Dynamic Bayesian Updating

• Dynamic Bayesian Updating is the sequential revision of probabilistic beliefs that uses a recursive framework to incorporate new evidence and transform priors into posteriors.
• It employs various formulations—including recursive, generalized, and composite approaches—to manage non-stationary data and evolving latent structures across diverse applications.
• Applied in areas like scheduling, portfolio selection, and structural health monitoring, its methodology underpins accurate forecasting and risk assessment using both exact and approximate computational strategies.

Dynamic Bayesian updating is the sequential revision of probabilistic beliefs as new evidence arrives, with the posterior from one update becoming the prior for the next. In the supplied literature, this idea appears in standard recursive Bayes, in generalized Bayesian or Gibbs posteriors when a full likelihood is unavailable or inconvenient, and in non-stationary settings where memory, forgetting, or adaptive transformation of prior information become part of the update mechanism. The updated object may be an activity-duration distribution, a latent graph trajectory, a dynamic linear model parameter, a distribution over markings in a condition/event net, a structural deterioration law, or a full probability measure over future observations (Khoshkonesh et al., 17 May 2026, Loyal, 24 Sep 2025, Nassar et al., 2022, Vovk, 2023).

1. Recursive, generalized, and composite formulations

The canonical form of dynamic Bayesian updating is recursive conditioning. In the construction scheduling framework, the duration belief for activity $i$ is updated from prior to posterior by

$$P(D_i \mid O_i)=\frac{P(O_i \mid D_i)\,P(D_i)}{P(O_i)},$$

and the posterior becomes the next prior as new observations arrive (Khoshkonesh et al., 17 May 2026). In dynamic linear models for portfolio selection, the same recursive structure appears in Kalman-filter-style updates,

$$m_t=m_{t-1}+A_t(r_t-f_t), \qquad C_t=C_{t-1}-A_t Q_t A_t',$$

so that $(\theta\mid D_t)\sim N(m_t,C_t)$ is the next period’s prior (Sikaria et al., 2019). In the diachronic forecasting formulation, standard conditioning is written as

$$P_n(x)=P_{n-1}(x\mid y_{n-1})=\frac{P_{n-1}(y_{n-1}x)}{P_{n-1}(y_{n-1})},$$

with $P_n$ a probability measure over the remaining future sequence (Vovk, 2023).

A second formulation replaces a fully specified likelihood with a generalized Bayesian or Gibbs posterior. For dynamic random dot product graphs, the posterior is defined through a least-squares loss,

$$\Pi_\lambda(X_{1:m}\mid Y_{1:m}) \propto \exp\!\left( -\lambda \sum_{t=1}^m \|Y_t - X_tX_t^\top\|_F^2 \right)\Pi(X_{1:m}),$$

where $\lambda>0$ is the learning rate. This posterior is the unique minimizer of a loss consisting of expected reconstruction error plus a Kullback–Leibler term relative to the prior, following Bissiri et al.’s generalized Bayes framework (Loyal, 24 Sep 2025). The same paper emphasizes that the learning rate controls how much the data dominate the prior, with $\lambda\to 0$ returning essentially the prior and $\lambda\to\infty$ concentrating around the empirical adjacency spectral embedding solution.

A third formulation concerns objective or noninformative priors when successive experiments relate to the same continuous parameter in different ways. Lewis argues that ordinary sequential updating can become order-dependent because the noninformative prior appropriate for experiment A may differ from the one appropriate for experiment B. The proposed correction is to treat the accumulated data as a composite experiment, multiplying likelihoods while adding Fisher information: $$P(D_i \mid O_i)=\frac{P(O_i \mid D_i)\,P(D_i)}{P(O_i)},$$0 This replaces “posterior-to-prior updating” by “composite-experiment updating” and is explicitly motivated by order-invariance and probability matching (Lewis, 2013). This suggests that dynamic Bayesian updating is not always exhausted by repeated multiplication by a new likelihood; in some objective-Bayesian settings, the prior geometry must also evolve.

