Blessings: Inference, Gratitude & Turbulence

• Blessings are regimes where extra structure, such as multiple causes or outcomes, reveals latent confounding and improves model identifiability.
• The concept spans diverse fields, including causal inference, robust regression, high-dimensional factor analysis, digital gratitude practices, and elastic turbulence.
• Researchers leverage blessings to transform seemingly adverse complexity into experimental and computational advantages using methods like deconfounders and proxy discovery.

Blessings is a polysemous term in contemporary research. Across several arXiv literatures, it most often denotes a contrastive condition in which a structure usually treated as difficult—many causes, many treatments and outcomes, high ambient dimension, or even adversarial distribution shift—becomes informative and therefore statistically advantageous. In a distinct human-computer interaction usage, blessings retain their ordinary sense as objects of gratitude that can be noticed, recorded, and reflected upon. In a fluid-mechanical usage, “blessings” names a collective spatio-temporally intermittent state composed of multiple localized coherent structures called narwhals (Wang et al., 2019, Bhattacharjee et al., 2024, Morozov et al., 3 Sep 2025).

1. Terminological range and recurrent logic

The recurrent logic behind most technical uses is that multiplicity can supply indirect observability. Dependence among many causes can reveal latent confounding; many measured coordinates can stabilize latent-factor estimation; and, in one regression setting, adversarial covariate shift can move data toward an optimal experimental design rather than away from learnability. This suggests a shared editorial shorthand: blessings are regimes in which extra structure acts as leverage rather than nuisance.

Domain	Meaning of “blessing”	Representative papers
Causal inference	Many observed causes or outcomes help reveal hidden confounding	(Wang et al., 2019, Wang et al., 2019, Wu et al., 2023)
High-dimensional inference	Increasing dimension improves factor-space or precision estimation	(Lam et al., 2010, Chattopadhyay et al., 2024)
Robust learning	Adversarial shift aids regression by concentrating data along a residual signal direction	(Liang, 2022)
Gratitude systems	Blessings are positive life events or conditions recorded in gratitude practice	(Bhattacharjee et al., 2024)
Elastic turbulence	A blessing is a multi-narwhal intermittent chaotic state	(Morozov et al., 3 Sep 2025)

The term is therefore not unified by subject matter but by a repeated inversion of the “curse” motif. Where classical formulations emphasize obstruction, the blessing formulation emphasizes recoverable structure.

2. Multiple causes as a source of causal identification

In causal inference, the phrase “blessings of multiple causes” refers to the claim that when many causes $A_1,\dots,A_m$ affect one outcome, the dependence structure among the causes can itself provide evidence about unobserved shared confounding. The graphical formulation treats the causes as potential proxies for latent shared causes, allowing identification of intervention distributions for subsets of causes, such as $P(y \mid \mathrm{do}(a_{\mathcal C}))$, under proxy-style assumptions and completeness conditions (Wang et al., 2019).

The deconfounder is the principal algorithmic expression of this idea. It first fits a probabilistic factor model to the causes, constructing a substitute confounder $\hat Z$ through a factorization of the form

$$\hat P(a_1,\dots,a_m,\hat z)=\hat P(\hat z)\prod_{j=1}^m \hat P(a_j\mid \hat z),$$

and then uses $\hat Z$ in downstream outcome modeling. The reply “The Blessings of Multiple Causes: A Reply to Ogburn et al. (2019)” rejects the claims that the original theory contained “foundational errors” or that its premise was incorrect, and restates the method as a two-stage procedure: jointly model the causes, infer a substitute confounder, and use that substitute confounder in causal analysis (Wang et al., 2019).

The reply specifies two essential requirements for the original identification results in Theorems 6–8. First, the joint distribution of causes $p(a)$ must be describable by a factor model. Second, the factor model must pinpoint the substitute confounder,

$Z \stackrel{a.s.}{=} f_\theta(\mathbf{A}),$

so that the causes determine $Z$ almost surely. In the original formulation, this is tied to the “consistency of the substitute confounder,” which is satisfied when the number of causes goes to infinity while $Z$ remains finite-dimensional (Wang et al., 2019).

