You can deploy this model and start earning money today!

Roy-SR-007/
BayeSymX

probabilistic-symbolic-regression

mcmc

symbolic uncertainty

automated-materials-discovery

probabilistic-scientific-machine-learning

equation-discovery

2509.19710

Roy-SR-007/feynman-equations

Roy-SR-007/oxide-perovskites

mit

Views

No views yet

`BayeSym𝕏`

`Baye`sian `Sym`bolic regression forests for e`𝕏`pression discovery

License: MIT

BayeSym𝕏 searches over forests of symbolic expression trees, ranks symbolic models across independent parallel MCMC chains, and reports an Occam's window set of learned equations. Based on Roy, S., Dey, P., Mallick, B. K., & Pati, D. Probabilistic Symbolic Regression for Equation Discovery via Operator-induced and Regularized Symbolic Forests. (2026+; https://arxiv.org/abs/2509.19710)

Repository provenance

This Hugging Face repository is sourced from and mirrors the canonical Roy-SR-007/BayeSymX GitHub repository. The authoritative source code, development history, issue tracker, and pull requests are maintained on GitHub. This Hugging Face mirror is provided for discoverability and convenient distribution.

Overview of the `BayeSym𝕏` model

For a symbolic forest containing $K$ symbolic trees representing expressions (constructed recursively from features and operators), BayeSym𝕏 models the response as

$$ y_i = \beta_0 + \sum_{j=1}^{K}\beta_j g(x_i; T_j) + \varepsilon_i, \qquad \varepsilon_i \sim \mathcal N(0, \sigma^2),; 1\leq i \leq n. $$

Each MCMC explored forest is scored by the joint marginal posterior (JMP) distribution over symbolic forests:

$$ \log \mathrm{JMP}(\mathrm{T}) = \log \Pi(\mathrm{T}) + \log p(y \mid \mathrm{T}, X), $$

where the symbolic tree prior is equipped with depth-dependent regularization to control symbolic expression complexity. Also, the distribution of (y| T, X) is obtained after integrating out model regression and noise parameters with respect to their corresponding prior. The Dirichlet prior over operator and feature weight vectors of each tree enables data-adaptive learning of operator and feature preferences.

BayeSymX probabilistic symbolic forest model overview

Highlights

Posterior sampling — executes parallel MCMC chains for efficient exploration of symbolic forest space.
Interpretable symbolic models — constructs compact symbolic expressions (equations) using different mathematical (binary and unary) operators.
Symbolic model selection — ranks using JMP over symbolic forests and retain an Occam's window set of expressions.
Compact final symbolic expressions — performs post-MCMC symbolic model refinement using BIC and SymPy.
Train/test predictive accuracy — reports train and test accuracy of learned expressions through RMSE, MAE, and $R^2$.
Posterior diagnostics — provides parallel-chain log-JMP trace plots.

BayeSymX training, ranking, refinement, and posterior-diagnostics workflow

Installation
Quick start
Main API
Prior parameters
Operator set
Symbolic tree moves
Result schema

Installation

Clone the repository and enter the project directory:

bash

git clone https://github.com/Roy-SR-007/BayeSymX.git
cd BayeSymX

Install the required packages (see below for details on required dependencies):

python -m pip install numpy pandas scipy matplotlib tqdm networkx sympy jupyter

Dependencies

Dependency	Used for
NumPy	arrays, linear algebra, random sampling, predictions, and metrics
pandas	result tables and tabular dataset loading
SciPy	marginal likelihood and tree prior calculations
Matplotlib	MCMC trace plots
tqdm	single- and multi-chain progress displays
NetworkX	graph/tree utilities
SymPy	symbolic conversion, simplification, factoring, and model size calculation
Jupyter	optional; required only to run `example.ipynb` interactively

json, pathlib, dataclasses, typing, math, random, copy, time, os, sys, shutil, queue, multiprocessing, and concurrent.futures come from the Python standard library.