2. Evolving latent structure and state representation

The literature treats dynamic Bayesian updating as a mechanism for learning many different kinds of evolving latent structure. In construction digital twins, the project is represented as a directed acyclic graph

$$P(D_i \mid O_i)=\frac{P(O_i \mid D_i)\,P(D_i)}{P(O_i)},$$1

each activity has stochastic duration $$P(D_i \mid O_i)=\frac{P(O_i \mid D_i)\,P(D_i)}{P(O_i)},$$2, and project completion time is the network-level random variable

$$P(D_i \mid O_i)=\frac{P(O_i \mid D_i)\,P(D_i)}{P(O_i)},$$3

Activity durations are modeled as lognormal random variables,

$$P(D_i \mid O_i)=\frac{P(O_i \mid D_i)\,P(D_i)}{P(O_i)},$$4

and are repeatedly revised as project evidence arrives (Khoshkonesh et al., 17 May 2026).

In dynamic random dot product graphs, the latent state is geometric. A sequence of adjacency matrices $$P(D_i \mid O_i)=\frac{P(O_i \mid D_i)\,P(D_i)}{P(O_i)},$$5 is modeled through node positions $$P(D_i \mid O_i)=\frac{P(O_i \mid D_i)\,P(D_i)}{P(O_i)},$$6 with

$$P(D_i \mid O_i)=\frac{P(O_i \mid D_i)\,P(D_i)}{P(O_i)},$$7

Temporal dependence is introduced by Gaussian random walk priors. For $$P(D_i \mid O_i)=\frac{P(O_i \mid D_i)\,P(D_i)}{P(O_i)},$$8,

$$P(D_i \mid O_i)=\frac{P(O_i \mid D_i)\,P(D_i)}{P(O_i)},$$9

and for $$m_t=m_{t-1}+A_t(r_t-f_t), \qquad C_t=C_{t-1}-A_t Q_t A_t',$$0,

$$m_t=m_{t-1}+A_t(r_t-f_t), \qquad C_t=C_{t-1}-A_t Q_t A_t',$$1

Inference is only identifiable up to orthogonal transformations, so posterior contraction is stated for $$m_t=m_{t-1}+A_t(r_t-f_t), \qquad C_t=C_{t-1}-A_t Q_t A_t',$$2 and for $$m_t=m_{t-1}+A_t(r_t-f_t), \qquad C_t=C_{t-1}-A_t Q_t A_t',$$3 up to rotation (Loyal, 24 Sep 2025).

Distributed monitoring problems use yet another state representation. In asynchronous dynamic Bayesian networks, each variable is a supernode containing subnodes $$m_t=m_{t-1}+A_t(r_t-f_t), \qquad C_t=C_{t-1}-A_t Q_t A_t',$$4, and continuous-time evolution is modeled with conditional intensity matrices $$m_t=m_{t-1}+A_t(r_t-f_t), \qquad C_t=C_{t-1}-A_t Q_t A_t',$$5. The transition over an interval $$m_t=m_{t-1}+A_t(r_t-f_t), \qquad C_t=C_{t-1}-A_t Q_t A_t',$$6 is

$$m_t=m_{t-1}+A_t(r_t-f_t), \qquad C_t=C_{t-1}-A_t Q_t A_t',$$7

Beliefs are updated locally and asynchronously by message passing between supernodes and subnodes, rather than by globally synchronized time slices (Pfeffer et al., 2012).

Complex dynamic Bayesian networks with non-Gaussian observations are represented cliquewise on junction trees. The state vector is partitioned as

$$m_t=m_{t-1}+A_t(r_t-f_t), \qquad C_t=C_{t-1}-A_t Q_t A_t',$$8

and each observed block depends on a clique-local linear predictor

$$m_t=m_{t-1}+A_t(r_t-f_t), \qquad C_t=C_{t-1}-A_t Q_t A_t',$$9

This permits local approximate updates followed by exact Gaussian propagation on the junction tree (Settimi et al., 2013).

