How to make decisions under uncertainty? Decision-making under uncertainty relies mainly on how efficiently we can extract useful knowledge from the data that were previously unknown to the decision-maker 1 . C. R. Rao, in his 1996 article 2 on 'Uncertainty, Statistics, and Creation of New Knowledge' provided an exquisite description of the mechanics of decision-making under uncertainty using a simple logical formula: Uncertainty of knowledge `Knowledge of uncertainty " Usable knowledge. (1)

A decision analyst confronts data X 1 , . . . , X n equipped with a tentative (imprecise and uncertain) probabilistic model f 0 pxq of the underlying phenomena. The challenge then boils down to effectively using the misspecified model f 0 pxq to learn from data and to apply that knowledge for informed decision-making. Rao's uncertainty principle suggests the following three-staged approach, which we call the 'model-building triad ': The purpose of this article is to describe a general statistical theory that permits us to implement this three-staged model-building procedure for data analysis and decision-making. 'All analysts approach data with preconceptions. The data never speak for themselves. Sometimes preconceptions are encoded in precise models. Sometimes they are just intuitions that analysts seek to confirm and solidify. A central question is how to revise these preconceptions in the light of new evidence.' - Heckman and Singer (2017) Empirical scientific inquiry typically starts with a simple yet believable model of reality (model-0) and aims to sharpen existing knowledge by gathering new observations. We observe a random sample X 1 , . . . , X n 9 " F 0 . By " 9 "" we mean that F 0 is an 'approximately correct' structured provisional model for X that is given to us by subject-matter experts. We like to extract new knowledge from the data by smartly leveraging existing knowledge5 that is encoded in the initial approximate model f 0 pxq.6

Creating knowledge-guided statistical models. The core mechanism of our process involves: (i)

inspecting whether the structured provisional model-0 is still a good fit in light of fresh data;

(ii) if not, then we like to know what's new in the data that cannot be tackled by the current model; and, finally, (iii) repair the current misspecified model in order to cope with the new reality. However, the question remains as to how can we design an inference machine that can offer these successively fine-grained insights? To address this question, we will describe a new statistical model building principle, called the 'density-sharpening principle.' We introduce a dyadic model with two interrelated subsystems that accommodates the decision maker's concern for misspecification of the starting expert-guided model. Definition 1 (Dyadic model). X be a general (discrete, continuous, or mixed) random variable with true unknown density f pxq and cdf F pxq. Let f 0 pxq represents a simple approximate model for X with cdf F 0 pxq, whose support includes the support of f pxq.7 Then the following dyadic density decomposition formula holds:

here dpu; F 0 , F q is defined as

where Q 0 puq " inftx : F 0 pxq ě uu for 0 ă u ă 1 is the quantile function. The function dpu; F 0 , F q is called 'comparison density' because it compares the initial model-0 f 0 pxq with the true f pxq and it integrates to one:

However, we will interpret the d-function as the density-sharpening function (DSF), since it plays the role of "sharpening" the initial model-0 to hedge against its potential misspecification. To simplify the notation, dpF 0 pxq; F 0 , F q of eq. ( 2) will be abbreviated as d 0 pxq. A few remarks on density-sharpening law:

1. The model building mechanism of Definition 1 provides a statistical process of transforming and refining a crude initial model into a useful one for better decision-making. 2. Note that if dpu; F 0 , F q ‰ 1, i.e., if dpu; F 0 , F q deviates from uniform distribution then change of probability assignment is needed to embrace the current scenario. The density sharpening mechanism of (2) prescribes how to revise the old probability assignments in light of new evidence. The density-sharpening law provides a mechanism of building a model f pxq for the data X 1 , . . . , X n by inheriting knowledge from the assumed working model f 0 pxq. To apply the formula (2), we need to estimate d 0 pxq from data.8 And we call this learning process 'comparison coding' because d 0 pxq codes how surprising the current situation is in light of the model-0 by contrasting expectations with reality. Since the density-sharpening function d 0 pxq :" dpF 0 pxq; F 0 , F q is a function of F 0 pxq, we can approximate it by a linear combination of polynomials that are function of F 0 pxq and orthonormal with respect to the base-model f 0 pxq. One such orthonormal system is the LPfamily of polynomials (Mukhopadhyay and Parzen, 2020, Mukhopadhyay, 2021, 2017), which can be constructed as follows. For an arbitrary continuous F 0 , define the first-order LP-basis function as standardized F 0 pxq:

