Bayesian Logistic Regression Calculator

Calculated Posterior Summary

Results are based on a Gaussian prior and a Laplace approximation around the MAP estimate.

Result shown above the form

Interpretation

Diagnostics

Posterior Covariance Matrix

Parameter	Intercept	Slope

Calculator Inputs

Enter binary data, prior settings, and a target predictor value for posterior prediction.

Prior mean: intercept

Prior mean: slope

Maximum iterations

Prior variance: intercept

Prior variance: slope

Convergence tolerance

Predict probability at x

Credible interval level (%)

Posterior simulation draws

Dataset (CSV lines: x,y)

Use one observation per line. The response y must be 0 or 1. Example: 1.75,1

The calculator estimates a two-parameter model: intercept and one predictor slope.

Example Data Table

This example reflects a binary outcome becoming more likely as the predictor increases.

Observation	x	y	Comment
1	-2.0	0	Low predictor, negative class observed
2	-1.5	0	Negative outcome remains likely
3	-1.0	0	Still below decision shift
4	-0.2	0	Borderline but negative
5	0.6	1	Positive outcome begins to appear
6	1.1	0	One noisy exception
7	1.8	1	Positive class more common
8	2.2	1	Higher x supports positive class
9	2.9	1	Strong signal at higher x
10	3.4	1	Very likely positive outcome

Formula Used

Likelihood: For each observation, p(y=1|x,β) = 1 / (1 + e^{-(β₀ + β₁x)}).

Prior: β ~ N(μ, Σ), where μ contains prior means and Σ is a diagonal matrix from prior variances.

Log posterior: log p(β|D) ∝ Σ[yη - log(1+e^η)] - 0.5(β-μ)^TΣ^-1(β-μ).

MAP update: Newton-Raphson uses gradient g = X^T(y-p) - Σ^-1(β-μ) and Hessian H = -X^TWX - Σ^-1.

Posterior covariance: Near the MAP estimate, Cov(β|D) ≈ (X^TWX + Σ^-1)^-1.

Predicted probability: For a chosen x*, p(y*=1|x*,D) is summarized from posterior draws using the covariance approximation.

How to Use This Calculator

Enter prior means and variances for the intercept and slope.
Paste the dataset as comma-separated lines with x and y values.
Choose the prediction point, credible interval level, iteration cap, and simulation draws.
Click Calculate posterior to estimate the MAP coefficients and uncertainty.
Review coefficient signs, posterior intervals, predicted probability, and diagnostics.
Export the calculated summary with the CSV or PDF buttons.

Why Bayesian Logistic Regression Helps

Bayesian logistic regression is useful when you want a probability model with explicit prior assumptions. Priors stabilize estimates in small samples, noisy data, or cases with partial separation. The posterior output adds uncertainty ranges around coefficients and predictions instead of returning only one fitted line. This calculator uses a practical Laplace approximation, which is fast for web use and informative for exploratory analysis, planning, quality checks, and educational demonstrations.

FAQs

1. What does this calculator estimate?

It estimates the posterior mode for an intercept and one slope in a binary logistic model, then approximates posterior uncertainty with a covariance matrix.

2. What prior distribution is used?

The tool uses independent normal priors for the intercept and slope. You set their means and variances directly in the form.

3. Why is the result called an approximation?

The full posterior is not integrated exactly. Instead, the calculator uses a Laplace approximation around the MAP estimate for speed and clarity.

4. Can I paste many observations?

Yes. Paste one observation per line using x,y format. The tool validates that y is binary and x is numeric.

5. What does the predicted probability represent?

It is the estimated probability that y equals 1 at your chosen predictor value, summarized from the approximate posterior distribution.

6. What if my model does not converge?

Try increasing prior strength, lowering the predictor scale, checking data quality, or raising the maximum iteration setting.

7. Can I use more than one predictor?

This page is designed for one predictor plus an intercept so the results remain transparent and easy to verify manually.

8. When should I prefer this method?

Use it when prior knowledge matters, sample sizes are modest, or uncertainty communication is more valuable than a single point estimate.