Prophet: Forecasting & Decision Models

• Prophet is a decomposable model for univariate time series and an offline benchmark in optimal stopping, integrating trend, seasonality, holiday, and residual components.
• It supports both linear and logistic growth by employing Fourier series for seasonality and a maximum a posteriori approach for efficient changepoint detection.
• Prophet is applied across diverse domains, from forecasting energy and water demand to evaluating online allocation, bridging prediction with structured decision-making.

Prophet is a technically overloaded term in current research literature. In time-series analysis, it most commonly denotes the open-source forecasting framework developed by Meta for univariate series with additive trend, seasonality, holiday, and residual components (Shapiro et al., 9 Jan 2026). In online algorithms and mechanism design, a prophet is an offline benchmark that knows realized future values and thereby defines prophet inequalities (Arsenis et al., 2020). The name also appears in derivative constructs, including NeuralProphet (Chhibber et al., 8 Jan 2026), the PROPHET benchmark for inferable future-event forecasting (Tao et al., 2 Apr 2025), the Bayesian Prophet framework for oracle-driven online decision making (Vera et al., 2019), and “Follow the Prophet” for delayed-feedback conversion prediction (Li et al., 2021).

1. Terminological scope

Within forecasting research, Prophet is presented as a decomposable model for univariate time series used in business, finance, utility demand, municipal infrastructure planning, and consumer-facing energy applications (Shapiro et al., 9 Jan 2026). Within theoretical computer science, “the prophet” denotes an omniscient comparator who knows all realizations in advance and therefore defines the offline optimum against which online rules are evaluated (Kleinberg et al., 2012). These senses are conceptually distinct: one is a forecasting framework; the other is a benchmark in optimal stopping and online allocation.

This distinction matters because several papers use the same word in ways that are explicitly unrelated. “Follow the Prophet” does not refer to Meta’s forecasting model; there, the prophet is an ideal conversion-rate predictor trained on fully matured labels in delayed-feedback advertising systems (Li et al., 2021). Likewise, PROPHET is a benchmark for retrieval-augmented future-event forecasting rather than a time-series forecaster (Tao et al., 2 Apr 2025). A plausible implication is that “Prophet” functions as a family name across multiple research programs rather than a single canonical method.

2. Prophet as a decomposable forecasting framework

As a forecasting framework, Prophet models a univariate series in additive form,

$y(t)=g(t)+s(t)+h(t)+\epsilon_t,$

where $g(t)$ is trend, $s(t)$ is seasonality, $h(t)$ is holiday or event structure, and $\epsilon_t$ is residual error (Shapiro et al., 9 Jan 2026). The framework supports both linear and logistic growth. For piecewise linear growth with changepoints,

$$g(t) = (k + \mathbf{a}(t)^T \boldsymbol{\delta}) t + (m + \mathbf{a}(t)^T \boldsymbol{\gamma}),$$

with $\gamma_j=-t_j\delta_j$. For logistic growth with carrying capacity $C$,

$$g(t) = \frac{C}{1 + \exp(-(k + \mathbf{a}(t)^T \boldsymbol{\delta})(t - (m + \mathbf{a}(t)^T \boldsymbol{\gamma})))}.$$

Seasonality is represented with Fourier series,

$$s(t) = \sum_{n=1}^{N} \left[ a_n \cos\left(\frac{2\pi n t}{P}\right) + b_n \sin\left(\frac{2\pi n t}{P}\right) \right],$$

and multiple seasonalities are summed across components such as yearly, weekly, and daily (Shapiro et al., 9 Jan 2026). Holiday effects are encoded with indicator variables,

$g(t)$0

with Gaussian regularization on $g(t)$1. Trend flexibility is controlled by a Laplace prior on changepoint adjustments,

$g(t)$2

where $g(t)$3 corresponds to changepoint_prior_scale. The paper describing Prophet as a reproducibility framework states that candidate changepoints are placed uniformly in the first $g(t)$4 of the history by default, that estimation is implemented in Stan, and that default fitting uses maximum a posteriori optimization rather than full Bayesian inference (Shapiro et al., 9 Jan 2026).

