We are currently solving the following issue within our repository. Here is the issue text:
--- BEGIN ISSUE ---
[Docs] Bernoulli likelihoods 
# 📚 Documentation/Examples

In the document for [Bernoulli likelihoods](https://docs.gpytorch.ai/en/stable/likelihoods.html), since the labels take value in {0, 1}, the likelihood should be 

$$p(Y=y | f) = \Phi ((2y-1)f)$$

instead of the currently displayed

$$p(Y=y|f) = \Phi(yf).$$

--- END ISSUE ---

Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
--- BEGIN FILES ---
Path: `gpytorch/likelihoods/bernoulli_likelihood.py`
Content:
```
1 #!/usr/bin/env python3
2 
3 import warnings
4 
5 import torch
6 
7 from ..distributions import base_distributions
8 from ..functions import log_normal_cdf
9 from .likelihood import _OneDimensionalLikelihood
10 
11 
12 class BernoulliLikelihood(_OneDimensionalLikelihood):
13     r"""
14     Implements the Bernoulli likelihood used for GP classification, using
15     Probit regression (i.e., the latent function is warped to be in [0,1]
16     using the standard Normal CDF :math:`\Phi(x)`). Given the identity
17     :math:`\Phi(-x) = 1-\Phi(x)`, we can write the likelihood compactly as:
18 
19     .. math::
20         \begin{equation*}
21             p(Y=y|f)=\Phi(yf)
22         \end{equation*}
23     """
24 
25     def forward(self, function_samples, **kwargs):
26         output_probs = base_distributions.Normal(0, 1).cdf(function_samples)
27         return base_distributions.Bernoulli(probs=output_probs)
28 
29     def log_marginal(self, observations, function_dist, *args, **kwargs):
30         marginal = self.marginal(function_dist, *args, **kwargs)
31         return marginal.log_prob(observations)
32 
33     def marginal(self, function_dist, **kwargs):
34         mean = function_dist.mean
35         var = function_dist.variance
36         link = mean.div(torch.sqrt(1 + var))
37         output_probs = base_distributions.Normal(0, 1).cdf(link)
38         return base_distributions.Bernoulli(probs=output_probs)
39 
40     def expected_log_prob(self, observations, function_dist, *params, **kwargs):
41         if torch.any(observations.eq(-1)):
42             # Remove after 1.0
43             warnings.warn(
44                 "BernoulliLikelihood.expected_log_prob expects observations with labels in {0, 1}. "
45                 "Observations with labels in {-1, 1} are deprecated.",
46                 DeprecationWarning,
47             )
48         else:
49             observations = observations.mul(2).sub(1)
50         # Custom function here so we can use log_normal_cdf rather than Normal.cdf
51         # This is going to be less prone to overflow errors
52         log_prob_lambda = lambda function_samples: log_normal_cdf(function_samples.mul(observations))
53         log_prob = self.quadrature(log_prob_lambda, function_dist)
54         return log_prob
55
```
--- END FILES ---

Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.

Here is an example:

```diff
diff --git a/examples/server_async.py b/examples/server_async.py
--- a/examples/server_async.py
+++ b/examples/server_async.py
@@ -313,4 +313,4 @@
 
 
 if __name__ == "__main__":
-    asyncio.run(run_async_server("."), debug=True)
+    asyncio.run(run_async_server(), debug=True)
diff --git a/examples/server_sync.py b/examples/server_sync.py
--- a/examples/server_sync.py
+++ b/examples/server_sync.py
@@ -313,5 +313,5 @@
 
 
 if __name__ == "__main__":
-    server = run_sync_server(".")
+    server = run_sync_server()
     server.shutdown()

```

diff --git a/gpytorch/likelihoods/bernoulli_likelihood.py b/gpytorch/likelihoods/bernoulli_likelihood.py
--- a/gpytorch/likelihoods/bernoulli_likelihood.py
+++ b/gpytorch/likelihoods/bernoulli_likelihood.py
@@ -18,8 +18,11 @@
 
     .. math::
         \begin{equation*}
-            p(Y=y|f)=\Phi(yf)
+            p(Y=y|f)=\Phi((2y - 1)f)
         \end{equation*}
+
+    .. note::
+        The labels should take values in {0, 1}.
     """
 
     def forward(self, function_samples, **kwargs):

