Module 2

Prediction Error & Precision

Belief updates in the Bayesian brain are driven by prediction errors(observation minus expectation) weighted by precision (inverse variance). This module introduces Kalman filtering, variational inference with the evidence lower bound (ELBO), and the cognitive role of precision as attention.

1. Kalman Filtering

A Kalman filter is the optimal Bayesian estimator for a linear Gaussian system. The posterior mean after each observation is:

\[ \hat\mu_t = \hat\mu_{t|t-1} + K_t\bigl(y_t - \hat\mu_{t|t-1}\bigr),\qquad K_t = \frac{P_{t|t-1}}{P_{t|t-1} + R} \]

The Kalman gain K is the precision of the likelihood relative to the prior. High-precision likelihood (low sensor noise) pulls the posterior toward the observation; high-precision prior (low system noise) pulls it toward the prediction. This ratio is the key computational lever modulated by attention and psychiatric conditions (M7).

2. Variational Inference & ELBO

When the posterior P(z|x) is intractable, variational inference approximates it with Q(z) from a tractable family, minimising KL divergence. Equivalent to maximising the evidence lower bound:

\[ \log P(x) \;\geq\; \mathbb{E}_{Q(z)}[\log P(x, z)] - \mathbb{E}_{Q(z)}[\log Q(z)] = \text{ELBO} \]

Under Gaussian approximation, the ELBO becomes a precision-weighted sum of squared prediction errors minus a KL-complexity term. M4 shows that this is identical to the negative free energy Friston proposed as the brain’s unifying objective.

3. Precision = Attention

Precision is inverse variance — how confident the brain is in a sensory channel or a top-down prediction. Attentional enhancement can be modelled as up-weighting the precision of attended sensory evidence (Feldman & Friston 2010). Conversely, aberrant precision on priors is proposed as the computational substrate of hallucinations in schizophrenia and atypical sensory precision in autism (M7).

Simulation: Kalman Filter Precision Regimes

Python

script.py47 lines

import numpy as np, matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# 1-D Kalman filter: prediction-error-weighted belief update
rng = np.random.default_rng(0)
T = 100
x_true = np.cumsum(rng.normal(0, 0.5, T))      # random walk

# Noisy sensor
sensor_noise = 2.0
y = x_true + rng.normal(0, sensor_noise, T)

# Two precision regimes: low-precision priors (flexible) vs high-precision (rigid)
def kf(y, q, r, x0=0.0, P0=1.0):
    x = np.zeros_like(y); P = np.zeros_like(y)
    x_ = x0; P_ = P0
    for t in range(len(y)):
        x_p = x_            # constant-velocity zero drift prior
        P_p = P_ + q        # prior variance grows
        # Kalman gain = prior variance / (prior + sensor variance)
        K = P_p / (P_p + r)
        # Update: posterior mean = prior + K * prediction error
        x_ = x_p + K * (y[t] - x_p)
        P_ = (1 - K) * P_p
        x[t] = x_; P[t] = P_
    return x, P

x_flex, _ = kf(y, q=1.0, r=sensor_noise**2)
x_rigid, _ = kf(y, q=0.05, r=sensor_noise**2)

fig, ax = plt.subplots(figsize=(12, 5), facecolor='#0a0a1a')
ax.set_facecolor('#111827'); ax.tick_params(colors='#cbd5e1')
for s in ax.spines.values(): s.set_color('#334155')
ax.plot(y, '.', color='#94a3b8', alpha=0.6, label='Noisy observation')
ax.plot(x_true, color='#fbbf24', lw=2, label='True hidden state')
ax.plot(x_flex, color='#38bdf8', lw=2, label='Flexible prior (high q)')
ax.plot(x_rigid, color='#a78bfa', lw=2, label='Rigid prior (low q)')
ax.set_xlabel('Time', color='#cbd5e1'); ax.set_ylabel('State', color='#cbd5e1')
ax.set_title('Precision trades off prior vs. likelihood',
             color='#c4b5fd', fontweight='bold')
ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')
ax.grid(True, color='#334155', alpha=0.3)
plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')
print('Kalman gain K depends on the ratio of prior and likelihood precisions')
print('High-precision (small q) prior ignores prediction error; rigid beliefs')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Key References

• Kalman, R. E. (1960). “A new approach to linear filtering and prediction problems.” J. Basic Eng., 82, 35–45.

• Feldman, H. & Friston, K. J. (2010). “Attention, uncertainty, and free-energy.” Front. Hum. Neurosci., 4, 215.

• Jordan, M. I. et al. (1999). “An introduction to variational methods for graphical models.” Mach. Learn., 37, 183–233.

• Kingma, D. P. & Welling, M. (2014). “Auto-encoding variational Bayes.” ICLR.

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