In practice, Prophet’s interface standardizes the data representation to ds and y, while allowing explicit analyst control over growth form, seasonality structure, holiday/event tables, multiplicative versus additive seasonal interaction, external regressors, cross-validation, and serialization (Shapiro et al., 9 Jan 2026). This suggests that Prophet’s technical identity is not only its additive statistical structure but also its standardized workflow.

3. Applied forecasting uses

Applied papers use Prophet primarily where interpretable long-run structure matters more than highly specialized local dynamics. In residential energy billing, it is placed on top of an IoT pipeline in which ESP32-based smart meters collect voltage and current, estimate energy in kWh, push the resulting history to Firebase, and expose usage to users through Blynk and a web interface; Prophet then forecasts future household electricity consumption for bill awareness, peak-demand management, cost reduction, and resource allocation (Nadar et al., 19 Jun 2025). In municipal demand planning, it is used for long-term citywide daily water-demand forecasting from reconstructed billing-period data, with an operational horizon of four years and a final reported MAPE of $g(t)$5 averaged across rolling-origin 5-fold cross-validation (VanBerlo et al., 2021). In business and financial analytics, it is positioned as a reproducibility-enabling baseline rather than a universal accuracy champion, outperforming ARIMA and Random Forest on a strongly seasonal retail demand dataset but underperforming them on Tesla stock prices (Shapiro et al., 9 Jan 2026). In monthly sales forecasting for a distribution company, it performs best on long-history, frequent-sales, high-importance item–organization series, while global models such as DeepAR+ and CNN-QR are stronger on short-history or sparse series (Zunic et al., 2021).

The reported configurations vary substantially across domains. The municipal water-demand study used piecewise linear trend, yearly and weekly seasonality, holiday effects, additive seasonality mode, changepoint_prior_scale = 0.001, seasonality_prior_scale = 0.01, and holidays_prior_scale = 0.01, with changepoints restricted to the first $g(t)$6 of training data so that late irregularities such as COVID-19 would not dominate long-range extrapolation (VanBerlo et al., 2021). The residential energy-billing paper emphasized preprocessing instead of benchmarked accuracy: missing-value handling through forward-fill, backward-fill, and interpolation; outlier detection with IQR and Z-score methods; resampling; moving averages and STL decomposition; feature engineering with candidate regressors such as weather, demographic information, economic factors, and events; and qualitative tuning of seasonality_mode, changepoint_prior_scale, holidays_prior_scale, and add_regressor, but without a concrete forecast horizon or numerical evaluation metrics (Nadar et al., 19 Jun 2025).

Context	Prophet setup	Reported outcome
Residential/home electricity billing (Nadar et al., 19 Jun 2025)	Additive Prophet over Firebase meter history; preprocessing-heavy pipeline	“Conceptually central but experimentally lightly documented”
Municipal water demand (VanBerlo et al., 2021)	Linear trend, yearly+weekly seasonality, holiday effects, additive mode	MAPE $g(t)$7 $g(t)$8
Business/financial analytics (Shapiro et al., 9 Jan 2026)	growth='linear', yearly+weekly seasonality, additive mode, U.S. holidays	Best on retail; weaker on Tesla
Monthly sales forecasting (Zunic et al., 2021)	Local holidays and price as information; modified Prophet parameters not reported	Strong on long-history frequent sellers

A recurrent theme across these applications is that Prophet is preferred when the signal resembles trend plus recurring seasonality plus relatively structured calendar effects. This is explicit in the municipal water-demand study, where yearly seasonality varied by at least $g(t)$9 of the trend component and holiday effects remained small, with no holiday parameter exceeding $s(t)$0 (VanBerlo et al., 2021). By contrast, the residential energy-billing paper framed Prophet as an engineering component inside a broader blockchain-enabled billing system and left its experimental layer preliminary (Nadar et al., 19 Jun 2025).

4. Prophet inequalities and the offline benchmark

In algorithmic theory, a prophet is an offline decision maker that sees realized values in advance. In the classical setting, independent nonnegative random variables $s(t)$1 are observed sequentially by a gambler, whereas the prophet obtains

$s(t)$2

With a predetermined order of observation, the tight prophet inequality is

$s(t)$3

and $s(t)$4 is best possible (Arsenis et al., 2020). In constrained-order prophet inequalities, where the gambler may choose one permutation from a fixed family $s(t)$5, the forward/reverse family $s(t)$6 has threshold prophet ratio exactly

$s(t)$7

and the paper identifies a “double plateau” phenomenon in which threshold ratios remain essentially at $s(t)$8 until $s(t)$9 grows to roughly logarithmic size and never exceed $h(t)$0 even when all $h(t)$1 permutations are allowed (Arsenis et al., 2020).

