A.3.3.PI:4.4 - Construct the prediction and its update

For an algebraic reconstruction, solve the observation and change equations for the hidden values. Check the admitted domain and substitute the result back. Derive a recurrence if the next prediction will use a rolling history. Include how to discard old information and incorporate the next reading.

For a set of possible states, apply the change rule to that set under the admitted inputs, then use the observation relation to obtain possible outputs. A simpler enclosing interval can be enough for a threshold question. Keep the direction of the bound that makes the decision valid.

For a finite hidden-state probability model, let Pij be the probability of transition from state i to state j, and let pi be the current probability of state i given the observations and actions already used. First predict the next-state weights:

predicted_pj = sum_i(pi * Pij)

When a new observation y arrives, multiply each predicted weight by the observation likelihood Lj(y), then divide by the sum of those products. For a discrete readout, Lj(y) is the probability of y in state j. For a continuous readout with a modeled density, use that density at y. Under the Markov and observation assumptions, the normalized weights summarize the history for the next prediction. Use the transition probabilities for the action actually taken when actions affect the model.

Supply the initial weights and observation likelihoods from the selected model. If the normalizing sum is zero, this update cannot supply posterior weights. Check the observation account, initial possibilities and numerical calculation before proceeding. The weights describe the current uncertainty about the modeled state.

For example, two equally weighted hidden states have Gaussian observation densities with means 0 and 1 and variance 1. An observation y=0 gives likelihoods proportional to 1 and exp(-1/2), so the first state’s updated probability is 1/(1+exp(-1/2)), about 0.622. Using the probability of a single value would give zero for both continuous distributions and prevent this valid update. A learned predictor offers another construction. Train or select it for the output, horizon, inputs and error that matter. A direct prediction several steps ahead and repeated application of an approximate one-step model can behave differently. Evaluate the intended use, including how the predictor receives the information available at each step.