameters in all directions at least as much as mutual information does. As expected, mutual information provides no constraint whatsoever in the direction of any diffeomorphic mode of the model, since
(46) 
The converse is also true: if there is no constraint on parameters along , then must be a diffeomorphic mode. This is because
(47) 
Because is positive almost everywhere, the right hand side of Eq. (47) can vanish only if does not differ between any two sequences that have the same value. There must therefore exist a function such that for all sequences . This is precisely the requirement in Eq. (13) that be a diffeomorphic mode.
However, except along diffeomorphic modes, we can generally expect that the constraints provided by likelihood and by mutual information will be of the same magnitude. This situation is illustrated in Fig. 7d. Indeed, in the next section we will see an explicit example where all nondiffeomorphic constraints imposed by mutual information are the same as those imposed by likelihood.
Before proceeding, we note that the relationship between the Hessians of likelihood and mutual information suggests an analogy to fluid mechanics. Consider a trajectory in parameter space given by , where is time and is a velocity vector pointing in the direction of motion. This motion in parameter space will induce a motion in the prediction that the model provides for every sequence . The set of sequence thus presents us with a dynamic cloud of “particles” moving about in space. At , the quantity will be proportional to the average kinetic energy of particles at location . The quantity will be proportional to the (per particle) kinetic energy of the bulk fluid element at , a quantity that does not count energy due to thermal motion. In this way we see that is a weighted tally of total kinetic energy, whereas corresponds to a tally of internal thermal energy only, the kinetic energy of bulk motion having been subtracted out.
7.3 Noiseaveraged likelihood
Noiseaveraged likelihood provides constraints in between those of likelihood, computed using the correct noise function, and those of mutual information. This is illustrated in Fig. 7e. Whereas mutual information provides no constraints whatsoever on the diffeomorphic modes of , noiseaveraged likelihood provides weak constraints in these directions. These soft constraints reflect the Hessian of in Eq. (9). The constraints along diffeomorphic modes, however, have an upper bound on how tight they can become in the limit. This is because such constraints only reflect our prior on the noise function, not the information we glean from data.
8 Worked example: Gaussian selection
The above principles can be illustrated in the following analytically tractable model of a massively parallel experiment, which we call the “Gaussian selection model.” In this model, our experiment starts with a large library of “DNA” sequences , each of which is actually a dimensional vector drawn from a Gaussian probability distribution^{9}^{9}9For the sake of simplicity we set the covariance matrix of this distribution equal to the identity matrix. The more general case of a nonidentity covariance matrix yields the same basic results. Also, we note that, while approximating discrete DNA sequences by continuous vectors might seem crude, it is only the marginal distributions that matter for the inference problem. Most of the quantities that one encounters in practice are computed by summing up contributions from a large number of different nucleotide positions. In such cases, the marginal distributions will often be nearly continuous and virtually indistinguishable from the marginal distributions one might obtain from a Gaussian sequence library.
(48) 
Here, is a dimensional vector defining the average sequence in the library. From this library we extract sequences into two bins, labeled and . We fill the bin with sequences sampled indiscriminately from the library. The bin is filled with sequences sampled from this library with relative probability
(49) 
where the activity is defined as the dot product of with a dimensional vector , i.e.,
(50) 
We use to denote the number of sequences in each bin , along with .
All of our calculations are performed in the limit where is large but for which is far larger. More specifically, we assume that everywhere that both and are significant. We use to denote the ratio
(51) 
and all of our calculations are carried out only to first order in . This model experiment is illustrated in Fig. 8.
Our goal is this: given the sampled sequences in the two bins, recover the parameters defining the sequencefunction relationship for . To do this, we adopt the following model for the sequencedependent activity :
(52) 
where is the dimensional vector we wish to infer. From the arguments above and in Kinney:2014ge (), it is readily seen that the magnitude of , i.e. , is the only diffeomorphic mode of the model: changing this parameter rescales , which preserves rank order.
8.1 Binspecific distributions
We can readily calculate the conditional sequence distribution for each bin , as well as the conditional distribution of model predictions. Because the sequences sampled for bin 0 are indiscriminately drawn from , we have
(53) 
The selected distribution of sequences is found to be
(54) 
The value of is found to be related to , , and via
(55) 
Appendix C provides an explicit derivation of Eq. (54) and Eq. (55).
