Regression with Input-dependent Noise: A Gaussian Process Treatment
Authors: Paul W. Goldberg, Christopher K. I. Williams, Christopher M. Bishop
Published in: Advances in Neural Information Processing Systems 10  (NIPS 1997) (Main Conference Track), 1997

Abstract: Gaussian processes  provide natural non-parametric prior distribu(cid:173) tions over regression functions.  In this paper we consider regression  problems where there is  noise  on  the output,  and  the variance  of  the  noise  depends  on  the  inputs.  If we  assume  that  the  noise  is  a  smooth  function  of the  inputs,  then  it  is  natural  to  model  the  noise variance using a second Gaussian process,  in addition to the  Gaussian process governing the noise-free output  value.  We  show  that prior uncertainty about the parameters controlling both pro(cid:173) cesses  can  be  handled  and  that  the  posterior  distribution  of  the  noise  rate can  be  sampled from  using  Markov  chain Monte  Carlo  methods.  Our results on a synthetic data set give a posterior noise  variance that well-approximates the true variance. 
1  Background and Motivation 
A very  natural approach to regression problems is  to place a  prior on  the kinds  of  function  that  we  expect,  and  then  after  observing the  data to obtain  a  posterior.  The prior can be obtained by placing prior distributions on the weights in a neural 
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P.  W  Goldberg,  C. K. L Williams and C.  M.  Bishop 
network, although we would argue that it is perhaps more natural to place priors di(cid:173) rectly over functions.  One tractable way of doing this is to create a  Gaussian process  prior.  This has the advantage that predictions can be made from the posterior using  only  matrix multiplication for  fixed  hyperparameters  and  a  global  noise level.  In  contrast, for  neural networks  (with fixed  hyperparameters and a global noise level)  it is necessary to use approximations or Markov chain Monte Carlo (MCMC)  meth(cid:173) ods.  Rasmussen  (1996)  has  demonstrated that predictions obtained with Gaussian  processes are as  good as or better than other state-of-the art predictors. 
In  much of the work on regression problems in the statistical and neural networks  literatures, it is assumed that there is  a global noise level, independent of the input  vector  x.  The book  by  Bishop  (1995)  and the  papers  by  Bishop  (1994),  MacKay  (1995)  and  Bishop  and  Qazaz  (1997)  have  examined  the  case  of input-dependent  (Such  models  are  said  to  noise  for  parametric  models  such  as  neural  networks.  heteroscedastic in the statistics literature.)  In this paper we  develop the treatment  of an input-dependent noise model for  Gaussian process regression, where the noise  is assumed to be Gaussian but its variance depends on x.  As  the noise level is non(cid:173) negative we  place a  Gaussian process  prior on  the  log  noise  level.  Thus  there  are  two Gaussian processes involved in making predictions:  the usual Gaussian process  for  predicting the function  values  (the y-process),  and another one  (the  z-process)  for  predicting the log  noise  level.  Below  we  present  a  Markov chain Monte Carlo  method for  carrying out inference with this model and demonstrate its performance  on a  test  problem. 
1.1  Gaussian processes 
A stochastic process  is  a  collection of random variables  {Y(x)lx  E  X} indexed by  a  set  X.  Often  X  will  be  a  space  such  as  'R,d  for  some dimension  d,  although  it  could be more general.  The stochastic process is specified by giving the probability  distribution  for  every  finite  subset  of  variables  Y(Xl), ... , Y(Xk)  in  a  consistent  manner.  A  Gaussian  process  is  a  stochastic  process  which  can  be fully  specified  by  its  mean  function  J.L(x)  =  E[Y(x)]  and  its  covariance  function  Cp(x,x')  =  E[(Y(x)-J.L(x»)(Y(x')-J.L(x'»]; any finite set of points will have a joint multivariate  Gaussian distribution.  Below we  consider Gaussian processes which have J.L(x)  ==  O.  This  assumes  that  any  known  offset  or  trend  in  the  data  has  been. removed.  A  non-zero  I' (x )  is  easily  incorporated  into  the  framework  at  the  expense  of extra  notational complexity. 
A  covariance  junction  is  used  to  define  a  Gaussian  process;  it  is  a  parametrised  function  from  pairs  of  x-values  to  their  covariance.  The  form  of  the  covariance  function that we  shall use for  the prior over functions  is  given by 
Cy(x(i),xU» =vyexp (-~ tWYl(x~i) _x~j»2) + Jy 8(i,j)