A central dispute concerns Lemma 4. The reply restates it as: no single-cause pre-treatment variable, single-cause post-treatment variable, or multi-cause post-treatment variable can be measurable with respect to a consistent substitute confounder. The argument is that a valid substitute confounder explains common variation across many causes, not idiosyncratic structure of one cause. If a single-cause variable were absorbed into $Z$, the factor model would become degenerate, contradicting the assumption of a non-degenerate probabilistic factor model. On this basis, the reply argues that the deconfounder does not “smuggle in” single-cause mediators, single-cause colliders, or single-cause M-bias structures (Wang et al., 2019).

The same reply also states the limits of the approach. It is not a black-box causal inference tool; it requires careful, domain-specific modeling of the causes. It does not allow unobserved single-cause confounders, involves a variance–bias trade-off, and is not recommended for causally dependent causes such as time series. A broader implication is that the “blessing” is conditional: multiplicity creates identification power only under restrictive structural assumptions rather than automatically (Wang et al., 2019).

3. Multiple treatments and multiple outcomes

A later extension generalizes the blessing idea from multiple causes with one outcome to multiple continuous treatments $P(y \mid \mathrm{do}(a_{\mathcal C}))$0 and multiple outcomes $P(y \mid \mathrm{do}(a_{\mathcal C}))$1 under unobserved confounding $P(y \mid \mathrm{do}(a_{\mathcal C}))$2. The core claim is that other treatments and other outcomes can serve as proxies for the treatment effect under study, so multiplicity on both sides of the bipartite treatment–outcome graph becomes an identification resource rather than merely additional complexity (Wu et al., 2023).

For a target effect $P(y \mid \mathrm{do}(a_{\mathcal C}))$3, the paper states that there exist admissible proxies $P(y \mid \mathrm{do}(a_{\mathcal C}))$4 and $P(y \mid \mathrm{do}(a_{\mathcal C}))$5 under mild sparsity conditions. These satisfy proximal conditions

$P(y \mid \mathrm{do}(a_{\mathcal C}))$6

and

$P(y \mid \mathrm{do}(a_{\mathcal C}))$7

Identification then proceeds through bridge-function machinery under standard completeness assumptions, yielding identifiability of $P(y \mid \mathrm{do}(a_{\mathcal C}))$8 (Wu et al., 2023).

The existence of proxies is tied to graph sparsity. The paper’s null-proxy assumption requires at least one missing edge in the treatment–outcome bipartite graph; it further remarks that if the graph has at most $P(y \mid \mathrm{do}(a_{\mathcal C}))$9 edges, the effect remains identifiable. This recasts “blessing” as a graph-theoretic property: incompleteness of the treatment–outcome adjacency pattern creates usable proxy structure (Wu et al., 2023).

A further contribution is automatic proxy discovery. Instead of prespecifying proxies, the paper proposes a hypothesis-testing-based causal discovery procedure for continuous treatments. The null hypothesis for an edge is

$\hat Z$0

After discretization and under a null-TV Lipschitzness assumption, the test statistic has asymptotic null law

$\hat Z$1

This permits edge testing and, consequently, data-driven proxy selection (Wu et al., 2023).

Empirically, the paper reports synthetic results and a sepsis application using MIMIC-III with treatments Vancomycin, Morphine Sulfate, and Norepinephrine and outcomes White blood cell count, Mean blood pressure, and Platelets. For causal discovery at $\hat Z$2, it reports $\hat Z$3, precision $\hat Z$4, and recall $\hat Z$5, versus POP values $\hat Z$6, $\hat Z$7, and $\hat Z$8. Here the blessing is explicitly operational: additional treatments and outcomes furnish internal redundancy that can be converted into valid proxies and improved effect estimation (Wu et al., 2023).

4. Blessings of dimensionality in latent-factor inference

In high-dimensional statistics, “blessing” denotes a regime in which increasing ambient dimension improves inference because many coordinates share a low-dimensional latent structure. Two related formulations make this precise: factor models for high-dimensional time series and pseudo-Bayesian covariance inference via factor analysis (Lam et al., 2010, Chattopadhyay et al., 2024).

