Bayesian family: scaled and clipped utility update

Read the equation

The numerator adds utility to the current state, the denominator attenuates that sum by 1+|utility|, and the final clamp restricts the output to [0,1]. For nonnegative utility, the un-clipped expression is a weighted average of x and 1. The expression contains no likelihood, prior distribution, or evidence normalization, so its bounded output alone is not a Bayesian posterior.

Terms & definitions

How it works

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Original context

## 3.5 Family 4: Bayesian Inference

```
f₄(x) = clamp₀₁((x + utility) / (1 + |utility|))
```

**Equation Type**: Posterior update: P(θ|D) ∝ P(D|θ)P(θ)

**Explanation**: Implements belief updating where the current state serves as prior and utility as likelihood. Division by (1 + |utility|) normalizes the posterior. Used for probabilistic reasoning.

## 3.6 Family 5: Learning / Weight Update

```
f₅(x) = x + 0.1 · dx_norm
```

**Equation Type**: Delta rule: W_{t+1} = W_t + ΔW

**Explanation**: Accumulates change signals over time, implementing basic learning through weight adjustment. This family stores learned patterns.

## 3.7 Family 6: Mutual Information

```
f₆(x) = 0.5 · (x + σ(dx_norm))

f₄(x) = clamp₀₁((x + utility) / (1 + |utility|))

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