<p dir="auto">If you scale the errors fed into curve_fit, the error returned does not change.  This is incorrect behavior.  The error (variance and covariance) for a final fit parameter should increase given an increased uncertainty in the actual measured data.</p>
<p dir="auto">I created a gaussian curve fitting routine.  I then show one version functioning as expected, and another version with error increased by a factor of 1e10.  The increase in 1e10 should be reflected in the variance/covariance for all fitted parameters.  It remains constant:</p>
<div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="#!/usr/bin/env python

import numpy as np
import matplotlib.pyplot as plt
import collections as col
import scipy as sp
import scipy.optimize as spopt

plt.close('all')

def _gaussvar(x,amp,mu,variance,bg=0):
    return amp*np.exp(-(x-mu)**2/(2*variance))+bg

def iter(factor):
    # Generate random numbers with gaussian
    # distribution, mu=0, sigma = 3
    x=np.random.randn(10000)*3

    # Bin the counts
    bins = 50
    h,edge = np.histogram(x,bins=bins)

    # Find the midpoints of the bins
    mids = edge + (edge[1]-edge[0])/2
    mids = mids[:-1]

    # Expected error in counts is sqrt(counts)
    sigma = np.sqrt(h)*1

    # Error of zero counts isn't zero, but less than one.
    # Use 0.5 as a guess.
    sigma[sigma==0] = 0.5

    # Factor scales error, which should cause
    # the pcov matrix to change
    sigma = sigma*factor

    # Fit the histogram to a gaussian
    # popt,pcov,red_chisq = mt.gaussfit(mids,h,sigma_y=sigma,plot=True,variance_bool=True)


    # Find initial guesses
    y=h
    x=mids
    amp = max(y)
    mu  = sum(x*y)/sum(y)
    variance = sum(x**2 * y)/sum(y)
    bg  = 0
    p0 = np.array((amp,mu,variance,bg))


    # Do actual curve fit
    func = _gaussvar
    popt,pcov = spopt.curve_fit(func,x,y,sigma=sigma,p0=p0)

    output = col.namedtuple('iterout',['popt','pcov'])
    out = output(popt,pcov)
    return out

num_samples       = 1000
variances_regular = np.ones(num_samples)
variances_large   = np.ones(num_samples)
means_regular     = np.ones(num_samples)
means_large       = np.ones(num_samples)
for i in np.linspace(1,num_samples,num_samples)-1:
    out                  = iter(1)
    means_regular[i]     = out.popt[2]
    variances_regular[i] = out.pcov[2,2]

    out                = iter(1e10)
    means_large[i]     = out.popt[2]
    variances_large[i] = out.pcov[2,2]

plt.hist(variances_regular,bins=20)
plt.hist(variances_large,bins=20)

plt.show()"><pre class="notranslate"><code class="notranslate">#!/usr/bin/env python

import numpy as np
import matplotlib.pyplot as plt
import collections as col
import scipy as sp
import scipy.optimize as spopt

plt.close('all')

def _gaussvar(x,amp,mu,variance,bg=0):
    return amp*np.exp(-(x-mu)**2/(2*variance))+bg

def iter(factor):
    # Generate random numbers with gaussian
    # distribution, mu=0, sigma = 3
    x=np.random.randn(10000)*3

    # Bin the counts
    bins = 50
    h,edge = np.histogram(x,bins=bins)

    # Find the midpoints of the bins
    mids = edge + (edge[1]-edge[0])/2
    mids = mids[:-1]

    # Expected error in counts is sqrt(counts)
    sigma = np.sqrt(h)*1

    # Error of zero counts isn't zero, but less than one.
    # Use 0.5 as a guess.
    sigma[sigma==0] = 0.5

    # Factor scales error, which should cause
    # the pcov matrix to change
    sigma = sigma*factor

    # Fit the histogram to a gaussian
    # popt,pcov,red_chisq = mt.gaussfit(mids,h,sigma_y=sigma,plot=True,variance_bool=True)


    # Find initial guesses
    y=h
    x=mids
    amp = max(y)
    mu  = sum(x*y)/sum(y)
    variance = sum(x**2 * y)/sum(y)
    bg  = 0
    p0 = np.array((amp,mu,variance,bg))


