▶Observation model
A linear signal is observed with Gaussian noise, whose standard deviation controls data uncertainty.
Markov Chain Monte Carlo: Parameter Estimation
Estimating parameters and their uncertainty from data
MCMC parameter estimation produces a distribution of plausible parameter values, rather than only one best fit. The spread of the samples quantifies uncertainty.
watchone idea
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The essential idea: sampling the posterior shows both which parameters fit the data and how certain those estimates are.
One Concept · One Example
What to notice
- The likelihood links parameters to observed data.
- Posterior samples form uncertainty intervals.
- No difficult normalization integral is required.
Fit a line, retain uncertainty
Posterior sampling
Metropolis moves through parameter space in proportion to how well each line explains the observations.
Sample plausible regression lines and inspect the trace
Compare chains as short as 50 and 200 steps with longer runs. The parameter trace reveals whether the chain has reached and explored its posterior region.
posterior meanplausible linestruth
PARAMETER TRACEslope a across all 200 steps
MCMC turns parameter fitting into uncertainty-aware inference by sampling the posterior distribution over model parameters.