prompt: You will be provided with a partial code base and an issue statement explaining a problem to resolve.
  <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
  
  $$p(Y=y|f) = \Phi(yf).$$
  
  </issue>
  <code>
  [start of gpytorch/likelihoods/bernoulli_likelihood.py]
  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 of gpytorch/likelihoods/bernoulli_likelihood.py]
  </code>
  
  I need you to solve this issue by generating a single patch file that I can apply directly to this repository using git apply. Please respond with a single patch file in the following format.
  <patch>
  diff --git a/file.py b/file.py
  --- a/file.py
  +++ b/file.py
  @@ -1,27 +1,35 @@
   def euclidean(a, b):
  -    while b:
  -        a, b = b, a % b
  -    return a
  +    if b == 0:
  +        return a
  +    return euclidean(b, a % b)
   
   
   def bresenham(x0, y0, x1, y1):
       points = []
       dx = abs(x1 - x0)
       dy = abs(y1 - y0)
  -    sx = 1 if x0 < x1 else -1
  -    sy = 1 if y0 < y1 else -1
  -    err = dx - dy
  +    x, y = x0, y0
  +    sx = -1 if x0 > x1 else 1
  +    sy = -1 if y0 > y1 else 1
   
  -    while True:
  -        points.append((x0, y0))
  -        if x0 == x1 and y0 == y1:
  -            break
  -        e2 = 2 * err
  -        if e2 > -dy:
  -            err -= dy
  -            x0 += sx
  -        if e2 < dx:
  -            err += dx
  -            y0 += sy
  +    if dx > dy:
  +        err = dx / 2.0
  +        while x != x1:
  +            points.append((x, y))
  +            err -= dy
  +            if err < 0:
  +                y += sy
  +                err += dx
  +            x += sx
  +    else:
  +        err = dy / 2.0
  +        while y != y1:
  +            points.append((x, y))
  +            err -= dx
  +            if err < 0:
  +                x += sx
  +                err += dy
  +            y += sy
  +
  +    points.append((x, y))
       return points
  
  </patch>
golden_diff: 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):
  
verification_info: {"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"}]}