Quick start

Consider the following toy example:

$$ y = 1.25 + 2.5\exp(x_0^3) - 0.8x_1^2\sin(x_0) + \varepsilon, \qquad \varepsilon \sim \mathrm{N}(0, \sigma^{2} = 0.15^2). $$

python

# import everything from BayeSymX directory
from BayeSymX import *

# simulated data generation
rng = np.random.default_rng(2026) # seed for data generation
X = rng.uniform(-2.0, 2.0, size=(2000, 2))
y = (
    1.25
    + 2.5 * np.exp(X[:, 0] ** 3)
    - 0.8 * X[:, 1] ** 2 * np.sin(X[:, 0])
    + rng.normal(0.0, 0.15, size=X.shape[0])
)

# Reproducible 90%/10% train-test split.
indices = rng.permutation(len(X))
train_indices = indices[:1800]
test_indices = indices[1800:]

# Configuring and fitting BayeSymX
result_json = run_bayesymx(
    X_train=X[train_indices],               # train data
    y_train=y[train_indices],
    X_test=X[test_indices],                 # optional test data
    y_test=y[test_indices],
    K=3,                                    # symbolic forest size
    maxdepth=3,                             # max. depth of symbolic trees
    seeds=[101, 202, 303, 404, 505],        # 5 parallel MCMC chains
    maxiter=2000,                           # MCMC iterations
    significant_digits=3,                   # signif. digits in constants    
    print_results=True,
    show_trace_plot=True,
)

# Convert the JSON output of results into Python dictionary
results = json.loads(result_json)

From symbolic tree structured expressions to compact scientific equations

Explore the complete runnable workflows in example.ipynb. Data used for Feynman equation experiments is stored in feynman_data. The oxide perovskite catalyst dataset is recorded in perovskites_data/processed_data.csv. Full results for Feynman equations and the oxide perovskite catalyst dataset are in Feynman_Equation_Results.pdf and Oxide_Perovskite_Results.pdf, respectively.

Main API

python

run_bayesymx(
    X_train,
    y_train,
    K,
    maxdepth,
    seeds,
    prior_params=None,
    add_intercept=True,
    wts_init=None,
    wts_prop=None,
    opset=None,
    ftset=None,
    move_weights=None,
    maxiter=1000,
    burnin=0,
    thin=1,
    n_jobs=None,
    show_progress=True,
    report_every=10,
    r=10,
    X_test=None,
    y_test=None,
    force_intercept=False,
    blr_prior=None,
    prior_variance=10.0,
    rcond=None,
    significant_digits=6,
    print_results=False,
    show_trace_plot=False,
    save_trace_plot=False,
    trace_plot_path=None,
    trace_plot_dpi=300,
)

Function arguments

Argument	Required	Default	Meaning
Required arguments
`X_train`	Yes	—	Train feature matrix with shape `(n_train, p)`
`y_train`	Yes	—	Train response vector with length `n_train`
`K`	Yes	—	Number of symbolic trees in every forest
`maxdepth`	Yes	—	Maximum depth allowed for each symbolic tree
`seeds`	Yes	—	Nonempty sequence containing one integer seed per MCMC chain
Model and prior controls
`prior_params`	No	`None`	Optional `(alpha_op, alpha_ft, alpha, delta, beta0, V0, a0, b0)` prior specification; default priors are constructed when omitted; see below
`add_intercept`	No	`True`	Include an intercept in the regression model
`wts_init`	No	`None`	Initial operator and feature sampling weights, specified as `[op_weights, feature_weights]`
`wts_prop`	No	`None`	Operator and feature proposal weights used during MCMC
`opset`	No	`None`	Operator dictionaries available to the trees; `None` uses the complete default operator set; see below
`ftset`	No	`None`	Feature names; defaults to `x0`, `x1`, … and must follow the column order of `X_train`
`move_weights`	No	`None`	Mapping or seven-element sequence controlling the probabilities of MCMC symbolic tree moves; see below
MCMC and parallel chain execution
`maxiter`	No	`1000`	Number of MCMC iterations performed by each chain
`r`	No	`10`	Number of top log-JMP forests to post-process and return
`burnin`	No	`0`	Number of initial iterations excluded from retained states
`thin`	No	`1`	Retain every `thin`-th iteration after burn-in
`n_jobs`	No	`None`	Number of worker processes; defaults to `min(number_of_chains, CPU_count)`
`show_progress`	No	`True`	Display the launch banner and live parallel-chain progress
`report_every`	No	`10`	Number of iterations between worker-to-parent progress reports
Test-data evaluation
`X_test`	No	`None`	Optional held-out feature matrix; must be supplied together with `y_test`
`y_test`	No	`None`	Optional held-out response vector; must be supplied together with `X_test`
Symbolic model refinement
`force_intercept`	No	`False`	Require the BIC-selected reduced model to retain the intercept
`blr_prior`	No	`None`	Optional `(beta0_full, V0_full, a0, b0)` prior for reduced-model Bayesian linear regression
`prior_variance`	No	`10.0`	Fallback coefficient-prior variance used during model reduction
`rcond`	No	`None`	Numerical cutoff passed to least-squares and pseudoinverse calculations
Results and formatting
`significant_digits`	No	`6`	Number of significant digits used in expressions and formatted numerical output
`print_results`	No	`False`	Print the styled raw- and final-model result tables
Trace plot controls
`show_trace_plot`	No	`False`	Display the combined parallel-chain log-JMP trace plot
`save_trace_plot`	No	`False`	Save the trace plot to a file
`trace_plot_path`	No	`None`	Output file or directory; defaults to `./bayesymx_parallel_log_jmp_traces.png` when saving
`trace_plot_dpi`	No	`300`	Resolution, in dots per inch, used when saving the trace plot