{"golden_diff": "diff --git a/gpytorch/likelihoods/bernoulli_likelihood.py b/gpytorch/likelihoods/bernoulli_likelihood.py\n--- a/gpytorch/likelihoods/bernoulli_likelihood.py\n+++ b/gpytorch/likelihoods/bernoulli_likelihood.py\n@@ -18,8 +18,11 @@\n \n     .. math::\n         \\begin{equation*}\n-            p(Y=y|f)=\\Phi(yf)\n+            p(Y=y|f)=\\Phi((2y - 1)f)\n         \\end{equation*}\n+\n+    .. note::\n+        The labels should take values in {0, 1}.\n     \"\"\"\n \n     def forward(self, function_samples, **kwargs):\n", "issue": "[Docs] Bernoulli likelihoods \n# \ud83d\udcda Documentation/Examples\r\n\r\nIn the document for [Bernoulli likelihoods](https://docs.gpytorch.ai/en/stable/likelihoods.html), since the labels take value in {0, 1}, the likelihood should be \r\n\r\n$$p(Y=y | f) = \\Phi ((2y-1)f)$$\r\n\r\ninstead of the currently displayed\r\n\r\n$$p(Y=y|f) = \\Phi(yf).$$\n", "before_files": [{"content": "#!/usr/bin/env python3\n\nimport warnings\n\nimport torch\n\nfrom ..distributions import base_distributions\nfrom ..functions import log_normal_cdf\nfrom .likelihood import _OneDimensionalLikelihood\n\n\nclass BernoulliLikelihood(_OneDimensionalLikelihood):\n    r\"\"\"\n    Implements the Bernoulli likelihood used for GP classification, using\n    Probit regression (i.e., the latent function is warped to be in [0,1]\n    using the standard Normal CDF :math:`\\Phi(x)`). Given the identity\n    :math:`\\Phi(-x) = 1-\\Phi(x)`, we can write the likelihood compactly as:\n\n    .. math::\n        \\begin{equation*}\n            p(Y=y|f)=\\Phi(yf)\n        \\end{equation*}\n    \"\"\"\n\n    def forward(self, function_samples, **kwargs):\n        output_probs = base_distributions.Normal(0, 1).cdf(function_samples)\n        return base_distributions.Bernoulli(probs=output_probs)\n\n    def log_marginal(self, observations, function_dist, *args, **kwargs):\n        marginal = self.marginal(function_dist, *args, **kwargs)\n        return marginal.log_prob(observations)\n\n    def marginal(self, function_dist, **kwargs):\n        mean = function_dist.mean\n        var = function_dist.variance\n        link = mean.div(torch.sqrt(1 + var))\n        output_probs = base_distributions.Normal(0, 1).cdf(link)\n        return base_distributions.Bernoulli(probs=output_probs)\n\n    def expected_log_prob(self, observations, function_dist, *params, **kwargs):\n        if torch.any(observations.eq(-1)):\n            # Remove after 1.0\n            warnings.warn(\n                \"BernoulliLikelihood.expected_log_prob expects observations with labels in {0, 1}. \"\n                \"Observations with labels in {-1, 1} are deprecated.\",\n                DeprecationWarning,\n            )\n        else:\n            observations = observations.mul(2).sub(1)\n        # Custom function here so we can use log_normal_cdf rather than Normal.cdf\n        # This is going to be less prone to overflow errors\n        log_prob_lambda = lambda function_samples: log_normal_cdf(function_samples.mul(observations))\n        log_prob = self.quadrature(log_prob_lambda, function_dist)\n        return log_prob\n", "path": "gpytorch/likelihoods/bernoulli_likelihood.py"}], "after_files": [{"content": "#!/usr/bin/env python3\n\nimport warnings\n\nimport torch\n\nfrom ..distributions import base_distributions\nfrom ..functions import log_normal_cdf\nfrom .likelihood import _OneDimensionalLikelihood\n\n\nclass BernoulliLikelihood(_OneDimensionalLikelihood):\n    r\"\"\"\n    Implements the Bernoulli likelihood used for GP classification, using\n    Probit regression (i.e., the latent function is warped to be in [0,1]\n    using the standard Normal CDF :math:`\\Phi(x)`). Given the identity\n    :math:`\\Phi(-x) = 1-\\Phi(x)`, we can write the likelihood compactly as:\n\n    .. math::\n        \\begin{equation*}\n            p(Y=y|f)=\\Phi((2y - 1)f)\n        \\end{equation*}\n\n    .. note::\n        The labels should take values in {0, 1}.\n    \"\"\"\n\n    def forward(self, function_samples, **kwargs):\n        output_probs = base_distributions.Normal(0, 1).cdf(function_samples)\n        return base_distributions.Bernoulli(probs=output_probs)\n\n    def log_marginal(self, observations, function_dist, *args, **kwargs):\n        marginal = self.marginal(function_dist, *args, **kwargs)\n        return marginal.log_prob(observations)\n\n    def marginal(self, function_dist, **kwargs):\n        mean = function_dist.mean\n        var = function_dist.variance\n        link = mean.div(torch.sqrt(1 + var))\n        output_probs = base_distributions.Normal(0, 1).cdf(link)\n        return base_distributions.Bernoulli(probs=output_probs)\n\n    def expected_log_prob(self, observations, function_dist, *params, **kwargs):\n        if torch.any(observations.eq(-1)):\n            # Remove after 1.0\n            warnings.warn(\n                \"BernoulliLikelihood.expected_log_prob expects observations with labels in {0, 1}. \"\n                \"Observations with labels in {-1, 1} are deprecated.\",\n                DeprecationWarning,\n            )\n        else:\n            observations = observations.mul(2).sub(1)\n        # Custom function here so we can use log_normal_cdf rather than Normal.cdf\n        # This is going to be less prone to overflow errors\n        log_prob_lambda = lambda function_samples: log_normal_cdf(function_samples.mul(observations))\n        log_prob = self.quadrature(log_prob_lambda, function_dist)\n        return log_prob\n", "path": "gpytorch/likelihoods/bernoulli_likelihood.py"}]}