Note that E 0 pT 1 pX; F 0 qq " 0 and Var 0 pT 1 pX; F 0 qq " 1. Next, apply Gram-Schmidt procedure on powers of the first-order LP-basis functions tT 2 1 px; F 0 q, T 3 1 px; F 0 q, . . .u to construct a higherorder LP orthogonal system T j px; F 0 q:

T 3 px; F 0 q " ? 7 20F 3 0 pxq ´30F 2 0 pxq `12F 0 pxq ´1(

T 4 px; F 0 q " ? 9 70F 4 0 pxq ´140F 3 0 pxq `90F 2 0 pxq ´20F 0 pxq `1( ,

and so on. Compute these polynomials by performing the Gram-Schmidt process numerically, which can be done using readily available computer packages like R or python.
Definition 2 (Comparison coding). Expand comparison density in the LP-orthogonal series d 0 pxq :" dpF 0 pxq; F 0 , F q " 1 `ÿj LPrj; F 0 , F s T j px; F 0 q. (9)

To estimate the unknown LP-Fourier coefficient, note that:

Replacing LPrj; F 0 , F s with its plug-in estimator in (9) we get

where

Although (11) provides a robust nonparametric comparison-coding procedure, it has one drawback: the estimated r d may be unsmooth due to the presence of a large number of small noisy LP-coefficients. To avoid unnecessary ripples in r d, we need to isolate the small number of non-zero LP-coefficients. Our denoising strategy goes as follows (Mukhopadhyay, 2023): sort the empirical Ă LPrj; F 0 , F s in descending order based on their absolute value and compute the penalized ordered sum of squares. This Ordered PENalization scheme will be referred as OPEN model-selection method:

OPENpmq " Sum of squares of top m LP coefficients ´γn n m.

Throughout, we use AIC penalty with γ n " 2. Find the m that maximizes the OPENpmq. Store the selected indices j in the set J . The OPEN-smoothed LP-coefficients will be denoted by x LP j . Finally, return the following smoothed estimate:

Remark 1 (The scientific value of sparse d). The DSF d 0 pxq is the bridge between the theoretical world (idealized model) and the empirical world (real observations). A meaningful way to measure the simplicity of a model is the number of "new" statistical parameters that it contains beyond the given scientific parameters-that is, the parsimony (number of parameters) of d 0 pxq.9 A sparse p d 0 provides an intelligent and parsimonious way to elaborate the model-0 (not an indiscriminate, brute-force elaboration) to produce a 'sophisticatedly simple' model. Simplicity is vital to make the model usable and interpretable by decisionmakers, who like to understand how to change the initial model to explain the data. Understanding the deficiency of the current model is an essential part of the process of iterative model building and refinement: Have we overlooked something? Where are our knowledge gaps? This section provides a comprehensive understanding and exploratory tool for representing and assessing potential model misspecifications. Also, see Supp. note A1, which explains the distinctions between parametric and nonparametric model uncertainties. Example 1. Consider the following scenario: Fig. 1 displays the data that a physicist just collected from an experiment. The blue curve is the physics-informed background distribution f 0 pxq, which, in this case, is an exponential distribution with λ 0 " 25, and the red curve is the true unknown probability distribution. The physicist is mainly interested in knowing whether there is any new physics hidden in the data, i.e., anything new in the data that was overlooked by existing theory. If so, what is it? How does the theory (f 0 ) relate to practice? This will help the physicist to come up with some scientific explanations and potential alternative theories. The Shape of Uncertainty. The researcher ran the density-sharpening algorithm of the previous section with m " 10, and the resulting p d 0 pxq is displayed in the right of Fig. 1 as a function of F 0 pxq. Few conclusions: (i) Model appraisal: The non-uniformity of p d tells us that the "shape of the data" is inconsistent with the presumed model-0. (ii) Model amendment:

The shape of p d also informs the scientist about the nature of deficiency of the old model-i.e., what are the most worrisome aspects of the presumed model? In this example, the most consequential unanticipated pattern is the presence of a prominent 'bump' (excess mass) around F ´1 0 p0.63q « 24.85, which might be indicative of new physics. This newly discovered pattern can now be used to improve the background exponential model. Remark 2 (Visual explanatory decision-aiding tool). One of the unique abilities of our exploratory learning is its ability to generate explanations on why and how the model-0 is incomplete10 . Thus, the graph of p dpu; F 0 , F q explicitly addresses decision-makers model misspecification concerns. It digs into the observations to uncover the "blind spots" of the current model that can ultimately drive discovery (locating novel hypotheses) and better decisions. A general measure of the degree of model misspecification is defined using the Csiszár information divergence class. Definition 3. For ψ : r0, 8q Þ Ñ R a convex function with ψp1q " 0, define the Csiszár class of statistical divergence measure between F and F 0 :

We prefer to represent it in terms of density-sharpening function as follows:

One can recover popular divergence measures by appropriately choosing the ψ-function:

• KL-divergence: ψpxq " x logpxq; I KL pF, F 0 q " ş d log d.

