Clean up the fitting_with_x_and_y_errors tutorial

1. Fix the mangled xerr/yerr in the plotting routine that caused an Error in matplotlib 3.6+
2. Switch to the bilby_mcmc sampler to avoid pip installation of nestle
3. Run over the example and check it still works

examples/tutorials/fitting_with_x_and_y_errors.ipynb

 %% Cell type:markdown id: tags:
 
 # Fitting a model to data with both x and y errors with `Bilby`
 
 Usually when we fit a model to data with a Gaussian Likelihood we assume that we know x values exactly. This is almost never the case. Here we show how to fit a model with errors in both x and y.
 
-Since we are using a very simple model we will use the `nestle` sampler.
-This can be installed using
-
-```console
-$ conda install -c conda-forge nestle
-```
-
-or
-
-```console
-$ pip install nestle
-```
 
 %% Cell type:code id: tags:
 
 ``` 
 import bilby
 import matplotlib.pyplot as plt
 import numpy as np
 
 %matplotlib inline
 ```
 
 %% Cell type:markdown id: tags:
 
 ### Simulate data
 
 First we create the data and plot it
 
 %% Cell type:code id: tags:
 
 ``` 
 # define our model, a line
 def model(x, m, c, **kwargs):
     y = m * x + c
     return y
 
 
 # make a function to create and plot our data
 def make_data(points, m, c, xerr, yerr, seed):
     np.random.seed(int(seed))
     xtrue = np.linspace(0, 100, points)
     ytrue = model(x=xtrue, m=m, c=c)
 
-    xerr = xerr * np.random.randn(points)
-    yerr = yerr * np.random.randn(points)
-    xobs = xtrue + xerr
-    yobs = ytrue + yerr
+    xerr_vals = xerr * np.random.randn(points)
+    yerr_vals = yerr * np.random.randn(points)
+    xobs = xtrue + xerr_vals
+    yobs = ytrue + yerr_vals
 
     plt.errorbar(xobs, yobs, xerr=xerr, yerr=yerr, fmt="x")
     plt.errorbar(xtrue, ytrue, yerr=yerr, color="black", alpha=0.5)
     plt.xlim(0, 100)
     plt.show()
     plt.close()
 
     data = {
         "xtrue": xtrue,
         "ytrue": ytrue,
         "xobs": xobs,
         "yobs": yobs,
         "xerr": xerr,
         "yerr": yerr,
     }
 
     return data
 
 
 data = make_data(points=30, m=5, c=10, xerr=5, yerr=5, seed=123)
 ```
 
 %% Cell type:markdown id: tags:
 
 ### Define our prior and sampler settings
 
 Now lets set up the prior and bilby output directory/sampler settings
 
 %% Cell type:code id: tags:
 
 ``` 
 # setting up bilby priors
 priors = dict(
     m=bilby.core.prior.Uniform(0, 30, "m"), c=bilby.core.prior.Uniform(0, 30, "c")
 )
 
-sampler_kwargs = dict(priors=priors, sampler="nestle", nlive=1000, outdir="outdir", verbose=False)
+sampler_kwargs = dict(priors=priors, sampler="bilby_mcmc", nsamples=1000, printdt=5, outdir="outdir", verbose=False, clean=True)
 ```
 
 %% Cell type:markdown id: tags:
 
 ### Fit with exactly known x-values
 
 Our first step is to recover the straight line using a simple Gaussian Likelihood that only takes into account the y errors. Under the assumption we know x exactly. In this case, we pass in xtrue for x
 
 %% Cell type:code id: tags:
 
 ``` 
 known_x = bilby.core.likelihood.GaussianLikelihood(
     x=data["xtrue"], y=data["yobs"], func=model, sigma=data["yerr"]
 )
 result_known_x = bilby.run_sampler(
     likelihood=known_x,
     label="known_x",
     **sampler_kwargs,
 )
 ```
 
 %% Cell type:code id: tags:
 
 ``` 
 _ = result_known_x.plot_corner(truth=dict(m=5, c=10), titles=True, save=False)
 plt.show()
 plt.close()
 ```
 
 %% Cell type:markdown id: tags:
 
 ### Fit with unmodeled uncertainty in the x-values
 
 As expected this is easy to recover and the sampler does a good job. However this was made too easy - by passing in the 'true' values of x. Lets see what happens when we pass in the observed values of x
 
 %% Cell type:code id: tags:
 
 ``` 
 incorrect_x = bilby.core.likelihood.GaussianLikelihood(
     x=data["xobs"], y=data["yobs"], func=model, sigma=data["yerr"]
 )
 result_incorrect_x = bilby.run_sampler(
     likelihood=incorrect_x,
     label="incorrect_x",
     **sampler_kwargs,
 )
 ```
 
 %% Cell type:code id: tags:
 
 ``` 
 _ = result_incorrect_x.plot_corner(truth=dict(m=5, c=10), titles=True, save=False)
 plt.show()
 plt.close()
 ```
 
 %% Cell type:markdown id: tags:
 
 ### Fit with modeled uncertainty in x-values
 
 This is not good as there is unmodelled uncertainty in our `x` values.
 Getting around this requires marginalisation of the true x values or sampling over them.
 See discussion in section 7 of https://arxiv.org/pdf/1008.4686.pdf.
 
 For this, we will have to define a new likelihood class.
 By subclassing the base `bilby.core.likelihood.Likelihood` class we can do this fairly simply.
 
 %% Cell type:code id: tags:
 
 ``` 
 class GaussianLikelihoodUncertainX(bilby.core.likelihood.Likelihood):
     def __init__(self, xobs, yobs, xerr, yerr, function):
         """
 
         Parameters
         ----------
         xobs, yobs: array_like
             The data to analyse
         xerr, yerr: array_like
             The standard deviation of the noise
         function:
             The python function to fit to the data
         """
         super(GaussianLikelihoodUncertainX, self).__init__(dict())
         self.xobs = xobs
         self.yobs = yobs
         self.yerr = yerr
         self.xerr = xerr
         self.function = function
 
     def log_likelihood(self):
         variance = (self.xerr * self.parameters["m"]) ** 2 + self.yerr**2
         model_y = self.function(self.xobs, **self.parameters)
         residual = self.yobs - model_y
 
         ll = -0.5 * np.sum(residual**2 / variance + np.log(variance))
 
         return -0.5 * np.sum(residual**2 / variance + np.log(variance))
 ```
 
 %% Cell type:code id: tags:
 
 ``` 
 gaussian_unknown_x = GaussianLikelihoodUncertainX(
     xobs=data["xobs"],
     yobs=data["yobs"],
     xerr=data["xerr"],
     yerr=data["yerr"],
     function=model,
 )
 result_unknown_x = bilby.run_sampler(
     likelihood=gaussian_unknown_x,
     label="unknown_x",
     **sampler_kwargs,
 )
 ```
 
 %% Cell type:code id: tags:
 
 ``` 
 _ = result_unknown_x.plot_corner(truth=dict(m=5, c=10), titles=True, save=False)
 plt.show()
 plt.close()
 ```
 
 %% Cell type:markdown id: tags:
 
 Success! The inferred posterior is consistent with the true values.