Bayesian time series and state-space models: dynamics, filtering, and forecasting

What you should be able to do after this chapter

write dynamic linear models as Bayesian state-space factorizations;
derive the posterior update in an AR(1) Gaussian model conceptually;
explain local level, local trend, and stochastic-volatility style structure;
distinguish filtering, smoothing, and forecasting;
connect state-space modeling to Kalman and particle methods.

1 Dynamic Bayesian modeling

Static models assume one latent parameter vector explains all observations. Time series models relax this by allowing latent states to evolve:

\[ x_t \sim p(x_t \mid x_{t-1},\theta), \qquad y_t \sim p(y_t \mid x_t,\theta). \]

This structure captures persistence, shocks, seasonal dynamics, and regime changes.

2 Dynamic linear models

A general dynamic linear model writes

\[ x_t = G_t x_{t-1} + w_t,\qquad w_t \sim \mathcal{N}(0,W_t), \]

\[ y_t = F_t^\top x_t + v_t,\qquad v_t \sim \mathcal{N}(0,V_t). \]

When $(G_t,F_t,W_t,V_t)$ are known, inference for the latent states is linear Gaussian and therefore tractable through Kalman filtering and smoothing.

3 Local level model

The simplest state-space model is

\[ x_t = x_{t-1} + w_t,\qquad w_t \sim \mathcal{N}(0,W), \]

\[ y_t = x_t + v_t,\qquad v_t \sim \mathcal{N}(0,V). \]

This model expresses a slowly evolving latent mean observed with noise.

3.1 Interpretation

$W$ controls state volatility;
$V$ controls measurement noise;
the ratio $W/V$ determines how rapidly the latent signal adapts to new data.

4 Bayesian AR(1) thinking

An AR(1) model can be written as

\[ y_t = \phi y_{t-1} + \varepsilon_t,\qquad \varepsilon_t \sim \mathcal{N}(0,\sigma^2), \]

or more explicitly in Bayesian regression form by conditioning on lagged responses:

\[ y_{2:T} \mid \phi,\sigma^2 \sim \mathcal{N}(X\phi,\sigma^2 I), \]

where $X=(y_1,\dots,y_{T-1})^\top$.

Thus Bayesian AR models can often be analyzed using the same regression logic developed earlier, though stationarity priors on $\phi$ may need to restrict $|\phi|<1$.

5 Forecasting

Bayesian forecasting requires integrating over state and parameter uncertainty. For a one-step forecast,

\[ p(y_{t+1} \mid y_{1:t}) = \int p(y_{t+1} \mid x_{t+1},\theta)\, p(x_{t+1},\theta \mid y_{1:t})\, dx_{t+1}\, d\theta. \]

This is more honest than plug-in forecasting because it propagates uncertainty in dynamics and parameters forward in time.

6 Smoothing

Filtering targets

\[ p(x_t \mid y_{1:t}), \]

while smoothing targets

\[ p(x_t \mid y_{1:T}). \]

Smoothing uses future observations as well as past observations to infer the latent state at time $t$, so it is always at least as informed as filtering when applied retrospectively.

7 Hidden Markov models and regime switching

When the latent state is discrete, one obtains a hidden Markov model. This is useful when the dynamics are better described by regimes than by continuous latent trajectories.

Examples:

bull versus bear market states;
animal behavioral modes;
low versus high volatility regimes.

The Bayesian issues then include transition probabilities, emission parameters, label switching, and state uncertainty.

8 Stochastic volatility

A simple stochastic volatility model writes

\[ y_t \mid h_t \sim \mathcal{N}(0,\exp(h_t)), \]

\[ h_t = \mu + \phi(h_{t-1}-\mu)+\eta_t,\qquad \eta_t \sim \mathcal{N}(0,\sigma_h^2). \]

This is a state-space model with nonlinear observation variance. It illustrates why particle methods and HMC-based latent-state inference become necessary beyond the Gaussian linear case.

9 `R` workflow

The local-level particle-filter example is in particle_filter_local_level.R. The HMM recursion lab is in hmm_forward_algorithm.R.