• Total variation divergence: ψpxq " |x ´1|; I TV pF, F 0 q " ş |d ´1|. • Squared Hellinger distance: ψpxq " p1 ´?xq 2 ; I H pF, F 0 q " ş p1 ´?dq 2 . • χ 2 -divergence: ψpxq " px ´1q 2 ;

One can quickly estimate the χ 2 -model misspecification index by expressing it in terms of LP-Fourier coefficients (applying Parseval's identity to equation 9):

I χ 2 pF, F 0 q quantifies the uncertainty of the preliminary model f 0 pxq in light of the given datai.e., whether f 0 pxq is catastrophically wrong or slightly wrong. Estimate it by plugging the empirical LP-coefficients ( 12) into (17). Since, under H 0 : F " F 0 , the sample LP-coefficients have the following limiting null distribution (see Theorem 2 of Mukhopadhyay 2017):

n r I χ 2 pF, F 0 q follows χ 2 m under null. One can use this to compute the p-value. Applying this measure to example 1, we get a p-value of practically zero-indicating that the background exponential model is badly damaged and should be repaired before making a decision. Remark 3.
It is interesting to contrast our theory with Hansen and Sargent (2022, Sec. 4 and 5.1), keeping in mind that their mpxq is exactly our sharpening function d 0 pxq. This reinforces our belief that our theory can be applied broadly to econometrics and decision-making under uncertainty. Definition 4. DSpF 0 , mq stands for Density-Sharpening of f 0 pxq using m-term LP-series approximated d 0 pxq, given by:

obtained by replacing ( 9) into (2). DSpF 0 , mq generates a relevant class of plausible models in the neighbourhood of the postulated f 0 pxq that are worthy of consideration. A few additional points on density-sharpening:

1. The DSpF 0 , mq-based density-sharpening principle provides a mechanism for exploring data by exploiting the uncertain background knowledge model. It starts with data and an approximate model f 0 pxq-and produces a more refined picture of reality following (18). 2. The process of density-sharpening suitably 'stretches' the theory-informed model to create a class of robust empirico-scientific models. Moreover, it shows how new models are born out of data-driven mutation of pre-existing ones. 3. The truncation point m indicates the radius of the neighborhood around the elicited f 0 pxq to create permissible models. DSpF 0 , mq models with higher m entertain alternative models of higher complexity. However, to maintain conceptual appeal and interpretability, it is advisable to focus on the vicinity of f 0 by choosing an m that is not too large. Substituting the smooth estimates x LPrj; F 0 , F s of eq. ( 14) into the formula (18), we get the most economical model (among competing alternatives around f 0 ) that best explains the empirical surprise. Example 2 (Glomerular filtration data). We are given glomerular filtration rates 12 for 211 kidney patients. The experiment was done at Dr. Bryan Myers' Nephrology research lab at Stanford University. The dataset was previously analyzed in Efron and Hastie (2016). The blue curve on the left plot of Fig. 2 shows the best-fitted lognormal (LN) distribution. We start our analysis by asking whether the parametric LN model needs to be refined to fit the data. The middle panel displays the density-sharpening function, which provides insights into the nature of misspecification of the LN model: the peak and the tails of the initial LN distribution need repairing; LN underestimates the peak and neglects the presence of heavier tails. The repaired LN model (displayed on right-hand side of Fig. 2) is given by

