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logging, gradients, and bibtex
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--- | ||
myst: | ||
html_meta: | ||
"property=og:title": "Gradient of a Statistical Model" | ||
"property=og:description": "Modules to compute gradient and Hessian of negative log-probabilities" | ||
"property=og:image": "https://spey.readthedocs.io/en/main/_static/spey-logo.png" | ||
jupytext: | ||
formats: ipynb,md:myst | ||
text_representation: | ||
extension: .md | ||
format_name: myst | ||
format_version: 0.12 | ||
jupytext_version: 1.8.2 | ||
kernelspec: | ||
display_name: Python 3 | ||
language: python | ||
name: python3 | ||
--- | ||
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# Gradient of a Statistical Model | ||
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````{margin} | ||
```{note} | ||
In previous versions gradient and Hessian was limited to internal computations only. | ||
``` | ||
```` | ||
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With version 0.1.6, Spey includes additional functionalities to extract gradient and Hessian information directly from the statistical model. The gradient and Hessian are defined as follows | ||
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$$ | ||
{\rm Gradient} = -\frac{d\log\mathcal{L}(\theta)}{d\theta}\quad , \quad {\rm Hessian} = -\frac{d^2\log\mathcal{L}(\theta)}{d\theta_i d\theta_j}\quad , \quad \mu ,\theta_i \in \theta \ . | ||
$$ | ||
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In order to access this information we will use `spey.math` module. | ||
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```{code-cell} ipython3 | ||
:tags: [hide-cell] | ||
import spey | ||
from spey.math import value_and_grad, hessian | ||
import numpy as np | ||
np.random.seed(14) | ||
``` | ||
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{py:func}`spey.math.value_and_grad` returns a function that computes negative log-likelihood and its gradient for a given statistical model and {py:func}`spey.math.hessian` returns a function that computes Hessian of negative log-likelihood. | ||
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Let us examine this on ``"default_pdf.uncorrelated_background"``: | ||
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```{code-cell} ipython3 | ||
pdf_wrapper = spey.get_backend("default_pdf.uncorrelated_background") | ||
data = [36, 33] | ||
signal_yields = [12.0, 15.0] | ||
background_yields = [50.0, 48.0] | ||
background_unc = [12.0, 16.0] | ||
stat_model = pdf_wrapper( | ||
signal_yields=signal_yields, | ||
background_yields=background_yields, | ||
data=data, | ||
absolute_uncertainties=background_unc, | ||
) | ||
``` | ||
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Here we constructed a two-bin statistical model with observations $36,\ 33$, signal yields $12,\ 15$ and background yields $50\pm12,\ 48\pm16$. We can construct the function that will return negative log probability and its gradient as follows | ||
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```{code-cell} ipython3 | ||
neg_logprob = value_and_grad(stat_model) | ||
``` | ||
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Notice that this function constructs a negative log-probability for the observed statistical model using the default data that we provided earlier. This can be changed using ``expected`` and ``data`` keywords. Now we can choose nuisance parameters and execute the function: | ||
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```{code-cell} ipython3 | ||
nui = np.random.uniform(0,1,(3,)) | ||
neg_logprob(nui) | ||
``` | ||
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```python | ||
(27.81902589793928, array([13.29067478, 6.17223275, 9.28814191])) | ||
``` | ||
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For this particular model, we have only two nuisance parameters, $\theta_i$, and signal strength, $\mu$, due to the structure of the statistical model. For Hessian, we can use the same formulation: | ||
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```{code-cell} ipython3 | ||
hess = hessian(stat_model) | ||
hess(nui) | ||
``` | ||
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```python | ||
array([[ 2.74153126, 1.21034187, 1.63326868], | ||
[ 1.21034187, 2.21034187, -0. ], | ||
[ 1.63326868, -0. , 2.74215326]]) | ||
``` |
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