Dynamic updating can also operate on symbolic hidden states. In condition/event nets, the hidden state is the marking $(\theta\mid D_t)\sim N(m_t,C_t)$0, while the observer maintains a probability distribution over markings

$(\theta\mid D_t)\sim N(m_t,C_t)$1

Transition requests and observed success or failure induce updates to that distribution, and the corresponding modular Bayesian networks are manipulated by structural rewrites (Cabrera et al., 2018). Taken together, these examples show that the “state” in dynamic Bayesian updating may be scalar, vector, path-valued, graph-valued, or combinatorial.

3. Evidence models and sequential assimilation

The evidence stream in dynamic Bayesian updating is frequently noisy, partial, and heterogeneous. In construction digital twins, the evidence includes BIM progress reports, drone imagery, IoT telemetry, computer vision tracking, productivity logs, site inspection reports, material delivery records, and weather/external condition data. The observation model is explicitly noisy: $(\theta\mid D_t)\sim N(m_t,C_t)$2 These observations are treated as imperfect signals about the underlying duration distribution rather than direct measurements of the true duration (Khoshkonesh et al., 17 May 2026).

In condition/event nets, the evidence is not a numerical measurement but the outcome of an intervention: a requested transition either succeeds, fails due to precondition, or fails due to postcondition. The paper defines update operators $(\theta\mid D_t)\sim N(m_t,C_t)$3, $(\theta\mid D_t)\sim N(m_t,C_t)$4, and $(\theta\mid D_t)\sim N(m_t,C_t)$5, and shows that

$(\theta\mid D_t)\sim N(m_t,C_t)$6

while

$(\theta\mid D_t)\sim N(m_t,C_t)$7

These are proved to be Bayesian conditioning in disguise (Cabrera et al., 2018).

In embedding learning, the evidence arrives batch by batch. The triplet-sampling framework assumes that class-specific embeddings follow a multivariate Gaussian,

$(\theta\mid D_t)\sim N(m_t,C_t)$8

with a Normal-Inverse-Wishart prior on $(\theta\mid D_t)\sim N(m_t,C_t)$9. New mini-batches produce updated class means and covariances, and these updated distributions are then used to sample anchor-positive-negative triplets. The operational update uses posterior expectations for $$P_n(x)=P_{n-1}(x\mid y_{n-1})=\frac{P_{n-1}(y_{n-1}x)}{P_{n-1}(y_{n-1})},$$0 and $$P_n(x)=P_{n-1}(x\mid y_{n-1})=\frac{P_{n-1}(y_{n-1}x)}{P_{n-1}(y_{n-1})},$$1, making the process explicitly incremental over batches (Sikaroudi et al., 2020).

Clinical and structural monitoring problems assimilate evidence at coarser temporal resolutions. In the dynamic survival-prediction study, new data arrive by update period, and Bayesian updating uses the previous model’s coefficients as priors with a forgetting factor $$P_n(x)=P_{n-1}(x\mid y_{n-1})=\frac{P_{n-1}(y_{n-1}x)}{P_{n-1}(y_{n-1})},$$2: $$P_n(x)=P_{n-1}(x\mid y_{n-1})=\frac{P_{n-1}(y_{n-1}x)}{P_{n-1}(y_{n-1})},$$3 The process repeats across periods in rapidly changing environments (Tanner et al., 2023). In vibration-based structural health monitoring, dynamic response time series are processed by output-only operational modal analysis, specifically stochastic subspace identification, to extract modal quantities such as identified eigenvalues and mode shapes. These become the observed data in a sequential Bayesian model updating problem for deterioration parameters, and the updated posterior is propagated into structural reliability and hazard calculations (Kamariotis et al., 2021). This suggests that dynamic Bayesian updating is often a two-stage procedure: raw measurements are first converted into an informative intermediate representation, and only then assimilated probabilistically.