We compute the distribution of model predictions for each bin as follows. For each bin , this distribution is defined as
(56) 
This can be analytically calculated for both of the bins owing to the Gaussian form of each sequence distribution. We find that
(57)  
(58) 
See Appendix C for details.
8.2 Noise function
To compute likelihood, we must posit a noise function . Based on our prior knowledge of the selection procedure, we choose so that
(59) 
where and are scalar parameters that we might or might not know a priori. This, combined with the normalization requirement, , gives
(60) 
This noise function is correct when and . The parameter is dual to the diffeomorphic mode , whereas the parameter is not dual to any diffeomorphic mode.
In the experimental setup used to motivate the Gaussian selection model, the parameter is affected by many aspects of the experiment, including the concentration of the protein used in the binding assay, the efficiency of DNA extraction from the gel, and the relative amount of PCR amplification used for the bin 0 and bin 1 sequences. In practice, these aspects of the experiment are very hard to control, much less predict. From the results in the previous section, we can expect that if we assume a specific value for and perform likelihoodbased inference, inaccuracies in this value for will distort our inferred model in an unpredictable (i.e., nondiffeomorphic) way. We will, in fact, see that this is the case. The solution to this problem, of course, is to infer alone by maximizing the mutual information ; in this case the values for and become irrelevant. Alternatively, one can place a prior on and , then maximize noiseaveraged likelihood . We now analytically explore the consequences of these three approaches.
8.3 Likelihood
Using the noise function in Eq. (60), the likelihood becomes a function of , , and . Computing in the and limits, we find that
(61) 
We now consider the consequences of various approaches for using to estimate . In each case, the inferred optimum will be denoted by a superscript ‘o.’ Standard likelihoodbased inference requires that we assume a specific value for and for , then optimize over alone by setting
(62) 
for each component . By this criteria we find that the optimal model is given by a linear combination of and :
(63) 
where is a scalar that solves the transcendental equation
(64) 
See Appendix C for the derivation of this result. Note that is determined only by the value of and not by the value of . Moreover, if and only if .
If our assumed noise function is correct, i.e., and , then
(65) 
Thus, maximizing likelihood will identify the correct model parameters. This exemplifies the general behavior illustrated in Fig. 7a.
If but , our assumed noise function will differ from the correct noise function only in a manner dual to the diffeomorphic mode . In this case we find that and
(66) 
is thus proportional but not equal to . This comports with our claim above that the diffeomorphic mode of the inferred model, i.e. , will be biased so as to compensate for the error in the dual parameter . This finding follows the behavior described in Fig. 7c.
If , however, . As a result, is a nontrivial linear combination of and , and will thus point in a different direction than . This is true regardless of the value of . This behavior is illustrated in Fig. 7b: errors in nondual parameters of the noise function will typically lead to errors in nondiffeomorphic parameters of the activity model.
We now consider the error bars that likelihood places on model parameters. Setting and expanding about , we find that
(67) 
where Note that all eigenvalues of are greater or equal to 1. Adopting the posterior distribution
(68) 
therefore gives a covariance matrix on of
(69) 
Thus, in all directions of space. Although will change somewhat if and deviate from and , this same scaling behavior will hold. Therefore, when the noise function is incorrect and is sufficiently large, the finite bias introduced into will cause to fall outside the inferred error bars.
8.4 Mutual information
In the limit, Eq. (4) simplifies to
(70) 
The lowest order term on the right hand side can be evaluated exactly using Eq. (57) and Eq. (58):
(71) 
See Appendix C for details. Note that the expression on the right is invariant under a rescaling of . This reflects the fact that is a diffeomorphic mode of the model defined in Eq. (52).
To find the model that maximizes mutual information, we set
(72) 
The optimal model must therefore be parallel to , i.e.
(73) 
Expanding about as above, we find that
(74) 
where is the component of perpendicular to ; see Appendix C. Therefore, if we use the posterior distribution to infer , we find uncertainties in directions perpendicular to of magnitude . These error bars are only slightly larger than those obtained using likelihood, and have the same dependence on . However, we find no constraint whatsoever on the component of parallel to . These results are illustrated by Fig. 7d.
8.5 Noiseaveraged likelihood
We can also compute the noiseaveraged likelihood, , in the case of a uniform prior on and , i.e. where is an infinitesimal constant. We find that
(75)  