For high-dimensional time series, the model is

$\hat Z$9

with observable $$\hat P(a_1,\dots,a_m,\hat z)=\hat P(\hat z)\prod_{j=1}^m \hat P(a_j\mid \hat z),$$0, latent factors $$\hat P(a_1,\dots,a_m,\hat z)=\hat P(\hat z)\prod_{j=1}^m \hat P(a_j\mid \hat z),$$1, loading matrix $$\hat P(a_1,\dots,a_m,\hat z)=\hat P(\hat z)\prod_{j=1}^m \hat P(a_j\mid \hat z),$$2, and white-noise idiosyncratic component $$\hat P(a_1,\dots,a_m,\hat z)=\hat P(\hat z)\prod_{j=1}^m \hat P(a_j\mid \hat z),$$3. Factor strength is indexed by $$\hat P(a_1,\dots,a_m,\hat z)=\hat P(\hat z)\prod_{j=1}^m \hat P(a_j\mid \hat z),$$4 through

$$\hat P(a_1,\dots,a_m,\hat z)=\hat P(\hat z)\prod_{j=1}^m \hat P(a_j\mid \hat z),$$5

The strong-factor case is $$\hat P(a_1,\dots,a_m,\hat z)=\hat P(\hat z)\prod_{j=1}^m \hat P(a_j\mid \hat z),$$6, equivalently $$\hat P(a_1,\dots,a_m,\hat z)=\hat P(\hat z)\prod_{j=1}^m \hat P(a_j\mid \hat z),$$7, so each factor loading vector has Euclidean norm of order $$\hat P(a_1,\dots,a_m,\hat z)=\hat P(\hat z)\prod_{j=1}^m \hat P(a_j\mid \hat z),$$8. Estimation is based on eigenanalysis of

$$\hat P(a_1,\dots,a_m,\hat z)=\hat P(\hat z)\prod_{j=1}^m \hat P(a_j\mid \hat z),$$9

Under strong factors, the estimator for the loading matrix and the resulting estimator for the precision matrix are weakly consistent in $\hat Z$0-norm with convergence rates independent of $\hat Z$1. The paper characterizes this as the setting in which the “curse” is canceled by the “blessings” in dimensionality. It also emphasizes that factor modeling helps little for covariance estimation but substantially for precision matrix estimation, especially when $\hat Z$2 and the sample covariance is singular (Lam et al., 2010).

In Bayesian covariance inference, the factor-analysis model writes

$\hat Z$3

with covariance

$\hat Z$4

The FABLE method—Factor Analysis with BLEssing of dimensionality—replaces Gibbs or other variants of MCMC by a first-stage SVD, treats the estimated factor-score matrix as fixed, fits $\hat Z$5 independent conjugate Bayesian regressions in parallel, and combines the draws into a pseudo-posterior for $\hat Z$6. For the canonical choice in the SVD stage, the estimated factor matrix reduces to

$\hat Z$7

where $\hat Z$8 contains the leading $\hat Z$9 left singular vectors (Chattopadhyay et al., 2024).

The blessing appears in the factor-subspace error rate,

$p(a)$0

which improves as either $p(a)$1 or $p(a)$2 grows, with explicit benefit from large $p(a)$3. The corresponding pseudo-posterior contraction rates for the low-rank part, diagonal part, and full covariance all contain a $p(a)$4 or $p(a)$5 term, and the paper interprets these as blessing-of-dimensionality terms (Chattopadhyay et al., 2024).

Uncertainty quantification requires calibration. Raw pseudo-posterior credible intervals can under-cover, so the paper introduces a coverage correction factor $p(a)$6, yielding CC-FABLE. In simulations with $p(a)$7 and $p(a)$8, CC-FABLE matched or beat competitors on relative spectral error and achieved nominal $p(a)$9 coverage after correction. In the GSE109125 gene expression dataset with $Z \stackrel{a.s.}{=} f_\theta(\mathbf{A}),$0 and $Z \stackrel{a.s.}{=} f_\theta(\mathbf{A}),$1, it achieved the best fitted log-likelihood, average predictive coverage around $Z \stackrel{a.s.}{=} f_\theta(\mathbf{A}),$2, and was about $Z \stackrel{a.s.}{=} f_\theta(\mathbf{A}),$3 faster than MGSP in the reported run. Most notably, when estimating a fixed covariance submatrix while adding extra genes, performance improved rapidly as more genes were included and then leveled off, an explicit empirical manifestation of the blessing of dimensionality (Chattopadhyay et al., 2024).