where f 0 pxq is LNpµ 0 , σ 0 q, with µ 0 " 4 and σ 0 " 0.24. The part in the square bracket comes from d 0 pxq, which provides recommendations on how to suitably elaborate the LN-model to capture the unexplained shape. The point of this example was to show how the densitysharpening principle (DSP) allows an analyst to explicitly perform model formulation, fitting, checking, and repairing-all seamlessly combined into one workflow. It is interesting to compare our d-sharp LN-model (the red curve) with the seven-parameter exponential family fit shown in Fig. 5.7 of Efron and Hastie (2016). The most noticeable difference lies in the right tail. Efron's seven-parameter exponential family model shows weird spikes on the extreme-right tail. The main reason for this is that it is based on polynomials of raw x: (x, x 2 , . . . , x 7 ), which are not robust. That is to say, these traditional bases are unbounded and highly sensitive to 'large' data points. In contrast, our LP-polynomials are functions of F 0 pxq, not raw x, and thus robust by design. The other operational difference between our approach and Efron's exponential family approach is that we model the "gap" between lognormal and the data, which is often far easier to approximate nonparametrically (only required one parameter, see eq. 19) than modeling the data from scratch. 13

12 Glomerular filtration rate (GFR) measures how much blood is filtered through the kidney to remove excess wastes and fluids. Low gfr value indicates that the kidneys are not functioning as well as they should. 13 There is an easy way to see that: compare the shapes of the histograms of the left two plots of Fig. 2. Charles Sanders Peirce (1837Peirce ( -1914) ) was the pioneer of abductive reasoning; see Stigler (1978) and Mukhopadhyay (2023) for more details on the Peircean view of statistical modeling. The goal of Abductive Inference Machine14 or AIM is to provide a learning framework that endows a model with this ability to learn, grow and change with new information. Remark 4. The density-sharpening process plays an essential role for abductive inference, which provides the computational machinery for generating novel hypotheses with explanatory merit and selecting specific ones for further examinations. Remark 5 (Abductive inference ‰ Hypothesis testing). Any scientific inquiry begins with observations and some initial hypotheses. Classical statistical inference develops tools to test the validity of the null model in light of the data. Since all scientific theories are incomplete, accepting or rejecting a particular hypothesis is a pointless exercise. The real question is not whether the null hypothesis is true or false. The real question is: how far is the reality from the postulated model? In which direction(s) should we search to find a better model? Density-sharpening law provides a process of progressive refinement of yesterday's hypothesis. We often neglect how we get rid of the things that are less important...And oftentimes, I think that's a more efficient way of dealing with information. - The mechanics of our model-building mimic the brain's cognitive process that uses existing knowledge to sieve out the new information for correcting the error (sharpening) of the earlier mental model. How should a decision maker acknowledge model misspecification in a way that guides the use of purposefully simplified models sensibly? This section demonstrates how practicing abductive inference based on the density-sharpening principle can enable better decision-making in highly uncertain environments.
Abduction is the process of generating and revising a model before choosing the optimal action. Notation. A decision-maker (DM) has to take an action a from the set of available actions

A " ta 1 , . . . , a q u based on observed outcome X 1 , . . . , X n from an unknown probability distri-bution f pxq, representing some natural or social phenomenon. The DM selects the optimal action that minimizes expected loss (or risk) under the assumed model-0:

where f 0 pxq is the DM's posited probability distribution over outcomes. However, as an abducer, the DM is completely aware that the uncertainty about the outcomes may not be fully captured by a single, rigidly-defined probability distribution f 0 pxq and thus wants to choose the best decision by accommodating the uncertainty of model-0. Decision making based on density sharpening principle. To account for the imperfect nature of model-0, the most natural thing to do is to work with an enlarged class of plausible distributions around the vaguely acceptable f 0 pxq:

within a certain reasonable neighbourhood, say M " 10. We like to use this enlarged class of distributions Γ M for robust decision-making. Two such strategies are discussed below. Method 1. A cautious DM selects an action by its expected loss under the least favourable distribution within the set Γ M : fa,M " arg sup

We call this an abductive-minimax procedure. Our proposal is partly inspired by the 'localminimax' idea of Hansen and Sargent (2001a,b). Method 2. We now describe another robust decision-making procedure that takes into account the uncertainty in the analyst's elicited probability model of future states. Two key concepts are: bootstrap model averaging and action-profile function. Step 1. We use bootstrap to explore f P Γ M in an intelligent way. Draw n samples with replacement from the original data. Denote the bootstrap empirical cdf as r F Step 2. Use f p1q ˚to select the best action from the given set of q-actions ta 1 , . . . , a q u. Denote the selected action as a p1q ˚. Step 3. Repeat steps 1-2, B times (say B " 1000 times). And return:

