Is this the same as exponential smoothing?

No. Exponential smoothing updates a forecast or summary using weighted observations; decayed MCMC selects historical latent variables for sampling updates.

Does it always use exponential decay?

No. Inverse-polynomial decay is central to the original history-length-independent convergence analysis.

Can it return uncertainty rather than one state estimate?

Yes. Its purpose is to approximate the filtering distribution, from which probabilities or summary estimates can be obtained.

AI Glossary / Graphs and symbolic AI / Advanced

Decayed MCMC

Definition

Decayed MCMC is an approximate filtering method that uses Markov chain Monte Carlo over possible state histories while updating recent states more frequently than distant ones. Its aim is to estimate the current hidden state efficiently as observations accumulate.

What it means

Filtering asks what a system is probably like now, given observations received so far. For example, noisy readings may indicate whether equipment is operating normally, but its true condition is not observed directly.

A model describes how hidden states evolve and how they generate observations. MCMC explores possible hidden trajectories consistent with that model and the evidence. The current-state values found in those trajectories help estimate the filtering distribution.

The “decay” concerns how often historical state variables are selected for computational updates. It does not mean multiplying old observations by smaller likelihood weights or automatically deleting them. Keeping those concepts separate is central to understanding the method.

How it works

For a fixed observation horizon T, maintain a candidate trajectory of hidden states. Select a time index using a distribution that favors recent times, then resample the corresponding hidden state conditional on its relevant neighbors and observation.

In a simple first-order hidden Markov model, an interior state's conditional distribution depends on its preceding state, following state and observation at that time. Boundary states require the appropriate prior or endpoint treatment. More complex dynamic Bayesian networks use the relevant Markov blanket.

Repeated updates generate a computational Markov chain over trajectories. This is different from the physical Markov process being estimated. The desired filtering answer is the marginal distribution of the final state, rather than an equally accurate description of every historical state.

The original research analyses inverse-polynomial selection decay. Under its finite-state mixing assumptions, it bounds convergence time for the current-state marginal independently of observation-history length. That is not a universal claim of constant total memory, immediate convergence or success for every continuous or deterministic model. The required work still depends on accuracy and model properties.

Three practical examples

A research monitoring system estimates a hidden equipment mode from a stream of noisy measurements.
A tracking experiment allocates more trajectory-update effort near the current position while retaining a possibility of revising earlier states.
A probabilistic-inference study compares recency-biased MCMC updates with uniform historical updates and particle filtering under a fixed computation budget.

Think of it this way

Think of revising a long incident timeline to understand what is happening now. Recent events receive most of your attention, but an older assumption can still be revisited when necessary.

The analogy should not suggest that old evidence is declared false or irrelevant. Decayed MCMC changes the allocation of sampling effort, while the target posterior remains defined by the model and included observations.

Worked example

Computing the update-selection probabilities

Suppose the trajectory currently contains four time positions. Give their ages, from newest to oldest, values a=0,1,2,3. Choose unnormalized weights w(a)=1/(a+1)².

The weights are 1, 1/4, 1/9 and 1/16. Their sum is 205/144. Dividing each by that sum gives selection probabilities 144/205, 36/205, 16/205 and 9/205, approximately 70.24%, 17.56%, 7.80% and 4.39%.

Over 1,000 independent choices of the time index, the expected selection counts are about 702, 176, 78 and 44. Actual counts vary. The oldest position still has a positive chance of being revised; a strict three-position window would give it zero.

These probabilities say where the algorithm spends update effort. They are not probabilities that the hidden state is faulty, and the 1,000 resulting trajectory samples are not automatically independent posterior samples. The state's conditional resampling and the chain's mixing behavior determine the quality of the filtering estimate.

Illustrative case study

A fictional sensor-research team, Alder Signals, compares filtering methods on simulated machine-mode data. Its initial report describes the decayed method as “forgetting old measurements,” even though all the measurements remain in the probabilistic target.

The team corrects the description and separately records the distribution used to select time positions. It then tests a slowly changing hidden process in which distant history remains influential. A decay schedule that rarely revisits older states can produce an apparently stable estimate before sufficient mixing has occurred.

For a small discrete version of the model, the researchers compare results with an exact filtering calculation. They report error and runtime across increasing history lengths, rather than claiming that a theoretical bound guarantees a chosen finite sampling budget is adequate.

Common misconceptions and mistakes

Decay means discarding old evidence. In the full-history method it biases update selection rather than replacing the target distribution with a truncated one.
Many updates imply many independent samples. MCMC samples can be strongly correlated.
A bound independent of history length means all resource use is constant. Storing evidence and trajectory information is a separate consideration.
Any recency schedule has the same guarantee. Fixed windows, exponential decay and inverse-polynomial decay have different behavior and assumptions.

Business use

Decayed MCMC is mainly useful as a specialized inference concept for streaming latent-state models. It illustrates how computation can focus on the part of a posterior that matters to the current question.

An implementation decision should compare accuracy, latency and memory against suitable alternatives on the intended model. Check slow mixing, abrupt changes and rare observations, and document any history truncation. If a model falls outside the assumptions of the convergence analysis, describe experimental evidence separately from the original theorem.

Check your understanding

A time position is selected for updates with probability 70%. Does that mean its hidden state has a 70% probability of being in a particular mode?

Answer: No. Selection probability controls computational effort. The posterior mode probability must be estimated from the properly sampled states.

Related terms

Learn next

Normalize the four inverse-square weights by hand, then compare them with uniform selection. Explain which distribution is the update schedule and which is the desired current-state posterior. Finally, identify why a hard history cutoff changes the approximation question.

Sources

Marthi, Pasula, Russell and Peres, Decayed MCMC Filtering, original UAI 2002 paper: https://arxiv.org/abs/1301.0584

Author-hosted publication summary: https://people.csail.mit.edu/pasula/papers/uai02.html

The four-position calculation and Alder Signals case are original illustrations; the finite-state theorem's assumptions are not extended here to arbitrary models.

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