5. Adversarial covariate shift: blessing in regression, curse in classification

In robust learning under covariate shift, the term “blessing” refers to a specific dynamical phenomenon rather than to multiplicity. The setting assumes invariant $Z \stackrel{a.s.}{=} f_\theta(\mathbf{A}),$4 and changing covariate marginal $Z \stackrel{a.s.}{=} f_\theta(\mathbf{A}),$5, with Bayes-optimal predictor fixed. The model class is infinite-dimensional linear,

$Z \stackrel{a.s.}{=} f_\theta(\mathbf{A}),$6

and the adversary perturbs $Z \stackrel{a.s.}{=} f_\theta(\mathbf{A}),$7 within a Wasserstein ball. The resulting dynamics define a sequential game between adversarial shift and subsequent learning (Liang, 2022).

For regression with

$Z \stackrel{a.s.}{=} f_\theta(\mathbf{A}),$8

the pointwise utility satisfies

$Z \stackrel{a.s.}{=} f_\theta(\mathbf{A}),$9

so the adversarial update is linear and rank-one along the residual signal $Z$0. Defining

$Z$1

the paper proves directional convergence

$Z$2

with exponential rate

$Z$3

The interpretation is that adversarial shift concentrates the design along the residual direction still needed for identification, making the shifted data an optimal experimental design for the next learner (Liang, 2022).

This blessing is formalized in the learner-reaction theorem. After adversarially shifted samples and one gradient step with $Z$4, the learner satisfies

$Z$5

Thus the worst-case shift can accelerate subsequent recovery of the Bayes model in regression (Liang, 2022).

The same paper establishes the opposing result for classification. Under logistic loss, adversarial dynamics converge toward a “curse direction” $Z$6 orthogonal to $Z$7 under the theorem’s orthogonality assumption, and only at subquadratic rate,

$Z$8

In the sequential game, the learner then makes essentially no progress in the signal direction. The paper’s title therefore uses “blessings and curses” literally: regression and classification exhibit qualitatively opposite asymptotic geometries under the same adversarial-shift paradigm (Liang, 2022).

6. Blessings as objects of gratitude in digital wellbeing research

In the gratitude literature, blessings are not a metaphor for identifiability or dimension; they are the substantive content of gratitude practice. “Actually I Can Count My Blessings” studies how a mobile application can support young adults in noticing, recording, organizing, and reflecting on what they are grateful for in everyday life through a user-centered design process (Bhattacharjee et al., 2024).

The study used a two-stage UCD process. The formative study involved 20 participants, recruited from a large European city and a large North American city, with mean age $Z$9 years and balanced gender composition. Participants used one of two existing apps—Delightful – Three Good Things on Android or Gratitude 365 on iOS—for 2 weeks, self-directed and without instructions, and then completed 15–35 minute semi-structured Zoom interviews. Thematic analysis with open coding yielded three main design themes: novices wanted more structure for gratitude journaling, practice fit best in private and often in the evening after work or school, and mood labeling could support self-awareness but risk burden if overused (Bhattacharjee et al., 2024).

These findings shaped a custom app deployed for 2 weeks to 26 participants, randomized 13 experiment and 13 control, with mean age $Z$0 years. Shared features included Dashboard, Profile, About, and Settings pages, and both conditions received daily reminders at 6 pm. The experiment condition introduced structured life-area options aligned with the PERMA model—Physical Wellbeing, Peace/Calm, Energizing Moments, Engagement/Flow, Connection, Accomplishment, Meaning/Fulfillment, and Other—each with “More Info” explanations and examples. The control condition used open-ended entries. Both groups rated mood before and after each activity on a single 1–5 scale, where 1 = very low mood, 3 = neutral, and 5 = very high mood (Bhattacharjee et al., 2024).

The quantitative comparisons are descriptive but informative. Word count per entry was $Z$1 in the experiment condition and $Z$2 in control. Mood increase per entry was $Z$3 versus $Z$4. Mean perceived motivation on a 1–7 scale was $Z$5 versus $Z$6, mean perceived engagement was $Z$7 versus $Z$8, and mean perceived usefulness was $Z$9 versus $Z$0. Experiment participants logged entries on $Z$1 of study days and control participants on $Z$2 (Bhattacharjee et al., 2024).

