# How Bayesian estimation of parameters of ODE system is done?

**URL:** <https://discourse.julialang.org/t/how-bayesian-estimation-of-parameters-of-ode-system-is-done/87092>\
**Category:** Probabilistic Programming\
**Tags:** documentation, tutorials, turing\
**Created:** [September 11, 2022, 6:28pm UTC](https://discourse.julialang.org/t/how-bayesian-estimation-of-parameters-of-ode-system-is-done/87092 "2022-09-11T18:28:34Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![vembha](https://avatars.discourse-cdn.com/v4/letter/v/5fc32e/32.png) [@vembha](https://discourse.julialang.org/u/vembha)\
**Post date:** [September 11, 2022, 6:28pm UTC](https://discourse.julialang.org/t/how-bayesian-estimation-of-parameters-of-ode-system-is-done/87092/1 "2022-09-11T18:28:34Z")

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Hello people,

I tried to find texts, journal articles, blog posts, etc., to understand how Bayesian estimation (for example, `NUTS`) works for ODE parameter estimation, but I could not. None of the material I found on the web deal with parameter estimation for systems of ODEs.

For instance, Turing’s website provides an excellent [tutorial](https://turing.ml/v0.21/tutorials/10-bayesian-differential-equations/) using the Lotka-Volterra model as to how to implement the framework for inferring parameters of an ODE model. More specifically, using the tutorial, what do these lines mean?

```julia
# Observations.
for i in 1:length(predicted)
    data[:, i] ~ MvNormal(predicted[i], σ^2 * I)
end

```

How is the process of ‘inferring the actual parameter values’ executed? How are the data simulated from solving the ODEs compared with the actual observed data, and what is estimated for such ‘comparison’?

Could someone post links or names of such material in this thread?

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**Author:** ![liamfdoherty](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/liamfdoherty/32/24516_2.png) [@liamfdoherty](https://discourse.julialang.org/u/liamfdoherty)\
**Post date:** [September 11, 2022, 6:47pm UTC](https://discourse.julialang.org/t/how-bayesian-estimation-of-parameters-of-ode-system-is-done/87092/2 "2022-09-11T18:47:49Z")

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This example shows Bayesian estimation of parameters from Lotka-Volterra:

[https://turing.ml/v0.21/tutorials/10-bayesian-differential-equations/](https://turing.ml/v0.21/tutorials/10-bayesian-differential-equations/)

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**Author:** ![vembha](https://avatars.discourse-cdn.com/v4/letter/v/5fc32e/32.png) [@vembha](https://discourse.julialang.org/u/vembha)\
**Post date:** [September 11, 2022, 7:24pm UTC](https://discourse.julialang.org/t/how-bayesian-estimation-of-parameters-of-ode-system-is-done/87092/3 "2022-09-11T19:24:49Z")

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Thank you, @liamfdoherty, for the quick response. But my question was more about the mathematical basics rather than the tutorials. I’ve updated the question for clarity.

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**Author:** ![empet](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/empet/32/221303_2.png) [@empet](https://discourse.julialang.org/u/empet)\
**Post date:** [September 11, 2022, 8:22pm UTC](https://discourse.julialang.org/t/how-bayesian-estimation-of-parameters-of-ode-system-is-done/87092/4 "2022-09-11T20:22:00Z")

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A google search, " Bayesian estimation of ODE parameters", returns a lot of references on this topic.

 ![Screenshot_20220911-231856_Chrome](https://global.discourse-cdn.com/julialang/original/3X/b/5/b5f9ad54d98120c0fe0e173eb8968243f746013f.jpeg)

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**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [September 11, 2022, 10:19pm UTC](https://discourse.julialang.org/t/how-bayesian-estimation-of-parameters-of-ode-system-is-done/87092/5 "2022-09-11T22:19:46Z")

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[https://book.sciml.ai/notes/16/](https://book.sciml.ai/notes/16/)