• The sample bootstrap distribution p A of optimal actions ta p1q ˚, . . . , a pBq ˚u-which we call the action profile of the decision problem. • Bootstrap systematically generates probable alternative models tf  (ii) Furthermore, in our scheme, bootstrap provides a way to automatically estimate the posterior probability of accepting each synthesized alternative model (light blue curves in Fig. 4), eliminating the need for arbitrary subjective probabilities over models. To compute the f , bootstrap-derived model posterior probabilities are used to weigh the various alternatives from the density-set Γ M ; see Supp. note A3. Step 4. This bootstrap scheme can also be used to approximately compute the least favourable distribution, defined in ( 22):

fa,M " argmax j ż L a pxq dF pjq ˚pxq. The decision-maker can use this estimated model to carry out the proposed abductive-minimax procedure (method 1). Step 5. Robust procedure 18 : A pragmatic 19 decision-maker chooses an action (or ranks the actions) that minimizes expected loss (or maximizes the expected utility) with respect to the averaged-distribution: p a robust :" argmin aPA ş L a pxq d F pxq. Our strategy prescribes action that is robust across a wide range of plausible alternative models. It could be especially powerful for dealing with "deep uncertainty" in making robust policies. For a comprehensive overview on this subject, see Marchau et al. (2019). Step 6. Quantifying the 'robustness' of the action (or decision rule): How much does the optimal action change when a model is selected from a reasonable neighborhood of the assumed initial opinion, i.e. from the Γ m class? The shape of the action profile distribution can be used to determine how robust the optimal action is to model perturbation. In particular, the entropy of the action profile distribution can be used to assess the robustness (or stability) of the inference to potential model misspecification:

18 Our philosophy of robustness is in complete agreement with Huber (1977), who advocated distributional robustness: "one would like to make sure that methods work well not only at the [idealized parametric] model itself, but also in a neighborhood of it." 19 Pragmatism is the logic of abduction. PrpA " a i q log PrpA " a i q. (24)

Uniform probability over possible actions yields maximum uncertainty-indicating that the decision is highly non-robust (sensitive) to model misspecification. Until now, we have assumed experts can precisely formulate their opinion in a probabilistic form f 0 pxq. However, for complex real-world decision-making problems, experts might only have incomplete information about the uncertainty distribution of the target variable. A decision-analyst often elicit their partial knowledge about an uncertain quantity as a set of quantile-probability (QP) pairs tx i , F px i qu, for i " 1, . . . , . The job of an analyst is to find a simple, flexible, and parameterizable density that honors the assessed percentiles. Model ambiguity due to incomplete information. The task of eliciting an expert's probability distribution from a small set of QP pairs is a vital yet nascent topic in decision analysis; see Powley (2013), Keelin andPowley (2011), Hadlock (2017). In this section, we present an algorithm called Q2D (stands for quantile to distribution) that provides a systematic approach to deduce a reliable expert distribution from arbitrary QP-specifications. Probability-gap Approximation. The main theoretical idea behind Q2D algorithm: Recall our DSpF 0 , mq model

Integrating from minus infinity to x on both sides, we have

where S j puq " T j pQ 0 puq; F 0 q is defined over the unit interval r0, 1s and Q 0 puq is the quantile function of the distribution f 0 .
This leads to

Given a set of arbitrary quantile-probability data px i , F px i qq, for i " 1, . . . , , we can rewrite (26) compactly as a matrix equation

where v i " F px i q ´F0 px i q, β i " LP j , and S 0 P R ˆm, S 0 ri, js " ş F 0 px i q 0 S j puq. The desired parameters are β " pβ 1 , . . . , β m q, where β j is shorthand for LPrj; F 0 , F s.

For m ď , we can uniquely estimate β using the least-square method

For large (say, ě 5), a better, more stable estimate can be found through regularization

where } ¨}p is the p norm, and λ ą 0 is the regularization parameter. The lasso (Tibshirani, 1996) penalized p β yields a sparse estimate and counters over-fitting. This penalized estimate provides a tradeoff between accuracy and interpretability.20 Finally, plug the estimated LP-Fourier coefficients β j into the primary equation ( 18) to get the expert distribution. Remark 8. The expert quantile specifications should not be viewed as a 'gold standard'they are nothing but a preliminary guess (prone to errors of judgment or hindsight bias) whose purpose is to steer the analyst in the right direction21 . For that reason, we recommend the Example 3 (Bimodal Distribution). We are given the following quantile judgments:

Quantile: x i -3.40 -2.53 -1.20 0 2.0 2.83 3.60

Probability: F px i q 0.04 0.15 0.39 0.50 0.75 0.90 0.97

In our Q2D algorithm, we choose F 0 (an initial approximate shape) to be normal distribution. To estimate the parameters µ 0 and σ 0 of the normal distribution, note that the quantile function Qpuq « µ 0 `σ0 Φ ´1puq. Thus one can quickly get a rough estimate by simply performing a linear regression22 on pΦ ´1pu i q, Qpu i qq; see Fig. 5. The estimated normal distribution is shown in the right panel, along with the Q2D-estimated density. Example 4 (U.S. Navy data). Fig. 6 shows a histogram of 122 repair times (in hours) for a component of a U.S. Navy weapons system. The dataset was analyzed in Law (2011). Imagine that for privacy and other reasons, we do not have access to the full data. The goal is to infer a probability distribution that faithfully represents the following quantiles:

Quantile: x i 0.12 1.30 3.00 7.00 26.17

Probability: F px i q 0.01 0.20 0.50 0.80 0.99

We start with exponential distribution as our initial guess, which is often taken as a 'default' distribution (model-0) in reliability analysis. For X " Exppλq, we have

MedianpXq " λ lnp2q, where λ " EpXq. From the quantile table we get λ " 3{ lnp2q " 4.32. Next, we apply the Q2D algorithm to derive the LP-parameters with f 0 " Expp4.32q. The resulting density sharpening function and the final d-sharp exponential are shown in Fig. 6. The red curve on the right plot shows an excellent fit to the data, which was derived by the Q2D algorithm simply by utilizing the five quantile-probability pairs. High-stakes decision-making (say, COVID-19 pandemic or climate change) is often based on multiple experts' opinions instead of putting all bets on a single rigidly-defined probability model. The challenge is to aid data-driven decision-making by appropriately combining several experts' models. We describe one possible way to build a 'consensus committee model ' that can be used as a possible model-0 within an abductive decision-making framework. Learning from multiple expert distributions. Given k competing probability models tf 01 , . . . , f 0k u, which may differ markedly in shape, define the following model-weights:

Relevance weight:

where LP j| is the LP-Fourier coefficients of the -th model:

Note that the relevance weight for the -th model is always 0 ă w ď 1, and

w " 1 if and only if LP j| " 0, @ j.

LP j| " 0 for all j when f 0 fully explain the data and there is no need to sharpen it further (i.e., d 0 " 1). In that sense, w 's are data-driven weights (which will keep changing as we get more and more fresh data), computed based on the degree of agreement between the observed data and expert model f 0 . Define mixture expert distribution as

where π " w { ř w . This model serves two purposes: it tries to resolve conflicting opinions based on data and at the same time encourages one to include as much diverse information as possible. Additional layer of model uncertainty. If the analyst believes that the correct model might not be among the collection of models being considered, then use the combined expert model f 0 mix pxq as a model-0 in the subsequent density-sharpening-based learning and abductive decision-making process. How should an analyst use imperfect models to learn from data?23 What should be the output of such an analysis that can ultimately aid informed decision-making? We address these questions by introducing a general inferential framework for statistical learning and decision-making under uncertainty-which builds on two core ideas: abductive thinking and density-sharpening principle. Some of the defining features of our approach for data analysis, scientific discovery, and decision-making are highlighted below:

‚ Data analysis and science of model management: No model is perfect, irrespective of how cunningly it is designed. The central problem of statistical model developmental process is to understand how a relatively simple model can evolve into a more complex and mature one in the presence of a new data environment.
The principle of density-sharpening assists this model evolution process (thereby helping empirical scientists to abduct): by abductively generating explanations on why the presumed model-0 is unfit for the data [playing the role of a quality inspector] and also providing recommendations on how to fix the misspecification issues [serving as a policy adviser] in order to make better decisions in new circumstances. ‚ Discovery and creation of new knowledge: Abductive data analysts are less interested in testing a particular working model. They are mainly interested in conceptual innovation: discovering new pursuitworthy hypotheses based on surprising empirical evidence. 24 The density-sharpening function dpu; F 0 , F q picks out 'what's new' in the data beyond the current scientific knowledge encoded in f 0 pxq, thereby helping the scientist to uncover new unexpected knowledge from the data using graphical tools. The density-sharpening principle (DSP) provides a learning mechanism that isolates the 'known' from the 'unknown' and allows us to focus on the newfound pattern in the data, which is the basis for knowledge-creation25 . ‚ Abductive inference and decision-making: The proposed theory of abductive decisionmaking tackles model uncertainty induced by imprecise, ambiguous, and incomplete knowledge about the underlying probabilistic structure. An abductive-decision support system automatically discovers and explicitly articulates the possible alternatives to the analysts, which forces them to rethink their choices before taking impulsive action; see Supp. note A5. This style of empirical reasoning and adaptive decision-making could be especially valuable in situations where strategic planners need to take quick action in the face of uncertainty, equipped with approximate subject-matter knowledge. All the datasets and R-code written for the analysis are available upon request to the author. The Author declares that there is no conflict of interest/competing interests. For more discussion on how our abductive statistical approach compares to the more traditional Bayesian statistical approach for model misspecification, robustness, and decisionmaking, see the Supplementary section. This supplementary section contains some additional notes on the connections and differences between the Bayesian statistical approach vs. the Abductive statistical approach to model misspecification, robustness, and decision-making. It is important to distinguish two main types of model uncertainty: parametric uncertainty and more general nonparametric shape uncertainty. 1) Parametric uncertainty is a classical scenario in which the model structure is assumed to be known but not the relevant parameter values. In this setup, it is implicitly assumed that the decision-maker is aware of the correct parameterized statistical models f 0 p¨; θq, which is misspecified in terms of only θ. 26 We can therefore think of it as a finite-dimensional (statistical search) problem, for which we have a number of legacy theories, including Bayesian inference. 2) Under nonparametric shape uncertainty, we work with much deeper or more severe uncertainties about the shape of the data-generating model. It is a far more challenging infinite-dimensional (statistical search) problem for which there is no established general theory. This paper addresses this concern by offering a 'general theory of nonparametric model revision' whose foundation stands on two pillars: the density-sharpening principle and abductive inference. In our theoretical framework, we only have access to a probability model f 0 pxq that approximately encodes the decision-maker's beliefs about the distribution of the observations. Our theory then provides a systematic method for searching a useful class of alternative models DSpF 0 , mq by 'correcting' (or sharpening) the hypothesized model f 0 in an automated data-driven manner. A2. Bayesian statistical approach vs. Abductive statistical approach 1) Bayesian inference is extremely effective for dealing with parametric model uncertainty. Bayes' law provides a principled method for updating beliefs about a model's parameters. Bayes' law. Given observed data x " px 1 , . . . , x n q and the parametrized likelihood function f 0 px; θq, Bayes' multiplicative rule updates belief about θ from the prior to posterior as follows r πpθ|xq 9 πpθq n ź i"1 tf 0 px i ; θqu. (A.1)

For more details on standard methods for Bayesian parametric modeling, see Box (1980). 27

Bayes decision rule. Optimal Bayes action is taken by minimizing the expected loss under the posterior:

âBayes :" argmin aPA ż Θ L a pθq πpθ|xq dθ. For more detailed treatment refer to the standard textbooks like Berger (2013, Chapter 4). 2) Abductive inference, on the other hand, is a powerful mode of statistical reasoning for nonparametric model uncertainty problems. In particular, the proposed density-sharpening law provides a systematic rule for updating the shape of a probability density model. The abductive decision analysts do not live in a fantasy world where decision-makers pretend to know the ideal parametrized model for the data. A new class of abductive inference-based decision-theoretic models called "dyadic models" are introduced in this paper (see Sec. 2), which allow the analyst to automatically generate a class of probable alternative models from data without imposing any prior structural constraints. The goal of the "model synthesis problem" is to answer the following question:

Given a set of observations, how to systematically go about searching for a model superior to the one the decision-maker initially guessed? 27 For nonparametric Bayesian modeling see Ghosal and Van der Vaart (2017). A5. Significance of abductive inference for decision making. 1) Adaptability. Reality always carries an element of surprise. To make effective decisions in a dynamic uncertain environment, it is critical to ensure that the decision model can withstand surprise; otherwise, it is unfit for use in the real world. The real advantage of using density-sharpening-based dyadic models is that they can recuperate from surprises through automated structural correction. As a result, an abductive decision rule based on dyadic models can adapt to surprises in the sense that if the true model deviates from the assumed one then still that decision works. This is achieved by averaging over the plausible alternative situations suggested by data, as described in section 3. 2) Explainability.
Another advantage of abductive inference is that it provides an interpretable and transparent explanation of why and how the real world differs from decisionmakers' initial belief about the model, which is of utmost importance when advising on decisions to policymakers. A6. Bayes, Smooth Bayes, Sharp Bayes, and Robust Bayes