The benchmark generalizes to richer feasibility systems. For any matroid, there exists an online algorithm such that

$h(t)$2

where $h(t)$3 is the maximum-weight feasible offline set under the realized weights (Kleinberg et al., 2012). For intersections of $h(t)$4 matroids, the same framework yields

$h(t)$5

and the $h(t)$6 dependence is tight up to constants (Kleinberg et al., 2012). For polymatroids, the $h(t)$7 factor survives in a fractional allocation model:

$h(t)$8

The proof proceeds by reduction to a block-structured matroid with correlated block weights and then extending the Kleinberg–Weinberg threshold method to that correlated setting (Duetting et al., 2013).

Subsequent work has recast prophet inequalities in more general frameworks. “Prophet Inequalities Made Easy” formulates sufficient conditions in terms of $h(t)$9-balanced prices and proves an extension theorem under which expected posted prices give welfare at least

$\epsilon_t$0

in the stochastic setting, thereby recovering and extending results for matroids, combinatorial auctions, knapsack, and sparse packing (Dütting et al., 2016). For submodular objectives, improved correlation-gap bounds and OCRS-based reductions lead to polynomial-time prophet inequalities for monotone and non-monotone submodular functions under any constraint admitting a greedy OCRS, with explicit constants such as $\epsilon_t$1 for monotone matroids and $\epsilon_t$2 for general non-negative submodular functions over matroids (Chekuri et al., 2021). A later LP framework represents online and offline allocation in reduced-form interim variables, recovers $\epsilon_t$3 for $\epsilon_t$4 nonnegative linear constraints, proves $\epsilon_t$5 for polymatroids, extends this to online polymatroids whose submodular function depends on realized rewards, and shows that a Minkowski sum of polyhedra inherits the minimum prophet factor of its summands (Bayrak et al., 7 Feb 2026).

5. Information-limited, oracle-augmented, and Bayesian prophet models

A major line of work studies how prophet guarantees change when the online algorithm has less or different information than full distributional knowledge. In the limited-information model, only a constant number of samples from each distribution are available. For $\epsilon_t$6-uniform matroids, the Rehearsal Algorithm achieves a single-sample competitive ratio

$\epsilon_t$7

matching the best asymptotic dependence known even under full information; the same paper also gives secretary-to-prophet reductions for graphic, transversal, and laminar matroids, and a constant-sample $\epsilon_t$8-competitive prophet inequality for constant-degree bipartite matching (Azar et al., 2013).

Oracle-augmented prophet inequalities instead endow the gambler with up to $\epsilon_t$9 queries to an oracle that answers YES if the current realization exceeds all future realizations and NO otherwise (Har-Peled et al., 2024). For the objective of maximizing the probability of selecting the maximum, the oracle model with $$g(t) = (k + \mathbf{a}(t)^T \boldsymbol{\delta}) t + (m + \mathbf{a}(t)^T \boldsymbol{\gamma}),$$0 calls is equivalent to the Top-$$g(t) = (k + \mathbf{a}(t)^T \boldsymbol{\delta}) t + (m + \mathbf{a}(t)^T \boldsymbol{\gamma}),$$1-of-$$g(t) = (k + \mathbf{a}(t)^T \boldsymbol{\delta}) t + (m + \mathbf{a}(t)^T \boldsymbol{\gamma}),$$2 model. For the ratio-of-expectations objective the equivalence fails, but the oracle model still transfers losslessly as a lower bound to Top-$$g(t) = (k + \mathbf{a}(t)^T \boldsymbol{\delta}) t + (m + \mathbf{a}(t)^T \boldsymbol{\gamma}),$$3-of-$$g(t) = (k + \mathbf{a}(t)^T \boldsymbol{\delta}) t + (m + \mathbf{a}(t)^T \boldsymbol{\gamma}),$$4. In the general independent, non-identically distributed case, the optimal oracle-model competitive ratio is

$$g(t) = (k + \mathbf{a}(t)^T \boldsymbol{\delta}) t + (m + \mathbf{a}(t)^T \boldsymbol{\gamma}),$$5

where $$g(t) = (k + \mathbf{a}(t)^T \boldsymbol{\delta}) t + (m + \mathbf{a}(t)^T \boldsymbol{\gamma}),$$6 is the unique positive solution to

$$g(t) = (k + \mathbf{a}(t)^T \boldsymbol{\delta}) t + (m + \mathbf{a}(t)^T \boldsymbol{\gamma}),$$7