    # Do actual curve fit
    func = _gaussvar
    popt,pcov = spopt.curve_fit(func,x,y,sigma=sigma,p0=p0)

    output = col.namedtuple('iterout',['popt','pcov'])
    out = output(popt,pcov)
    return out

num_samples       = 1000
variances_regular = np.ones(num_samples)
variances_large   = np.ones(num_samples)
means_regular     = np.ones(num_samples)
means_large       = np.ones(num_samples)
for i in np.linspace(1,num_samples,num_samples)-1:
    out                  = iter(1)
    means_regular[i]     = out.popt[2]
    variances_regular[i] = out.pcov[2,2]

    out                = iter(1e10)
    means_large[i]     = out.popt[2]
    variances_large[i] = out.pcov[2,2]

plt.hist(variances_regular,bins=20)
plt.hist(variances_large,bins=20)

plt.show()
</code></pre></div>
<p dir="auto">See also:<br>
<a target="_blank" rel="noopener noreferrer nofollow" href="https://camo.githubusercontent.com/20fa166fa0165bcdeabab6c7e4005e9e37ae5c3737cfc2a7251ef7b958ff493e/68747470733a2f2f662e636c6f75642e6769746875622e636f6d2f6173736574732f3435383837312f313334363734342f36626233336166652d333661352d313165332d383639312d3261666166376531353332362e6a7067"><img src="https://camo.githubusercontent.com/20fa166fa0165bcdeabab6c7e4005e9e37ae5c3737cfc2a7251ef7b958ff493e/68747470733a2f2f662e636c6f75642e6769746875622e636f6d2f6173736574732f3435383837312f313334363734342f36626233336166652d333661352d313165332d383639312d3261666166376531353332362e6a7067" alt="figure_1" data-canonical-src="https://f.cloud.github.com/assets/458871/1346744/6bb33afe-36a5-11e3-8691-2afaf7e15326.jpg" style="max-width: 100%;"></a></p>
<p dir="auto"><a href="https://github.com/joelfrederico/GaussFitDemo">https://github.com/joelfrederico/GaussFitDemo</a></p>
<p dir="auto">In the context of <a class="issue-link js-issue-link" data-error-text="Failed to load title" data-id="937950752" data-permission-text="Title is private" data-url="https://github.com/scipy/scipy/issues/14360" data-hovercard-type="issue" data-hovercard-url="/scipy/scipy/issues/14360/hovercard" href="https://github.com/scipy/scipy/issues/14360">#14360</a> &amp; <a class="issue-link js-issue-link" data-error-text="Failed to load title" data-id="1053213136" data-permission-text="Title is private" data-url="https://github.com/scipy/scipy/issues/15040" data-hovercard-type="pull_request" data-hovercard-url="/scipy/scipy/pull/15040/hovercard" href="https://github.com/scipy/scipy/pull/15040">#15040</a>, we cleaned up a lot of our module API and deprecated those that are supposed to be private, for removal that's scheduled for SciPy 2.0</p>
<p dir="auto">This was primarily (only?) done for the first-level modules though, not further down the tree. The added test <a href="https://github.com/scipy/scipy/blob/main/scipy/_lib/tests/test_public_api.py">infrastructure</a> contains:</p>
<div class="highlight highlight-source-python notranslate position-relative overflow-auto" dir="auto" data-snippet-clipboard-copy-content="# The PRIVATE_BUT_PRESENT_MODULES list contains modules that look public (lack
# of underscores) but should not be used.  For many of those modules the
# current status is fine.  For others it may make sense to work on making them
# private, to clean up our public API and avoid confusion.
# These private modules support will be removed in SciPy v2.0.0
PRIVATE_BUT_PRESENT_MODULES = [
    ..."><pre class="notranslate"><span class="pl-c"># The PRIVATE_BUT_PRESENT_MODULES list contains modules that look public (lack</span>
<span class="pl-c"># of underscores) but should not be used.  For many of those modules the</span>
<span class="pl-c"># current status is fine.  For others it may make sense to work on making them</span>
<span class="pl-c"># private, to clean up our public API and avoid confusion.</span>
<span class="pl-c"># These private modules support will be removed in SciPy v2.0.0</span>
<span class="pl-v">PRIVATE_BUT_PRESENT_MODULES</span> <span class="pl-c1">=</span> [
    ...</pre></div>
<p dir="auto">IMO, without tracking what needs to be done (deprecate and make private, or keep public) and then doing it, we won't be in a position to actually execute this for SciPy 2.0, hence I'm opening this issue. Ideally we decide per module what needs to be done, and then do it, so we can actually remove what we'd like to remove when the time comes.</p>
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<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.sparse.spfuncs</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.sparse.sputils</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.spatial.ckdtree</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.spatial.kdtree</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.spatial.qhull</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.spatial.transform.rotation</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.special.add_newdocs</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.special.basic</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.special.cython_special</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.special.orthogonal</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.special.sf_error</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.special.specfun</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.special.spfun_stats</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.stats.biasedurn</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.stats.kde</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.stats.morestats</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.stats.mstats_basic</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.stats.mstats_extras</code></li>
<li class="task-list-item"><input type="checkbox" id="" disabled="" class="task-list-item-checkbox"> <code class="notranslate">scipy.stats.mvn</code></li>
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</ul>