The study’s main design conclusions are that structured life-area prompts help users reflect more deeply, the 6 pm reminder fit many routines, mood labeling should be balanced and customizable, and passive engagement still matters. Some participants benefited even when they did not type an entry: simply receiving the notification and mentally reflecting on positive aspects of life was meaningful. The paper also reports transfer beyond the app, including discussing gratitude with a spouse, sharing reflections with friends, and becoming more mindful of support from family. In this literature, “counting blessings” thus refers to a structured gratitude practice whose efficacy depends on scaffolding, privacy, timing, and flexible engagement modes rather than on journaling volume alone (Bhattacharjee et al., 2024).

7. Blessings in elastic turbulence

In viscoelastic fluid mechanics, “blessings” is a technical name for a dynamical state rather than a metaphorical evaluation. “Narwhals and their blessings” defines blessings as “spatio-temporal intermittent states made up of several localised narwhal solutions” in pressure-driven viscoelastic channel flow. The usage is a collective noun borrowed from zoological naming: narwhals are the elementary exact coherent structures, and a blessing is their collective arrangement (Morozov et al., 3 Sep 2025).

The governing equations use the simplified Phan-Thien–Tanner constitutive model for dilute polymer solutions between parallel plates. The principal fields are the polymer conformation tensor $Z$3, the velocity $Z$4, and the pressure $Z$5, with control parameters including the Weissenberg number $Z$6, Reynolds number $Z$7, viscosity ratio $Z$8, shear-thinning parameter $Z$9, and polymer diffusivity $P(y \mid \mathrm{do}(a_{\mathcal C}))$00. In the low-$P(y \mid \mathrm{do}(a_{\mathcal C}))$01 regime studied, the authors emphasize that the velocity is largely slaved to the polymer stress through a Stokes-like balance, so the stress/conformation tensor is the true dynamical variable (Morozov et al., 3 Sep 2025).

In two-dimensional channel flow, a narwhal is a traveling-wave solution localized around the channel centerline, with a distinctive body-and-tusk stress pattern. The laminar base state is linearly stable for the parameter ranges studied, yet sufficiently strong finite-amplitude perturbations can converge to a traveling wave through what the authors describe as a subcritical bifurcation from infinity, with critical value given by the saddle-node Weissenberg number $P(y \mid \mathrm{do}(a_{\mathcal C}))$02. Narwhals are characterized by a pair of localized stagnation points, a solitary pair of vortices near the tusk-body junction, and a filamentary polymer-stress structure centered near the channel midline (Morozov et al., 3 Sep 2025).

The paper’s main point is that these 2D narwhals are linearly unstable in three-dimensional domains. When extended spanwise and perturbed by small noise, they evolve into a fully 3D chaotic state that the paper identifies as the onset of elastic turbulence. At onset, the dynamics are spatio-temporally intermittent: localized stress-rich structures appear near the midplane, other regions remain comparatively laminar, structures split, merge, or relaminarize, and the number of coherent structures fluctuates in time. In long-box simulations the paper gives examples with three, four, and five narwhals in one domain, and explicitly calls such states blessings (Morozov et al., 3 Sep 2025).

The proposed transition scenario is sequential: the laminar state is stable to infinitesimal perturbations; finite-amplitude perturbations excite a localized narwhal-like structure; in 3D such structures become unstable; their nonlinear interaction produces a chaotic intermittent state; and at higher $P(y \mid \mathrm{do}(a_{\mathcal C}))$03 the chaotic state contains multiple localized structures, namely blessings. Diagnostics include the midplane polymer stretch $P(y \mid \mathrm{do}(a_{\mathcal C}))$04, the total kinetic energy $P(y \mid \mathrm{do}(a_{\mathcal C}))$05, spatial distributions of $P(y \mid \mathrm{do}(a_{\mathcal C}))$06, repeated localized stress peaks, and the fluctuating count of those peaks. The jump in kinetic energy at transition is reported as only a few percent, whereas the polymer stretch can increase by $P(y \mid \mathrm{do}(a_{\mathcal C}))$07 relative to the laminar value (Morozov et al., 3 Sep 2025).

This usage also defines a contrast with Newtonian turbulence. The strongest fluctuations are near the channel midplane rather than near the wall; velocity fluctuations are modest; polymer-stress fluctuations are large; and coherent structures are localized and center-mode based. Blessings are therefore the multi-structure realization of an elasticity-driven, inertialess, stress-dominated turbulent organization, rather than a near-wall inertial cycle (Morozov et al., 3 Sep 2025).