This yields the asymptotic behavior $$g(t) = (k + \mathbf{a}(t)^T \boldsymbol{\delta}) t + (m + \mathbf{a}(t)^T \boldsymbol{\gamma}),$$8 (Har-Peled et al., 2024).

The uncertain-acceptance variant modifies the action itself. Each item has a realized value $$g(t) = (k + \mathbf{a}(t)^T \boldsymbol{\delta}) t + (m + \mathbf{a}(t)^T \boldsymbol{\gamma}),$$9 and a Bernoulli acceptance variable $\gamma_j=-t_j\delta_j$0; if the decision maker attempts item $\gamma_j=-t_j\delta_j$1, she receives $\gamma_j=-t_j\delta_j$2 only when $\gamma_j=-t_j\delta_j$3, otherwise she continues (Emile et al., 23 Mar 2026). The paper defines three benchmarks: the online decision maker $\gamma_j=-t_j\delta_j$4, the value-aware decision maker $\gamma_j=-t_j\delta_j$5 who sees all $\gamma_j=-t_j\delta_j$6 but not the $\gamma_j=-t_j\delta_j$7, and the full prophet $\gamma_j=-t_j\delta_j$8 who sees both. The key reduction sets $\gamma_j=-t_j\delta_j$9 and proves that $C$0 for the classical prophet inequality instance $C$1. Consequently, the worst-case competitive ratios all equal $C$2:

$C$3

where $C$4, $C$5, and $C$6. Yet if $C$7 uniformly, then

$C$8

so the value-aware decision maker strictly beats the $C$9 barrier against the full prophet (Emile et al., 23 Mar 2026).

“The Bayesian Prophet” shifts the focus from multiplicative competitive ratios to additive regret against the offline optimum (Vera et al., 2019). It introduces compensated coupling, with sample-path decomposition

$$g(t) = \frac{C}{1 + \exp(-(k + \mathbf{a}(t)^T \boldsymbol{\delta})(t - (m + \mathbf{a}(t)^T \boldsymbol{\gamma})))}.$$0

where $$g(t) = \frac{C}{1 + \exp(-(k + \mathbf{a}(t)^T \boldsymbol{\delta})(t - (m + \mathbf{a}(t)^T \boldsymbol{\gamma})))}.$$1 is the event that the online action disagrees with the offline optimum and $$g(t) = \frac{C}{1 + \exp(-(k + \mathbf{a}(t)^T \boldsymbol{\delta})(t - (m + \mathbf{a}(t)^T \boldsymbol{\gamma})))}.$$2 is the exact compensation required to force agreement. The resulting Bayes Selector greedily minimizes an oracle estimate $$g(t) = \frac{C}{1 + \exp(-(k + \mathbf{a}(t)^T \boldsymbol{\delta})(t - (m + \mathbf{a}(t)^T \boldsymbol{\gamma})))}.$$3 of the disagreement probability $$g(t) = \frac{C}{1 + \exp(-(k + \mathbf{a}(t)^T \boldsymbol{\delta})(t - (m + \mathbf{a}(t)^T \boldsymbol{\gamma})))}.$$4. For online packing and online matching with finite types, this framework yields constant expected regret, independent of horizon and resource scale (Vera et al., 2019).

