Fits with shared parameters

We demonstrate how to simultaneously fit two datasets with different models that shares a common parameter.

from iminuit import Minuit
from iminuit.cost import UnbinnedNLL
from iminuit.util import describe
from matplotlib import pyplot as plt
import numpy as np
from numba_stats import norm

# generate two data sets which are fitted simultaneously
rng = np.random.default_rng(1)

width = 2.0
data1 = rng.normal(0, width, size=1000)
data2 = rng.normal(5, width, size=1000)

# use two pdfs with different names for non-shared parameters,
# so that they are fitted independently

def pdf1(x, μ_1, σ):
    return norm.pdf(x, μ_1, σ)

def pdf2(x, μ_2, σ):
    return norm.pdf(x, μ_2, σ)

# combine two log-likelihood functions by adding them
lh = UnbinnedNLL(data1, pdf1) + UnbinnedNLL(data2, pdf2)

print(f"{describe(lh)=}")

describe(lh)=['μ_1', 'σ', 'μ_2']

The σ parameter is shared between the data sets, while the means of the two normal distributions are independently fitted.

def plot(cost, xe, minuit, ax, **style):
    signature = describe(cost)
    data = cost.data

    values = minuit.values[signature]
    errors = minuit.errors[signature]

    cx = (xe[1:] + xe[:-1]) / 2

    ym = np.diff(norm.cdf(xe, *values)) * np.sum(w)
    t = []
    for n, v, e in zip(signature, values, errors):
        t.append(f"${n} = {v:.3f} ± {e:.3f}$")
    ax.plot(cx, ym, label="\n".join(t), **style)

m = Minuit(lh, μ_1=1, μ_2=2, σ=1)

fig, ax = plt.subplots(1, 2, figsize=(14, 5))

hists = [np.histogram(lhi.data, bins=50) for lhi in lh]

# draw data and model with initial parameters
for lhi, (w, xe), axi in zip(lh, hists, ax):
    cx = (xe[1:] + xe[:-1]) / 2
    axi.errorbar(cx, w, np.sqrt(w), fmt="ok", capsize=0, zorder=0)
    plot(lhi, xe, m, axi, ls="--")

m.migrad()

# draw model with fitted parameters
for lhi, (w, xe), axi in zip(lh, hists, ax):
    plot(lhi, xe, m, axi)
    axi.legend()

The dashed line shows the initial model before the fit, the solid line shows the model after the fit. Note that the σ parameter is shared.