# Is it possible to correlate two distributions in a BayesDistribution?

**URL:** <https://openturns.discourse.group/t/is-it-possible-to-correlate-two-distributions-in-a-bayesdistribution/398>\
**Category:** Python usage\
**Created:** [November 22, 2024, 10:34am UTC](https://openturns.discourse.group/t/is-it-possible-to-correlate-two-distributions-in-a-bayesdistribution/398 "2024-11-22T10:34:06Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![al2543](https://avatars.discourse-cdn.com/v4/letter/a/c4cdca/32.png) [@al2543](https://openturns.discourse.group/u/al2543)\
**Post date:** [November 22, 2024, 10:34am UTC](https://openturns.discourse.group/t/is-it-possible-to-correlate-two-distributions-in-a-bayesdistribution/398/1 "2024-11-22T10:34:06Z")

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Hi there,

I am exploring the use of `BayesDistribution` within OpenTURNS to model the dependency between two variables. My goal is to capture the correlation between these variables, and I’m curious if OpenTURNS allows for the integration of a correlation matrix or copula mechanism directly within `BayesDistribution`.

Previously, I’ve applied copulas to correlate distributions, followed by a filtering criterion to eliminate samples where X1 exceeds the X2 by 19 units. This method was part of a custom Monte Carlo simulation I developed.

Now I am looking into Subset Simulation to improve efficiency, which means I need to get rid of the filter. This let me to the `BayesDistribution`. However, I’m unsure how to incorporate the correlation between samples using this distribution.

Is there a methodology within OpenTURNS that would allow me to correlate samples within the context of `BayesDistribution`? Alternatively, if `BayesDistribution` isn’t suited for this purpose, could you recommend a more appropriate approach that aligns with the capabilities of Subset Simulation and maintains the correlation structure?

I appreciate any insights or suggestions 🙂

I used this code to describe the dependency, which is exactly what I need. Just that the samples are not correlated…

```python
X2_distribution = ot.Normal(-5.0, 1.0)
parameter_for_X1_distribution = ot.SymbolicFunction(
    ["X2"], ["25", "5", "0", "min(-X2 + 19, 25)"])
X1_base_distribution = ot.TruncatedNormal()
combined_2d_distribution = ot.BayesDistribution(
    X1_base_distribution, X2_distribution, parameter_for_X1_distribution
)

sample = combined_2d_distribution.getSample(10_000)

graph = combined_2d_distribution.drawPDF()
cloud = ot.Cloud(sample)
cloud.setColor("red")
graph.add(cloud)
view = viewer.View(graph)

```

which looks like this:

 ![image](https://global.discourse-cdn.com/free1/uploads/openturns/original/1X/af9c5f08aae59b96867cac548518ae66cb3191bb.png)

I am quite new to OpenTURNS, if there is a similar post I missed, please let me know - and excuse me if this is a stupid question 🙂

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<div class="post-metadata">

**Author:** ![schueller](https://yyz2.discourse-cdn.com/free1/user_avatar/openturns.discourse.group/schueller/32/3_2.png) [@schueller](https://openturns.discourse.group/u/schueller)\
**Post date:** [November 26, 2024, 11:07am UTC](https://openturns.discourse.group/t/is-it-possible-to-correlate-two-distributions-in-a-bayesdistribution/398/2 "2024-11-26T11:07:59Z")

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Hello,

That’s an interesting way to ordering by conditioning, didnt thought of that !  
We have the distribution MaximumEntropyOrderStatisticsDistribution for order relation, but you cannot set a shift.  
A possibility is to mesh the domain X1\>X2+19 then truncate the correlated distribution on it using the TruncatedOverMesh:

```auto
import openturns as ot
import openturns.viewer as otv

ot.ResourceMap.SetAsString("Contour-DefaultColorMapNorm", "rank")
R = ot.CorrelationMatrix(2)
R[1, 0] = -0.8
normal2d = ot.Normal([25,-5], [5,1], R)

levelSet = ot.LevelSet(ot.SymbolicFunction(['x1', 'x2'], ['x1-x2-19']), ot.Greater(), 0.0)
mesh = ot.LevelSetMesher([10, 10]).build(levelSet, normal2d.getRange())
print(mesh)
combined_2d_distribution = ot.TruncatedOverMesh(normal2d, mesh)
sample = combined_2d_distribution.getSample(10000)

graph = combined_2d_distribution.drawPDF()
cloud = ot.Cloud(sample)
cloud.setColor("red")
cloud.setPointStyle("dot")
graph.add(cloud)

view = otv.View(graph)
otv.View.ShowAll()

```

 ![Figure_1](https://global.discourse-cdn.com/free1/uploads/openturns/original/1X/baf9f5cfd1ce9b7e1e800cc8eecf8979f2059e62.png)

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<div class="post-metadata">

**Author:** ![al2543](https://avatars.discourse-cdn.com/v4/letter/a/c4cdca/32.png) [@al2543](https://openturns.discourse.group/u/al2543)\
**Post date:** [December 2, 2024, 7:25pm UTC](https://openturns.discourse.group/t/is-it-possible-to-correlate-two-distributions-in-a-bayesdistribution/398/3 "2024-12-02T19:25:36Z")

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Great! Thank you!

I was looking into it, is there any literature on how this works/ which algorithms are used for this?

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<div class="post-metadata">

**Author:** ![schueller](https://yyz2.discourse-cdn.com/free1/user_avatar/openturns.discourse.group/schueller/32/3_2.png) [@schueller](https://openturns.discourse.group/u/schueller)\
**Post date:** [February 10, 2025, 5:45pm UTC](https://openturns.discourse.group/t/is-it-possible-to-correlate-two-distributions-in-a-bayesdistribution/398/4 "2025-02-10T17:45:29Z")

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I dont think so, but it uses optimization and numerical integration.

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<div class="post-metadata">

**Author:** ![al2543](https://avatars.discourse-cdn.com/v4/letter/a/c4cdca/32.png) [@al2543](https://openturns.discourse.group/u/al2543)\
**Post date:** [February 11, 2025, 3:53pm UTC](https://openturns.discourse.group/t/is-it-possible-to-correlate-two-distributions-in-a-bayesdistribution/398/5 "2025-02-11T15:53:20Z")

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