These labs complement the filtering chapter by showing how model structure and computation fit together in dynamic settings.

10 Diagnostics and pitfalls

10.1 Confusing observation noise and state noise

These play distinct inferential roles and should not be conflated.

10.2 Overfitting dynamic variation

Allowing very large state variance can make the latent state chase noise.

10.3 Ignoring forecast calibration

Good in-sample fit does not guarantee good predictive performance.

10.4 Forgetting initial-state assumptions

Short series can be sensitive to the prior or distribution assumed for $x_0$.

11 Exercises

Write the local-level model as a state-space factorization and identify all conditional independences.
Explain why AR(1) regression can be viewed as a special case of Bayesian regression with lagged predictors.
Distinguish filtering, smoothing, and forecasting both mathematically and substantively.
Describe how stochastic volatility differs from a local-level Gaussian model.
Explain why hidden-state models are natural for regime-switching phenomena.

12 Summary

Bayesian time-series modeling extends the regression worldview into dynamic latent structure:

states evolve;
observations are conditionally independent given states;
forecasting, filtering, and smoothing become distinct inferential tasks;
the computational strategy depends strongly on whether the model is linear Gaussian, discrete-state, or nonlinear.

13 Further reading

Murphy (2023) chapters 8 and 29 give the most complete modern treatment of state-space models, covering the Kalman filter, the forward-backward recursions for hidden Markov models, and the smoothing algorithms in a single notation.

Doucet et al. (2001) and Del Moral (2004) are the references for the nonlinear and non-Gaussian cases where exact filtering is unavailable; the computational machinery is developed in sequential and online Bayesian computation.

Gelman et al. (2013) chapter 21 is shorter and better on the modelling questions — how to choose between a local level and a local trend, and how to check a dynamic model’s predictions honestly.

References

Del Moral, Pierre. 2004. Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer.

Doucet, Arnaud, Nando de Freitas, and Neil Gordon. 2001. Sequential Monte Carlo Methods in Practice. Springer.

Gelman, Andrew, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. Bayesian Data Analysis. 3rd ed. CRC Press.

Murphy, Kevin P. 2023. Probabilistic Machine Learning: Advanced Topics. MIT Press.

