re-execute blackbox numpy notebook (#496)

* re-execute blackbox numpy notebook

* simplify and fix final comparison

examples/case_studies/blackbox_external_likelihood_numpy.myst.md

@@ -368,7 +368,7 @@ with pm.Model() as pymodel:
     theta = pt.as_tensor_variable([m, c])
 
     # use a Normal distribution
-    pm.Normal("likelihood", mu=(m * x + c), sd=sigma, observed=data)
+    y = pm.Normal("likelihood", mu=(m * x + c), sigma=sigma, observed=data)
 
     idata = pm.sample()
 
@@ -412,38 +412,14 @@ az.plot_pair(idata, **pair_kwargs, ax=ax);
 We can now check that the gradient Op works as expected. First, just create and call the `LogLikeGrad` class, which should return the gradient directly (note that we have to create a [PyTensor function](http://deeplearning.net/software/pytensor/library/compile/function.html) to convert the output of the Op to an array). Secondly, we call the gradient from `LogLikeWithGrad` by using the [PyTensor tensor gradient](http://deeplearning.net/software/pytensor/library/gradient.html#pytensor.gradient.grad) function. Finally, we will check the gradient returned by the PyMC model for a Normal distribution, which should be the same as the log-likelihood function we defined. In all cases we evaluate the gradients at the true values of the model function (the straight line) that was created.
 
 ```{code-cell} ipython3
-# test the gradient Op by direct call
-pytensor.config.compute_test_value = "ignore"
-pytensor.config.exception_verbosity = "high"
+ip = pymodel.initial_point()
+print(f"Evaluating dlogp of model at point\n    {ip}")
 
-var = pt.dvector()
-test_grad_op = LogLikeGrad(data, x, sigma)
-test_grad_op_func = pytensor.function([var], test_grad_op(var))
-grad_vals = test_grad_op_func([mtrue, ctrue])
+grad_vals_custom = opmodel.compile_dlogp()(ip)
+grad_vals_pymc = pymodel.compile_dlogp()(ip)
 
-print(f'Gradient returned by "LogLikeGrad": {grad_vals}')
-
-# test the gradient called through LogLikeWithGrad
-test_gradded_op = LogLikeWithGrad(my_loglike, data, x, sigma)
-test_gradded_op_grad = pt.grad(test_gradded_op(var), var)
-test_gradded_op_grad_func = pytensor.function([var], test_gradded_op_grad)
-grad_vals_2 = test_gradded_op_grad_func([mtrue, ctrue])
-
-print(f'Gradient returned by "LogLikeWithGrad": {grad_vals_2}')
-
-# test the gradient that PyMC uses for the Normal log likelihood
-test_model = pm.Model()
-with test_model:
-    m = pm.Uniform("m", lower=-10.0, upper=10.0)
-    c = pm.Uniform("c", lower=-10.0, upper=10.0)
-
-    pm.Normal("likelihood", mu=(m * x + c), sigma=sigma, observed=data)
-
-    gradfunc = test_model.logp_dlogp_function([m, c], dtype=None)
-    gradfunc.set_extra_values({"m_interval__": mtrue, "c_interval__": ctrue})
-    grad_vals_pymc = gradfunc(np.array([mtrue, ctrue]))[1]  # get dlogp values
-
-print(f'Gradient returned by PyMC "Normal" distribution: {grad_vals_pymc}')
+print(f'\nGradient of model using a custom "LogLikeWithGrad":\n    {grad_vals_custom}')
+print(f'Gradient of model using a PyMC "Normal" distribution:\n    {grad_vals_pymc}')
 ```
 
 We could also do some profiling to compare performance between implementations. The {ref}`profiling` notebook shows how to do it.
@@ -454,6 +430,7 @@ We could also do some profiling to compare performance between implementations.
 
 * Adapted from [Jørgen Midtbø](https://github.com/jorgenem/)'s [example](https://discourse.pymc.io/t/connecting-pymc-to-external-code-help-with-understanding-pytensor-custom-ops/670) by Matt Pitkin both as a [blogpost](http://mattpitkin.github.io/samplers-demo/pages/pymc-blackbox-likelihood/) and as an example notebook to this gallery in August, 2018 ([pymc#3169](https://github.com/pymc-devs/pymc/pull/3169) and [pymc#3177](https://github.com/pymc-devs/pymc/pull/3177))
 * Updated by [Oriol Abril](https://github.com/OriolAbril) on December 2021 to drop the Cython dependency from the original notebook and use numpy instead ([pymc-examples#28](https://github.com/pymc-devs/pymc-examples/pull/28))
+* Re-executed by Oriol Abril with pymc 5.0.0 ([pymc-examples#496](https://github.com/pymc-devs/pymc-examples/pull/496))
 
 +++