Simulate Example: AU2 (MCMC Importance Sampling)

This page documents a simulate run with importance sampling (-i) where the proposal distribution is built with Markov chain Monte Carlo (MCMC). It uses the same problem as the Importance Sampling example, which uses the default kernel density estimation (KDE) method, so the two runs can be compared directly.

About the tables on this page

• Table values are rounded to 4 significant figures for readability. Very small/large values use scientific notation. Refer to the Excel/JSON result files for full precision.

Run Context

Problem module:

• problems/AU2Problem.py

Recorded result set:

• results/2026-10-31/08-51-07/AU2-effe7.xlsx
• results/2026-10-31/08-51-07/AU2-effe7.json
• results/2026-10-31/08-51-07/AU2-effe7.py
• results/2026-10-31/08-51-07/AU2-effe7.pickle
• results/2026-10-31/08-51-07/AU2-effe7.pdf
• results/2026-10-31/08-51-07/profile-effe7.yaml

Profile and run mode from saved profile:

• Profile used: custom
• run_type: simulate
• include_mc: true
• mc_with_is: true
• is_method: mcmc

For results-folder and filename conventions, see CLI Result Files.

Equivalent command:

brinkpy simulate custom -i

Profile Customization

The importance sampling method is set in the profile, not on the command line. This run uses a profile named custom with these importance sampling settings (the values for the MCMC options are the defaults):

reliability_options:
  is_method: mcmc
  is_kde_bandwidth: scott
  is_mcmc_chains: 5
  is_mcmc_chain_length: 500
  is_mcmc_thinning: 3
  is_mcmc_burnin: 200
  is_mcmc_fit_method: kde

Each chain runs is_mcmc_burnin + is_mcmc_chain_length × is_mcmc_thinning steps, here 200 + 500 × 3 = 1,700 steps. Brinkpy discards the burn-in steps and keeps every third step after that, so each chain contributes 500 samples.

With is_mcmc_fit_method: kde, the MCMC samples are fitted with a KDE to form the proposal distribution, so is_kde_bandwidth also applies. To create and edit a profile, see Profiles. For every option, see Importance Sampling Settings and MCMC Importance Sampling Settings.

Problem File Used

This run uses AU2Problem.py, the series system of two limit state functions in two standard normal variables. See Problem File Used on the KDE example page for the definition and the ISplot key.

Extracted Results Worksheet Tables

The tables below are transcribed from the Results worksheet in AU2-effe7.xlsx. For the meaning of each column, see Results Tables Reference.

Header Information

Field	Value
Problem	AU2
Request ID	64af472980134deeba4d62e0b19effe7
Run time	00 min 03.86 sec

Deterministic Variables

var_name	value
c	5

Stochastic Variables Definition

var_name	var_type	mean	std	param1	param2
T1	Normal	0	1	0	1
T2	Normal	0	1	0	1

Monte Carlo Results (with Importance Sampling)

beta	pf	cv	max_cv	size	%_removed	cycles	auto_size	mc_with_is	method	bandwidth	chains	length	burn_in	thinning
4.7745	9.008e−7	0.04931	0.05	8,000	0	4	True	True	mcmc+kde	0.2714	5	500	200	3

The method value mcmc+kde combines the sampling method (is_method) and the fitting method (is_mcmc_fit_method). The columns chains, length, burn_in, and thinning appear only for MCMC runs.

Monte Carlo Variable Statistics and Correlations (Proposal Samples)

Statistics describe the proposal samples

As with any importance sampling run, these statistics describe the samples drawn from the proposal distribution, not the original input distributions. The histograms in Figures 6 and 7 also show the proposal samples.

var_name	mean	std	%_oob	cor(T1)	cor(T2)
T1	−0.0117	3.3723	0	1.0000	0.8350
T2	1.0203	3.7686	0	0.8350	1.0000

MCMC Diagnostics

This table appears in the worksheet under Importance sampling method-specific results (mcmc+kde). It summarizes the norm of the MCMC samples in standard normal space (u-space), that is, each sample's distance from the origin.

mean	sd	eti89_lb	eti89_ub	ess_bulk	ess_tail	r_hat	mcse_mean	mcse_sd
5.1391	0.1181	5.0157	5.3576	240.5	342.5	1.026	0.007857	0.009308

For what each column means and how to use it, see MCMC Diagnostics.

Notes Reported by Brinkpy

1. Validation: Stochastic variables definition and limit state function validation required 1 function call.
2. Monte Carlo: Creation of importance sampling proposal distribution required 8,615 calls to the limit state function.
3. Monte Carlo: Completed 4 cycles with 2.00e+03 samples per cycle.
4. Monte Carlo: Load and resistance histograms are not available because the limit state function does not return load and resistance values.

Comparison with the KDE Run

	KDE (example)	MCMC (this page)
beta	4.7775	4.7745
pf	8.873e−7	9.008e−7
cv	0.02978	0.04931
Samples (size)	6,000	8,000
Limit state function calls to build the proposal	109,980	8,615
Run time	8.93 sec	3.86 sec

Both methods estimate nearly the same probability of failure. For this problem, MCMC needed about 13 times fewer limit state function calls to build the proposal distribution. That matters most when each limit state function call is expensive.

Interpretation Snapshot

• The probability of failure (pf ≈ 9.01 × 10⁻⁷, beta ≈ 4.77) agrees with the KDE run, and cv = 0.0493 is below max_cv = 0.05. Brinkpy added sampling cycles until the target was met, which took 4 cycles instead of 3.
• The MCMC samples are used only to build the proposal distribution. The probability of failure comes from the weighted importance samples, and cv reports its precision. Chain convergence still matters: if the chains miss part of the failure region, the proposal distribution may not cover it.
• Brinkpy calculates r_hat but does not enforce a limit. Here r_hat = 1.026, above the 1.01 often recommended in the MCMC literature. To improve convergence, increase is_mcmc_chain_length, is_mcmc_burnin, or is_mcmc_chains in the profile and run again. Figures 1 and 2 show why a longer burn-in can help: chain 4 starts at a norm of about 6.2, well above the level where the chains settle, and takes several steps to come down.
• Figures 4 and 5 show that the chains found all three parts of the failure region: the arc along the top (component 1) and the two branches in the upper-right and lower-left corners (component 2).

Generated Figures

The PDF result file for this run is saved as results/2026-10-31/08-51-07/AU2-effe7.pdf.

Figures 1 and 2: Trace of Sample Norms in u-space

Each row shows one chain. The left plot traces the norm of the chain's samples at each MCMC step, with the burn-in steps shaded gray. The right plot shows the distribution of the norms after burn-in. Well-mixed chains move up and down around a stable level, without long trends.

Figure 3: Chain Convergence Diagnostics

The rank plot shows, for each chain, how its samples rank among the samples of all chains. When the chains have converged to the same distribution, the bars of every chain are roughly level. The autocorrelation plot shows how strongly each sample depends on earlier samples in the same chain. Faster decay toward zero means more independent information per sample, which gives a larger effective sample size (ess_bulk, ess_tail).

Figure 4: IS Proposal Diagnostics (u-space)

The proposal samples in standard normal space, colored by the proposal probability density. The red curve is the limit state surface.

Figure 5: IS Proposal Diagnostics (x-space)

The same proposal samples in the original variable space (T1, T2). Because both variables are standard normal, this figure matches Figure 4.

Figure 6: IS Histogram — T1

Figure 7: IS Histogram — T2

Figure 8: Limit State Function Histogram (Importance Sampling)