--- title: "Bayesian time series and state-space models: dynamics, filtering, and forecasting" --- ::: {.callout-note} ## What you should be able to do after this chapter - write dynamic linear models as Bayesian state-space factorizations; - derive the posterior update in an AR(1) Gaussian model conceptually; - explain local level, local trend, and stochastic-volatility style structure; - distinguish filtering, smoothing, and forecasting; - connect state-space modeling to Kalman and particle methods. ::: ## Dynamic Bayesian modeling Static models assume one latent parameter vector explains all observations. Time series models relax this by allowing latent states to evolve: $$ x_t \sim p(x_t \mid x_{t-1},\theta), \qquad y_t \sim p(y_t \mid x_t,\theta). $$ This structure captures persistence, shocks, seasonal dynamics, and regime changes. ## Dynamic linear models A general dynamic linear model writes $$ x_t = G_t x_{t-1} + w_t,\qquad w_t \sim \mathcal{N}(0,W_t), $$ $$ y_t = F_t^\top x_t + v_t,\qquad v_t \sim \mathcal{N}(0,V_t). $$ When $(G_t,F_t,W_t,V_t)$ are known, inference for the latent states is linear Gaussian and therefore tractable through Kalman filtering and smoothing. ## Local level model The simplest state-space model is $$ x_t = x_{t-1} + w_t,\qquad w_t \sim \mathcal{N}(0,W), $$ $$ y_t = x_t + v_t,\qquad v_t \sim \mathcal{N}(0,V). $$ This model expresses a slowly evolving latent mean observed with noise. ### Interpretation - $W$ controls state volatility; - $V$ controls measurement noise; - the ratio $W/V$ determines how rapidly the latent signal adapts to new data. ## Bayesian AR(1) thinking An AR(1) model can be written as $$ y_t = \phi y_{t-1} + \varepsilon_t,\qquad \varepsilon_t \sim \mathcal{N}(0,\sigma^2), $$ or more explicitly in Bayesian regression form by conditioning on lagged responses: $$ y_{2:T} \mid \phi,\sigma^2 \sim \mathcal{N}(X\phi,\sigma^2 I), $$ where $X=(y_1,\dots,y_{T-1})^\top$. Thus Bayesian AR models can often be analyzed using the same regression logic developed earlier, though stationarity priors on $\phi$ may need to restrict $|\phi|<1$. ## Forecasting Bayesian forecasting requires integrating over state and parameter uncertainty. For a one-step forecast, $$ p(y_{t+1} \mid y_{1:t}) = \int p(y_{t+1} \mid x_{t+1},\theta)\, p(x_{t+1},\theta \mid y_{1:t})\, dx_{t+1}\, d\theta. $$ This is more honest than plug-in forecasting because it propagates uncertainty in dynamics and parameters forward in time. ## Smoothing Filtering targets $$ p(x_t \mid y_{1:t}), $$ while smoothing targets $$ p(x_t \mid y_{1:T}). $$ Smoothing uses future observations as well as past observations to infer the latent state at time $t$, so it is always at least as informed as filtering when applied retrospectively. ## Hidden Markov models and regime switching When the latent state is discrete, one obtains a hidden Markov model. This is useful when the dynamics are better described by regimes than by continuous latent trajectories. Examples: - bull versus bear market states; - animal behavioral modes; - low versus high volatility regimes. The Bayesian issues then include transition probabilities, emission parameters, label switching, and state uncertainty. ## Stochastic volatility A simple stochastic volatility model writes $$ y_t \mid h_t \sim \mathcal{N}(0,\exp(h_t)), $$ $$ h_t = \mu + \phi(h_{t-1}-\mu)+\eta_t,\qquad \eta_t \sim \mathcal{N}(0,\sigma_h^2). $$ This is a state-space model with nonlinear observation variance. It illustrates why particle methods and HMC-based latent-state inference become necessary beyond the Gaussian linear case. ## `R` workflow The local-level particle-filter example is in [particle_filter_local_level.R](../../labs/r/particle_filter_local_level.R). The HMM recursion lab is in [hmm_forward_algorithm.R](../../labs/r/hmm_forward_algorithm.R). These labs complement the filtering chapter by showing how model structure and computation fit together in dynamic settings. ## Diagnostics and pitfalls ### Confusing observation noise and state noise These play distinct inferential roles and should not be conflated. ### Overfitting dynamic variation Allowing very large state variance can make the latent state chase noise. ### Ignoring forecast calibration Good in-sample fit does not guarantee good predictive performance. ### Forgetting initial-state assumptions Short series can be sensitive to the prior or distribution assumed for $x_0$. ## Exercises 1. Write the local-level model as a state-space factorization and identify all conditional independences. 2. Explain why AR(1) regression can be viewed as a special case of Bayesian regression with lagged predictors. 3. Distinguish filtering, smoothing, and forecasting both mathematically and substantively. 4. Describe how stochastic volatility differs from a local-level Gaussian model. 5. Explain why hidden-state models are natural for regime-switching phenomena. ## Summary Bayesian time-series modeling extends the regression worldview into dynamic latent structure: - states evolve; - observations are conditionally independent given states; - forecasting, filtering, and smoothing become distinct inferential tasks; - the computational strategy depends strongly on whether the model is linear Gaussian, discrete-state, or nonlinear. ## Further reading @murphy2023 chapters 8 and 29 give the most complete modern treatment of state-space models, covering the Kalman filter, the forward-backward recursions for hidden Markov models, and the smoothing algorithms in a single notation. @doucet2001 and @delmoral2004 are the references for the nonlinear and non-Gaussian cases where exact filtering is unavailable; the computational machinery is developed in [sequential and online Bayesian computation](../computation/sequential-and-online-bayes.qmd). @gelman2013 chapter 21 is shorter and better on the modelling questions — how to choose between a local level and a local trend, and how to check a dynamic model's predictions honestly.