Markov Chain Monte Carlo

Example

Simulating the “Forest Ranger”

Signal Detection Theory (SDT)is widely applied in psychophysics, where we model an observer’s ability to distinguish signal from noise.

1. Setting the model

Human perception is noisy and prone to internal or external interference, so we define:

• Noise ($N$): when there is no smoke, factors like water vapor or shifting shadows create sensory noise. The noise distribution can be modeled as $\mathcal{N}(\mu_{\text{noise}}, \sigma_{\text{noise}}^2)$.
• Signal + Noise ($S+N$): when actual smoke is present, the distribution of signal + noise is also a normal distribution, $\mathcal{N}(\mu_{\text{signal+noise}}, \sigma_{\text{signal+noise}}^2)$. Here, $\mu_{\text{signal+noise}} > \mu_{\text{noise}}$.
• Decision criterion ($c$): Forest ranger's internal threshold. If the perceived smoke intensity exceeds $c$, the ranger sounds the alarm.

2. Monte Carlo iteration

We use a computer to act as the “virtual ranger” and conduct 10,000 simulated trials:

• Random sampling: in each trial, randomly draw a value from either the $N$ or $S+N$ distribution. This represents the instantaneous sensory strength perceived by the ranger.
• Decision making: compare the sampled value against $c$ to determine the response.

3. Statistical output (results analysis)

Aggregating 10,000 trials gives the probabilities of the four possible outcomes:

Interactive visualization

Drag the sliders to change the distribution means and the criterion. Shaded regions show the four outcomes. Click the button to run 10,000 actual samples and compare empirical to theoretical rates.