4. Exact updates, approximations, and computational machinery

Some dynamic Bayesian updates are analytically tractable because they use conjugate structure. The batch-incremental triplet method uses a Normal-Inverse-Wishart prior for a Gaussian likelihood, so the posterior remains Normal-Inverse-Wishart and the update is closed form. The paper further shows that the marginal posterior for the mean is multivariate Student-$$P_n(x)=P_{n-1}(x\mid y_{n-1})=\frac{P_{n-1}(y_{n-1}x)}{P_{n-1}(y_{n-1})},$$4, while the marginal posterior for the covariance is inverse Wishart (Sikaroudi et al., 2020). In dynamic generalized linear junction trees, local non-Gaussian observations are handled by choosing a conjugate family for the transformed latent variable $$P_n(x)=P_{n-1}(x\mid y_{n-1})=\frac{P_{n-1}(y_{n-1}x)}{P_{n-1}(y_{n-1})},$$5, updating within that family, and then mapping back to a Gaussian approximation for $$P_n(x)=P_{n-1}(x\mid y_{n-1})=\frac{P_{n-1}(y_{n-1}x)}{P_{n-1}(y_{n-1})},$$6 by moment matching. The approximation quality is assessed with the Hellinger metric and theoretical bounds (Settimi et al., 2013).

Other settings require approximate posterior computation because exact closed form is unavailable or computationally unattractive. In construction scheduling, evolving lognormal duration models with nonconjugate observation structure are updated by recursive maximum a posteriori estimation,

$$P_n(x)=P_{n-1}(x\mid y_{n-1})=\frac{P_{n-1}(y_{n-1}x)}{P_{n-1}(y_{n-1})},$$7

with $$P_n(x)=P_{n-1}(x\mid y_{n-1})=\frac{P_{n-1}(y_{n-1}x)}{P_{n-1}(y_{n-1})},$$8 (Khoshkonesh et al., 17 May 2026). In multimodal structural model updating, the likelihood itself is approximated in latent space by a multimodal variational autoencoder. The trained surrogate unimodal encoders approximate $$P_n(x)=P_{n-1}(x\mid y_{n-1})=\frac{P_{n-1}(y_{n-1}x)}{P_{n-1}(y_{n-1})},$$9 and $P_n$0, and posterior sampling is then carried out with Metropolis–Hastings using the VAE-based likelihood approximation (Itoi et al., 2024).

Sampling-based computation is central in structural dynamics. Bayesian finite element model updating uses posterior sampling because the posterior is high-dimensional, nonlinear, and lacks a closed form. The compared methods are Metropolis-Hastings, Slice Sampling, and Hybrid Monte Carlo. The reported conclusion is that both MH and HMC perform better than Slice Sampling when Young’s modulus is chosen as the updating parameter, and that HMC gives better results than MH and Slice sampling techniques when the area moment of inertias and section areas are updated (Boulkaibet et al., 2011). The more recent AM-SGHMC method augments SGHMC with neural networks that learn state-dependent diagonal $P_n$1 and $P_n$2 matrices, normalized energy and gradient inputs, and bounded-prior transformations. Because the inputs and outputs are designed to be scale-invariant across same-type structural problems, the trained sampler can be directly applied to various Bayesian updating problems of the same type of structure without further training (Meng et al., 28 Apr 2026).

Dynamic graph inference emphasizes scalability. The generalized Bayesian dynamic RDPG paper develops a Gibbs sampler whose precision matrix is banded and sparse, and reports that the cost of sampling latent positions scales as

$P_n$3

where $P_n$4 is the number of observed edges in the dynamic network, rather than $P_n$5 (Loyal, 24 Sep 2025). This suggests a general computational theme: dynamic Bayesian updating usually couples a recursive probabilistic rule with an approximation strategy tailored to the geometry, sparsity, or dimensionality of the latent state.

5. Non-stationarity, misspecification, and the limits of naive recursion

Several papers argue that naive recursive Bayes can fail in changing environments because it keeps accumulating evidence from outdated regimes. BAM states the problem directly: in non-stationary environments, a Bayesian never forgets. Its update rule introduces a memory buffer $P_n$6 and a readout weight vector

$P_n$7

and defines

$P_n$8

With binary weights $P_n$9, BAM chooses which past observations to remember and which to forget; it thereby generalizes recursive Bayes, power priors, exponential forgetting, and Bayesian unlearning (Nassar et al., 2022).