6. Derived names, adjacent systems, and common confusions

Several recent works use “Prophet” in derivative titles without denoting Meta’s forecasting package. “Follow the Prophet” studies delayed-feedback online conversion-rate prediction and defines the prophet as a well-trained CVR model built on fully matured labels $$g(t) = \frac{C}{1 + \exp(-(k + \mathbf{a}(t)^T \boldsymbol{\delta})(t - (m + \mathbf{a}(t)^T \boldsymbol{\gamma})))}.$$5 (Li et al., 2021). The method constructs horizon-specific tasks

$$g(t) = \frac{C}{1 + \exp(-(k + \mathbf{a}(t)^T \boldsymbol{\delta})(t - (m + \mathbf{a}(t)^T \boldsymbol{\gamma})))}.$$6

and a policy network with weights $$g(t) = \frac{C}{1 + \exp(-(k + \mathbf{a}(t)^T \boldsymbol{\delta})(t - (m + \mathbf{a}(t)^T \boldsymbol{\gamma})))}.$$7, yielding

$$g(t) = \frac{C}{1 + \exp(-(k + \mathbf{a}(t)^T \boldsymbol{\delta})(t - (m + \mathbf{a}(t)^T \boldsymbol{\gamma})))}.$$8

The aggregation policy is trained to imitate the historical prophet by assigning each sample to the task whose prediction is closest to the prophet’s prediction. This use of “prophet” is explicitly stated to be unrelated to Facebook/Meta Prophet (Li et al., 2021).

PROPHET, in all capitals, is a benchmark for retrieval-augmented future-event forecasting rather than a time-series forecaster (Tao et al., 2 Apr 2025). Its central claim is that forecasting questions should be inferable from their associated news evidence. The paper defines Causal Intervened Likelihood,

$$g(t) = \frac{C}{1 + \exp(-(k + \mathbf{a}(t)^T \boldsymbol{\delta})(t - (m + \mathbf{a}(t)^T \boldsymbol{\gamma})))}.$$9

to measure how strongly an article supports the correct answer, then filters questions into inferable $$s(t) = \sum_{n=1}^{N} \left[ a_n \cos\left(\frac{2\pi n t}{P}\right) + b_n \sin\left(\frac{2\pi n t}{P}\right) \right],$$0 and insufficiently supported $$s(t) = \sum_{n=1}^{N} \left[ a_n \cos\left(\frac{2\pi n t}{P}\right) + b_n \sin\left(\frac{2\pi n t}{P}\right) \right],$$1 subsets. The released benchmark contains $$s(t) = \sum_{n=1}^{N} \left[ a_n \cos\left(\frac{2\pi n t}{P}\right) + b_n \sin\left(\frac{2\pi n t}{P}\right) \right],$$2 questions in $$s(t) = \sum_{n=1}^{N} \left[ a_n \cos\left(\frac{2\pi n t}{P}\right) + b_n \sin\left(\frac{2\pi n t}{P}\right) \right],$$3 and $$s(t) = \sum_{n=1}^{N} \left[ a_n \cos\left(\frac{2\pi n t}{P}\right) + b_n \sin\left(\frac{2\pi n t}{P}\right) \right],$$4 in $$s(t) = \sum_{n=1}^{N} \left[ a_n \cos\left(\frac{2\pi n t}{P}\right) + b_n \sin\left(\frac{2\pi n t}{P}\right) \right],$$5, each paired with $$s(t) = \sum_{n=1}^{N} \left[ a_n \cos\left(\frac{2\pi n t}{P}\right) + b_n \sin\left(\frac{2\pi n t}{P}\right) \right],$$6 news articles, and evaluates forecasting systems with Brier Score (Tao et al., 2 Apr 2025).

NeuralProphet is presented in one later application paper as a Prophet-inspired framework with trend, seasonality, holiday, extra-regression, and autoregression components, combined with an MLP/DNN pipeline for a proposed “NP-DNN” stock-market prediction system (Chhibber et al., 8 Jan 2026). That paper also states a decomposition

$$s(t) = \sum_{n=1}^{N} \left[ a_n \cos\left(\frac{2\pi n t}{P}\right) + b_n \sin\left(\frac{2\pi n t}{P}\right) \right],$$7

but its dataset, task definition, and model specification are not fully aligned with standard Prophet-style forecasting practice. A cautious reading is therefore necessary: NeuralProphet is clearly framed as an extension of Prophet, but the specific application described there is not a stable reference for Prophet methodology (Chhibber et al., 8 Jan 2026).

Across these strands, “Prophet” names either a decomposable forecasting framework, an omniscient offline benchmark, or an inherited title for systems built around idealized foresight. The literature therefore uses the term less as a single definition than as a recurring research motif linking prediction, hindsight optimality, and structured decision making.