/**
 * Backward Algorithm for Hidden Markov Models
 *
 * Computes backward probabilities β(t,i) = P(oₜ₊₁...oₜ | qₜ=i, λ)
 * Uses same scaling factors as forward algorithm for consistency
 */
import { BackwardResult } from '../../schemas/BackwardResultSchema';
import { EmissionParams } from '../../schemas/EmissionParamsSchema';
/**
 * Backward Algorithm with scaling
 *
 * @param observations - T x D matrix of observations
 * @param transitionMatrix - N x N transition probability matrix
 * @param emissionParams - Emission parameters for each state
 * @param scalingFactors - Scaling factors from forward algorithm
 * @returns Backward probabilities
 *
 * @example
 * ```typescript
 * const forwardResult = forward(observations, transitionMatrix, emissionParams, initialProbs);
 * const backwardResult = backward(observations, transitionMatrix, emissionParams, forwardResult.scalingFactors);
 * ```
 */
export declare function backward(observations: number[][], transitionMatrix: number[][], emissionParams: EmissionParams[], scalingFactors: number[]): BackwardResult;