Dynamic transformation of prior knowledge in streams takes a different route. TPS does not merely initialize a prior once; it repeatedly transforms external prior knowledge $$\Pi_\lambda(X_{1:m}\mid Y_{1:m}) \propto \exp\!\left( -\lambda \sum_{t=1}^m \|Y_t - X_tX_t^\top\|_F^2 \right)\Pi(X_{1:m}),$$0 into model parameters through

$$\Pi_\lambda(X_{1:m}\mid Y_{1:m}) \propto \exp\!\left( -\lambda \sum_{t=1}^m \|Y_t - X_tX_t^\top\|_F^2 \right)\Pi(X_{1:m}),$$1

The posterior at minibatch $$\Pi_\lambda(X_{1:m}\mid Y_{1:m}) \propto \exp\!\left( -\lambda \sum_{t=1}^m \|Y_t - X_tX_t^\top\|_F^2 \right)\Pi(X_{1:m}),$$2 is based on the transition prior $$\Pi_\lambda(X_{1:m}\mid Y_{1:m}) \propto \exp\!\left( -\lambda \sum_{t=1}^m \|Y_t - X_tX_t^\top\|_F^2 \right)\Pi(X_{1:m}),$$3 and the current minibatch likelihood under $$\Pi_\lambda(X_{1:m}\mid Y_{1:m}) \propto \exp\!\left( -\lambda \sum_{t=1}^m \|Y_t - X_tX_t^\top\|_F^2 \right)\Pi(X_{1:m}),$$4. The paper also states that some existing streaming approaches can forget initial knowledge at a rate of $$\Pi_\lambda(X_{1:m}\mid Y_{1:m}) \propto \exp\!\left( -\lambda \sum_{t=1}^m \|Y_t - X_tX_t^\top\|_F^2 \right)\Pi(X_{1:m}),$$5, which motivates repeatedly re-encoding prior knowledge over time (Bach et al., 2020). In the clinical survival-updating study, the forgetting factor $$\Pi_\lambda(X_{1:m}\mid Y_{1:m}) \propto \exp\!\left( -\lambda \sum_{t=1}^m \|Y_t - X_tX_t^\top\|_F^2 \right)\Pi(X_{1:m}),$$6 plays a related role by discounting older information when new data arrive (Tanner et al., 2023).

Misspecification introduces a different limitation. For dependent data and possibly wrong models, posterior concentration is shown to occur around hypotheses with the smallest Kullback–Leibler divergence rate from the truth, not necessarily around a true model. The key dynamical assumption is a relative asymptotic equipartition property,

$$\Pi_\lambda(X_{1:m}\mid Y_{1:m}) \propto \exp\!\left( -\lambda \sum_{t=1}^m \|Y_t - X_tX_t^\top\|_F^2 \right)\Pi(X_{1:m}),$$7

with sieve-like control on the prior ensuring that the posterior suppresses suboptimal hypotheses exponentially (Shalizi, 2009). This suggests that in complex dynamic settings the practical target of Bayesian updating may be the best available approximation rather than exact recovery.

Adjacent literature also clarifies what dynamic Bayesian updating does not imply. In dynamic persuasion, delaying information provision does not affect the sender’s ex-ante value under divisible updating, and with a Bayesian receiver one-shot and two-step persuasion are equivalent in value. Strict gains or losses from gradual disclosure arise only when the receiver updates non-Bayesianly, as in Grether’s $$\Pi_\lambda(X_{1:m}\mid Y_{1:m}) \propto \exp\!\left( -\lambda \sum_{t=1}^m \|Y_t - X_tX_t^\top\|_F^2 \right)\Pi(X_{1:m}),$$8–$$\Pi_\lambda(X_{1:m}\mid Y_{1:m}) \propto \exp\!\left( -\lambda \sum_{t=1}^m \|Y_t - X_tX_t^\top\|_F^2 \right)\Pi(X_{1:m}),$$9 rule (Kobayashi, 17 Aug 2025). This is a useful contrast: temporal staging alone does not create a dynamic effect if the updating rule itself is dynamically consistent.

