Updates MMM comparison and adds benchmark

Refines the comparison of PyMC-Marketing with other MMM packages, highlighting feature differences and standardizing terminology.

Adds a detailed performance benchmark against Google Meridian, showcasing PyMC-Marketing's superior speed, accuracy, and scalability.

Includes updated recommendations for choosing the appropriate MMM library based on user needs and requirements.

Author: lfiaschi

docs/source/guide/mmm/comparison.md

@@ -1,6 +1,6 @@
 # How We Compare
 
-Given the popularity of the Media Mix Modelling (MMM) approach, numerous packages are available. Below is a concise comparison highlighting how PyMC-Marketing stands against other popular options:
+Given the popularity of the Media Mix Modelling (MMM) approach, numerous packages are available. Below is a concise comparison highlighting how the features of PyMC-Marketing stands against other popular options:
 
 | Feature                       | PyMC-Marketing | Robyn          | Orbit KTR | Meridian*               |
 |-------------------------------|:--------------:|:--------------:|:---------:|:----------------------:|
@@ -9,29 +9,32 @@ Given the popularity of the Media Mix Modelling (MMM) approach, numerous package
 | **Foundation**                | PyMC           | -              | STAN/Pyro | TensorFlow Probability |
 | **Company**                   | PyMC Labs      | Meta           | Uber      | Google                 |
 | **Open source**               | ✅              | ✅              | ✅         | ✅                      |
-| **Model Building**            | 🏗️ Build      | 🏗️ Build      | 🏗️ Build | 🏗️ Build               |
 | **Out-of-Sample Forecasting** | ✅              | ❌              | ✅         | ❌                      |
 | **Budget Optimizer**          | ✅              | ✅              | ❌         | ✅                      |
 | **Time-Varying Intercept**    | ✅              | ❌              | ✅         | ✅                      |
 | **Time-Varying Coefficients** | ✅              | ❌              | ✅         | ❌                      |
-| **Custom Priors**             | ✅              | ❌              | ❌         | ✅                      |
-| **Custom Model Terms** | ✅              | ❌                        | ❌         | ❌                             |
+| **Custom Priors**             | ✅              | NA              | ❌         | ✅                      |
+| **Custom Model Terms**        | ✅              | ❌              | ❌         | ❌                      |
 | **Lift-Test Calibration**     | ✅              | ✅              | ❌         | ✅                      |
-| **Geographic Modeling**       | ✅              | ❌              | ❌         | ✅                      |
+| **Hierachical Geographic Modeling**       | ✅              | ❌              | ❌         | ✅                      |
+| **Standardized Database Connectors** | ✅ (with Fivetran) | ❌ | ✅ | ✅ (limited to Google ecosystem) |
 | **Unit-Tested**               | ✅              | ❌              | ✅         | ✅                      |
-| **MLFlow Integration**       | ✅              | ❌              | ❌         | ✅                                |
-| **GPU Sampling Accelleration**| ✅              | -               | ❌         | ✅                      |
-| **Consulting Support**        | Provided by Authors              | Third-party agency               |   Third-party agency        |  Third-party agency                      |
+| **MLFlow Integration**       | ✅              | ❌              | ❌         | ✅                      |
+| **Multiple Sampling Backends**| ✅              | NA               | ❌         | ✅                      |
+| **GPU Sampling Acceleration**| ✅              | NA               | ❌         | ✅                      |
+| **Consulting Support**        | Provided by Authors | Third-party agency | Third-party agency | Third-party agency |
 
 *\*Meridian has been released as successor of Lightweight-MMM, which has been deprecated by Google*
 
-Last updated: 2025-08-07
+Last updated: 2025-10-17
 
 ---
 ### Key Takeaway
 Four of the five major libraries for MMM models implement different flavors of Bayesian models. While they share a broadly similar statistical foundation, they differ in API flexibility, underlying technology stack, and implementation approach.
 
-PyMC-Marketing stands out as the most widely used library by PyPI downloads (see plot below), offering unmatched flexibility and a comprehensive set of advanced features. This makes it ideal for teams looking for a highly customizable, state-of-the-art solution. However, its breadth and depth also make it the most sophisticated option, which may require a steeper learning curve. Other libraries have their own strengths—for example, Google Meridian features a more opinionated API and seamless integration with the Google ecosystem, which can be advantageous for organizations already embedded in Google's stack.
+PyMC-Marketing stands out as the most widely used library by PyPI downloads (see plot below), offering unmatched flexibility and a comprehensive set of advanced features. This makes it ideal for teams looking for a highly customizable, state-of-the-art solution. 
+
+However, its breadth and depth also make it the most sophisticated option, which may require a steeper learning curve. Other libraries have their own strengths — for example, Robyn is popular among the R community and provides extensive tutorials and documentation.
 
 Your optimal choice should depend primarily on:
 
@@ -41,6 +44,10 @@ Your optimal choice should depend primarily on:
 
 ![MMM Downloads Analysis](./mmm_downloads_analysis.png)
 
+## Detailed Performance Benchmark
+
+When it comes to Bayesian Media Mix Modeling the two most used options are PyMC-Marketing and Google Meridian. Our comprehensive technical benchmark comparing PyMC-Marketing against Google Meridian across realistic datasets (from startup to enterprise scale) reveals PyMC-Marketing's superior performance: **2-20x faster sampling**, **40% lower error** in channel contribution estimates, and **successful scaling** to large enterprise datasets where Meridian fails to converge. PyMC-Marketing's flexible sampling backends (NumPyro, BlackJAX, Nutpie) provide significant advantages over Meridian's fixed TensorFlow Probability implementation. See our [detailed benchmark analysis](https://www.pymc-labs.com/blog-posts/pymc-marketing-vs-google-meridian) for complete results and open-source methodology.
+
 ## Our Recommendation
 
 ### Choose Meta Robyn if:
@@ -53,12 +60,12 @@ Your optimal choice should depend primarily on:
 
 - You want a simplified (albeit less flexible) API to build models across geographies
 - Direct integration with the Google advertising ecosystem is important
-- You want strong integration with other Google products such as Collab
+- You can allow for reduced predictive accuracy and explainability
 
 ### Choose PyMC-Marketing if:
 
 - Maximum flexibility for complex, unique business requirements is necessary
 - You need advanced statistical modeling capabilities (e.g., Gaussian Processes)
 - Production ready setup and integration into broader data science workflows is important (MLflow)
 - You prefer independence from major ad publishers and networks
-- Professional consulting support is desirable
+- Professional indipendent consulting support is desirable info@pymc-labs.com