[Paper Review] On the error probability of stochastic decision and stochastic decoding

Jun Muramatsu, Shigeki Miyake|arXiv (Cornell University)|Jan 18, 2017

Error Correcting Code Techniques10 references3 citations

TL;DR

This paper establishes that the error probability of a stochastic decision based on the posterior distribution is at most twice that of the optimal maximum a posteriori (MAP) decision. By using i.i.d. sampling from the posterior and selecting the most probable sequence, the stochastic decision's error probability converges exponentially to the MAP error rate, enabling efficient stochastic decoders for source and channel codes.

ABSTRACT

This paper investigates the error probability of a stochastic decision and the way in which it differs from the error probability of an optimal decision, i.e., the maximum a posteriori decision. This paper calls attention to the fact that the error probability of a stochastic decision with the a posteriori distribution is at most twice the error probability of the maximum a posteriori decision. It is shown that, by generating an independent identically distributed random sequence subject to the a posteriori distribution and making a decision that maximizes the a posteriori probability over the sequence, the error probability approaches exponentially the error probability of the maximum a posteriori decision as the sequence length increases. Using these ideas as a basis, we can construct stochastic decoders for source/channel codes.

Motivation & Objective

• To analyze the error probability of stochastic decisions compared to the optimal maximum a posteriori (MAP) decision.
• To establish theoretical bounds on the error probability of stochastic decision rules relative to MAP decisions.
• To develop a framework for constructing stochastic decoders for source and channel coding using Markov Chain Monte Carlo (MCMC) sampling from the posterior distribution.
• To demonstrate that the error probability of stochastic decoding approaches the MAP error rate exponentially with sequence length.
• To provide a practical algorithm for constrained random number generation that samples from the posterior distribution under linear constraints.

Proposed method

• Uses a stochastic decision rule defined by a conditional distribution $ q_{\widehat{X}|Y} $, where $ \widehat{X} $ is drawn from the posterior $ p_{X|Y} $, forming a Markov chain $ X \leftrightarrow Y \leftrightarrow \widehat{X} $.
• Applies a Markov Chain Monte Carlo (MCMC) method to generate samples from the posterior distribution under linear constraints $ A\boldsymbol{x} = \boldsymbol{c} $, using Gibbs sampling over variable blocks.
• Employs a constrained-random-number generation algorithm that iteratively resamples variables to maximize the log-posterior $ \Lambda = \sum_j \log \mu_{X_j|Y_j}(x_j|y_j) $, tracking the maximum-likelihood sequence $ \boldsymbol{x}_{\max} $.
• Uses the systematic matrix form $ A = [\underline{A} \mid I] $ to represent linear constraints, enabling efficient computation of variable updates via $ v_{i,j}(x) = x_i + a_{i-n+l,j}[x_j - x] $.
• Defines a probability distribution $ \nu(x'_j) $ for resampling variable $ x_j $, conditioned on its neighbors in the support set $ \mathcal{I}_j $, ensuring the constraint is preserved.
• Proves convergence of the MCMC chain to the true posterior distribution: $ \lim_{k\to\infty} d(P_k(\cdot|\boldsymbol{y},\boldsymbol{c}), p_{\boldsymbol{X}|\boldsymbol{Y}\boldsymbol{C}}(\cdot|\boldsymbol{y},\boldsymbol{c})) = 0 $.

Experimental results

Research questions

• RQ1 What is the worst-case ratio between the error probability of a stochastic decision and that of the optimal MAP decision?
• RQ2 How does the error probability of a stochastic decoder based on i.i.d. sampling from the posterior compare to the MAP error rate?
• RQ3 Can a stochastic decoder be constructed that asymptotically achieves the error performance of the MAP decoder?
• RQ4 What is the convergence behavior of MCMC-based sampling from the posterior under linear constraints in coding problems?
• RQ5 How can a constrained random number generator be designed to sample from the posterior distribution efficiently in coding applications?

Key findings

• The error probability of a stochastic decision based on the posterior distribution is at most twice the error probability of the optimal MAP decision.
• By generating an i.i.d. sequence from the posterior and selecting the most probable sequence, the error probability of the stochastic decision approaches the MAP error probability exponentially fast with increasing sequence length.
• The MCMC-based constrained-random-number generation algorithm converges to the true posterior distribution: $ \lim_{k\to\infty} d(P_k(\cdot|\boldsymbol{y},\boldsymbol{c}), p_{\boldsymbol{X}|\boldsymbol{Y}\boldsymbol{C}}(\cdot|\boldsymbol{y},\boldsymbol{c})) = 0 $.
• The algorithm ensures that each updated sequence $ \boldsymbol{x} $ satisfies the linear constraint $ A\boldsymbol{x} = \boldsymbol{c} $, preserving feasibility throughout the sampling process.
• The method enables the construction of stochastic decoders for both source and channel codes by approximating the MAP decision via sampling from the posterior.
• The convergence is guaranteed under arbitrary initial feasible sequences and holds for any joint distribution $ p_{\boldsymbol{Y}\boldsymbol{C}} $.