6. Forecasts, decisions, and evaluation of evolving beliefs

Dynamic Bayesian updating is typically used not for posterior estimation alone but for forecast generation, risk quantification, and decision support. In construction scheduling, posterior duration distributions are propagated by Monte Carlo simulation to produce

$\lambda>0$0

These quantify expected completion time, completion-time variance, delay probability, and activity criticality (Khoshkonesh et al., 17 May 2026). In dynamic random dot product graphs, the generalized posterior predictive distribution supports $\lambda>0$1-step-ahead network forecasts and credible intervals for future edge probabilities (Loyal, 24 Sep 2025). In finance, updated AR or VAR parameter posteriors feed directly into dynamic mean-variance portfolio rules and optimal efficient portfolios under uncertain investment lifetime (Sikaria et al., 2019). In clinical survival prediction, updated models are assessed out-of-sample with the C-index with inverse probability of censoring weighting, calibration intercept, calibration slope, and Brier score with IPCW (Tanner et al., 2023). In structural health monitoring, posterior deterioration parameters update structural reliability, hazard, repair timing, expected total life-cycle cost, value of information, and value of partial perfect information (Kamariotis et al., 2021).

Domain	Updated quantity	Reported outputs
Construction digital twins	Activity durations $\lambda>0$2	$\lambda>0$3, $\lambda>0$4, $\lambda>0$5, $\lambda>0$6
Dynamic random dot product graphs	$\lambda>0$7, $\lambda>0$8, $\lambda>0$9	$\lambda\to 0$0-step-ahead generalized posterior predictive distribution, credible intervals for future edge probabilities
Multi-period portfolio selection	AR or VAR parameters	Optimal efficient portfolios
Clinical survival prediction	Survival model updates across periods	C-index, calibration intercept, calibration slope, Brier score
Structural health monitoring	Deterioration parameters and reliability	Hazard, repair timing, VoI, VPPI
Time-updating forecast evaluation	Model bankroll or credibility	Market consensus probabilities, implied model credibilities

The evaluation of evolving Bayesian beliefs is itself a subject of study. One line of work models forecast models as canonical Kelly bettors and interprets bankroll as Bayesian credibility. In the multinomial case, the update

$\lambda\to 0$1

is presented as a direct mathematical and conceptual analogue to Bayes’ theorem, with the ratio $\lambda\to 0$2 playing the role of a likelihood ratio or Bayes factor contribution (Beuoy, 10 Feb 2026). Another line develops game-theoretic betting protocols for a forecaster whose time-$\lambda\to 0$3 belief is a probability measure over the remaining future, showing that game-theoretic test martingales are equivalent to visible measure-theoretic test martingales in finite settings and extending the framework to $\lambda\to 0$4-steps-ahead forecasting and nearly optimal sequential decision making (Vovk, 2023). This suggests that dynamic Bayesian updating is simultaneously an inference rule, a forecasting device, and an object of sequential calibration and betting-based scrutiny.

Across the supplied literature, the recurrent structure is a closed loop of prior, new evidence, posterior, forecast, and next update. In construction scheduling, this loop is written explicitly as “Prior $\lambda\to 0$5 Likelihood from new data $\lambda\to 0$6 Posterior $\lambda\to 0$7 Monte Carlo schedule simulation $\lambda\to 0$8 Updated risk metrics $\lambda\to 0$9 Next update” (Khoshkonesh et al., 17 May 2026). A plausible implication is that this loop is the most stable encyclopedia-level characterization of dynamic Bayesian updating across domains: it is Bayesian learning made explicitly temporal, repeatedly operationalized, and tied to current decisions rather than a single terminal inference.