Default prior parameters

When prior_params=None, BayeSym𝕏 constructs the following prior parameters automatically:

Prior parameter	Default value	Meaning
`alpha_op`	`np.ones(len(opset))`	Symmetric Dirichlet concentration parameters for operator usage
`alpha_ft`	`np.ones(len(ftset))`	Symmetric Dirichlet concentration parameters for feature usage
`alpha`	`0.95`	Initial depth-dependent splitting probability
`delta`	`1.20`	Rate at which the splitting probability decreases with tree depth
`beta0`	`np.zeros(K + int(add_intercept))`	Prior mean of the regression coefficients
`V0`	`10.0 * np.eye(K + int(add_intercept))`	Prior covariance matrix for the regression coefficients
`a0`	`0.05`	Prior shape-related parameter for the model variance
`b0`	`0.05`	Prior scale-related parameter for the model variance

Default full operator set

When opset=None, BayeSym𝕏 considers the following full set of operators:

Operator	Arity	Evaluation
`add`	2	$x_1+x_2$
`mul`	2	$x_1\times x_2$
`neg`	1	$-x$
`inv`	1	protected $1/x$, with near-zero values stabilized
`sin`	1	$\sin(x)$
`cos`	1	$\cos(x)$
`exp`	1	$\exp(x)$, with its argument clipped for numerical safety
`sq`	1	$x^2$
`cu`	1	$x^3$
`sqrt_op`	1	protected $\sqrt{\lvert x \rvert}$

Supply a restricted set when domain knowledge supports it, as below:

python

result_json = run_bayesymx(
    ...,
    opset=[add, mul, sin, sq],
    ...
)

Symbolic tree moves

The MCMC proposal kernel supports the following seven symbolic tree moves:

grow
prune
change_feature
change_operator
subtree_replace
delete_node
insert_node

When move_weights=None, BayeSym𝕏 considers the following default move weights:

python

move_weights = {
    "grow": 1.0,
    "prune": 1.0,
    "change_feature": 1.0,
    "change_operator": 1.0,
    "subtree_replace": 1.0,
    "delete_node": 1.0,
    "insert_node": 1.0,
}

# optionally you can set the move_weights also as
move_weights = [1, 1, 1, 1, 1, 1, 1]

Result schema

run_bayesymx() returns a JSON string of results with the following hierarchy:

text

result
├── parallel_chains_runtime_seconds
├── trace_plot_path
├── ranking_scale                  # "log_JMP"
├── selection_criterion            # "BIC"
├── significant_digits
├── top_r_requested
├── top_r_returned
└── models[]
    ├── rank
    ├── source
    │   ├── chain_id
    │   ├── chain_seed
    │   └── iteration
    ├── raw_model
    │   ├── expression
    │   ├── log_JMP
    │   ├── train_metrics          # RMSE, MAE, R2
    │   └── test_metrics           # metrics or null
    └── final_model
        ├── expression
        ├── BIC
        ├── effective_K
        ├── model_size
        ├── train_metrics
        └── test_metrics

Maintainers and developers

Somjit Roy and Pritam Dey.

BayeSym𝕏 is distributed under the terms of the MIT License.

Roy-SR-007/BayeSymX

BayeSym𝕏

Bayesian Symbolic regression forests for e𝕏pression discovery

Overview of the BayeSym𝕏 model

Highlights

Contents

Installation

Dependencies

Quick start

From symbolic tree structured expressions to compact scientific equations

Main API

Function arguments

Default prior parameters

Default full operator set

Symbolic tree moves

Result schema

Maintainers and developers

Roy-SR-007/
BayeSymX

`BayeSym𝕏`

`Baye`sian `Sym`bolic regression forests for e`𝕏`pression discovery

Overview of the `BayeSym𝕏` model