As an educated guess at the data-generation process, a decision analyst handpicks a class of parametric models tf 0 px; θq : θ P Θu with quantile function Q 0 pu; θq and cdf F 0 px; θq. Definition 5 (Parametrized sharpening kernel). Define the sharpening kernel between the true generating process f pxq and the assumed parametrized f 0 px; θq as d θ :" d θ pu; F, F 0 p¨; θqq " f pQ 0 pu; θqq f 0 pQ 0 pu; θq; θq , 0 ă u ă 1 (A.4)

The corresponding sample estimate is given by r d θ :" d θ pu; r F , F 0 p¨; θqq " r f pQ 0 pu; θqq f 0 pQ 0 pu; θq; θq , 0 ă u ă 1 (A.5)

where r d θ : Θ ˆr0, 1s Ñ r0, 8q, which connects information in data to parameters of interest. With this definition in hand, we now present an important result. see, for example, Mukhopadhyay and Fletcher (2018). Instead of making critical decisions based solely on analysts' vague subjective specifications, this allows for prior misspecification. Notes A6 and A7 showcase how concept density-sharpening principles can unify both the classical and the most advanced versions of Bayesian inference using common terminology and notation -a novel contribution in and of itself. whatKeynes (1937) called 'uncertain knowledge.' Those who ignore experts' knowledge and only trust data are empirical-fools. Those who ignore data and only trust their gut-instinct are emotional-fools(Tversky and Kahneman, 1974). Expert decision-makers always use empirically-guided intuition by appropriately combining both data and available knowledge. Model amendment principle: the starting model f 0 pxq is incomplete but not useless. It contains valuable background knowledge. Rather than throwing this vital information, we want to build a model by smartly taking clues from it. The goal is to amend model-0, not to abandon it completely. As for notation: by F 0 pxq, we denote the cumulative distribution function (cdf) of the starting model-0; f 0 pxq is the probability density function (pdf) and quantile function is denoted by Q 0 puq. The expectation with respect to f 0 pxq will be abbreviated as E 0 . For dealing with truly zero-probability events seeColetti and Scozzafava (2002, Ch. 11)

To keep the theory of estimation simple, we will mainly focus on the X continuous case. A detailed account for the discrete case can be found inMukhopadhyay (2021). It selects only a handful of reasonable (rival) hypotheses out of a vast collection of possibilities. By 'reasonable,' we mean hypotheses with a high OPEN(m) score, which balances complexity (number of parameters of d 0 ) and accuracy (of explaining the surprising phenomenon). Explanation-based statistical reasoning is at the core of abductive inference, as discussed later. It brings our theory close to Gilbert Harman's "Inference to the best explanation" idea; seeHarman (1965). This is an area that merits further research. They are not traditional pattern recognition (or matching) engine, they are pattern discovery engine. Jordana Cepelewicz (2019) To Pay Attention, the Brain Uses Filters, Not a Spotlight Quanta Magazine, https://www.quantamagazine.org/to-pay-attention-the-brain-uses-filters-not-a-spotlight-20190924. The importance of model uncertainty in economics, finance, and business is beautifully illustrated inHansen and Sargent (2014), although from a different perspective. This is also known as bagging(Breiman, 1996) or bootstrap smoothing(Efron, 2014). Note that due to regularization, Eq. (29) can even tackle cases with m ą . In such scenarios, the OLS is ill-posed, with an infinite number of solutions. Winkler (1967) emphasized that the expert does not have some 'true' density function waiting to be elicited, only a 'satisficing' initial distribution that the policymaker is 'content to live with at a particular moment of time.' This technique will work for any location-scale family f 0 pxq, e.g. normal, Laplace, logistic, etc. The challenge of learning from uncertain knowledge is also a fundamental issue in the development of intelligent systems. A largely unexplored topic relative to the vast literature on hypothesis testing. As noted by George E. P.Box (2001): "Much of what we have been doing is adequate for testing but not adequate for discovery." Curious readers are invited to read the paper "Nobel Turing Challenge: creating the engine for scientific discovery" by Hiroaki Kitano, where he argued that the single-most-important mission of AI is to accelerate scientific discovery. As a result, decision theory based on parametric uncertainty is predicated on a highly restrictive model uncertainty assumptions.