ML and MAP Decoding Fundamentals

Fundamentals of ML and MAP Decoding

1. Introduction

In digital communication:

• A transmitter sends a symbol s ∈ 𝒮 (from a finite set of possible symbols).
• The channel adds noise, so the receiver observes y.
• The goal of the receiver is to decode y to the most likely transmitted symbol s.

This is where ML and MAP decoding come in.

2. Maximum Likelihood (ML) Decoding

Idea: Choose the symbol s that maximizes the likelihood of receiving y, assuming all symbols are equally likely.

Mathematically:

ŝ_ML = argmax_{s ∈ 𝒮} P(y | s)

Where P(y|s) = probability of observing y given that s was transmitted (likelihood function).

Intuition: Pick the symbol that makes the received signal y “most probable” based on the channel.

Notes:

• ML decoding does not consider prior probabilities of symbols.
• Common in AWGN channels: it reduces to minimum Euclidean distance decoding for equally likely symbols.

3. Maximum a Posteriori (MAP) Decoding

Idea: Choose the symbol s that maximizes the posterior probability given the observation y.

Mathematically:

ŝ_MAP = argmax_{s ∈ 𝒮} P(s | y)

By Bayes’ theorem:

P(s | y) = (P(y | s) P(s)) / P(y)

Since P(y) is constant for all symbols:

ŝ_MAP = argmax_{s ∈ 𝒮} P(y | s) P(s)

Where:

• P(s) = prior probability of symbol s
• P(y|s) = likelihood

Intuition: MAP combines channel observation and prior knowledge of symbol probabilities.

Notes:

• If all symbols are equally likely: P(s) = const → MAP = ML.
• MAP is Bayesian optimal, minimizing the probability of error when priors are known.

4. Comparison Table

Feature	ML Decoding	MAP Decoding
Goal	Maximize likelihood P(y | s)	Maximize posterior P(s | y)
Uses prior	No	Yes, P(s)
Optimality	Optimal if symbols equally likely	Optimal in Bayesian sense
Simplification	Often Euclidean distance minimization	Likelihood × Prior weighting

5. Intuition

• ML: “Which symbol would most likely produce what I received?”
• MAP: “Considering what I know about symbol probabilities, which symbol is most probable given the received signal?”

Think of ML as purely observation-driven and MAP as observation + prior knowledge-driven.

6. Example Setup

Suppose we have a binary communication system:

• Transmitted symbols: S = {0, 1}
• Channel: Binary Symmetric Channel (BSC) with crossover probability p = 0.1
• Observed symbol at receiver: y ∈ {0, 1}
• Goal: Decide which symbol was transmitted.

6.1. Maximum Likelihood (ML) Decoding

• Assume all symbols are equally likely: P(0) = P(1) = 0.5

Likelihoods:

P(y = 0 | s = 0) = 0.9
P(y = 0 | s = 1) = 0.1

ML rule: choose s that maximizes P(y|s)

Case 1: Receiver sees y = 0

P(y=0|s=0) = 0.9 > P(y=0|s=1) = 0.1
ŝ_ML = 0

Case 2: Receiver sees y = 1

P(y=1|s=1) = 0.9 > P(y=1|s=0) = 0.1
ŝ_ML = 1

ML just picks the symbol most likely to produce the received bit, assuming equal probability of 0 and 1.

6.2. Maximum a Posteriori (MAP) Decoding

• Now, suppose priors are unequal: P(0) = 0.8, P(1) = 0.2

Posterior:

P(s|y) ∝ P(y|s) * P(s)

Case 1: Receiver sees y = 0

P(0|0) ∝ 0.9 × 0.8 = 0.72
P(1|0) ∝ 0.1 × 0.2 = 0.02
ŝ_MAP = 0

Case 2: Receiver sees y = 1

P(0|1) ∝ 0.1 × 0.8 = 0.08
P(1|1) ∝ 0.9 × 0.2 = 0.18
ŝ_MAP = 1

Notice how MAP incorporates priors. If the prior was more extreme (e.g., P(0)=0.99), MAP could decode y=1 as 0, while ML would still pick 1.

Summary

• ML ignores priors, MAP uses them.
• ML = MAP when all symbols are equally likely.
• In practical communication, MAP can reduce probability of error when symbol probabilities are unequal.

Further Reading

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