[R] subject: Log likelihood above 0

[Ravi Varadhan]

Likelihood is a function of the parameters, conditioned upon the data.  It is not the same as a probability density function.  Terms or factors which do not involve parameters can be omitted from the likelihood function.  For continuous random variables, the density function can be in (0, Inf).  Therefore, the likelihood function can assume any value between 0 and Inf.  Hence the log-likelihood can be in (-Inf, Inf).  

When the random variable is discrete, the density or probability mass function cannot be greater than 1.   Hence the likelihood cannot be greater than 1, in which case, the log-likelihood cannot be positive.

Ravi.
____________________________________________________________________

Ravi Varadhan, Ph.D.
Assistant Professor,
Division of Geriatric Medicine and Gerontology
School of Medicine
Johns Hopkins University

Ph. (410) 502-2619
email: rvaradhan at jhmi.edu


----- Original Message -----
From: Daniel Haugstvedt <daniel.haugstvedt at gmail.com>
Date: Tuesday, October 5, 2010 9:16 am
Subject: [R] subject: Log likelihood above 0
To: r-help at r-project.org


> Hi -
>  
>  In an effort to learn some basic arima modeling in R i went through
>  the tutorial found at
>  
>  
>  One of the examples gave me a log likelihood of 77. Now I am simply
>  wondering if this is the expected behavior? Looking in my text book
>  this should not be possible. I have actually spent some time on this
>  but neither the documentation ?arima or google gave me a satisfying
>  answer.
>  
>  
>  
>  Data and code:
>  
>  gTemp.raw = c(-0.11, -0.13, -0.01, -0.04, -0.42, -0.23, -0.25, -0.45,
>  -0.23, 0.04, -0.22, -0.55
>  , -0.40,  -0.39, -0.32, -0.32, -0.27, -0.15, -0.21, -0.25, -0.05,
>  -0.05, -0.30, -0.35
>  , -0.42,  -0.25, -0.15, -0.41, -0.30, -0.31, -0.21, -0.25, -0.33,
>  -0.28, -0.02,  0.06
>  , -0.20,  -0.46, -0.33, -0.09, -0.15, -0.04, -0.09, -0.16, -0.11,
>  -0.15,  0.04, -0.05
>  ,  0.01,  -0.22, -0.03,  0.03,  0.04, -0.11,  0.05, -0.08,  0.01,
>  0.12,  0.15, -0.02
>  ,  0.14,   0.11,  0.10,  0.06,  0.10, -0.01,  0.01,  0.12, -0.03,
>  -0.09, -0.17, -0.02
>  ,  0.03,   0.12, -0.09, -0.09, -0.18,  0.08,  0.10,  0.05, -0.02,
>  0.10,  0.05,  0.03
>  , -0.25,  -0.15, -0.07, -0.02, -0.09,  0.00,  0.04, -0.10, -0.05,
>  0.18, -0.06, -0.02
>  , -0.21,   0.16,  0.07,  0.13,  0.27,  0.40,  0.10,  0.34,  0.16,
>  0.13,  0.19,  0.35
>  ,  0.42,   0.28,  0.49,  0.44,  0.16,  0.18,  0.31,  0.47,  0.36,
>  0.40,  0.71,  0.43
>  ,  0.41,   0.56,  0.70,  0.66,  0.60)
>  
>  gTemp.ts = ts(gTemp.raw, start=1880, freq=1)
>  
>  gTemp.model = arima(diff(gTemp.ts), order=c(1,0,1))
>  
>  
>  
>  Results:
>  
>  > gTemp.model
>  
>  Call:
>  arima(x = diff(gTemp.ts), order = c(1, 0, 1))
>  
>  Coefficients:
>           ar1      ma1         intercept
>         0.2695  -0.8180     0.0061
>  s.e.  0.1122   0.0624     0.0030
>  
>  sigma^2 estimated as 0.01680:  log likelihood = 77.05,  aic = -146.11
>  
>  ______________________________________________
>  R-help at r-project.org mailing list
>  
>  PLEASE do read the posting guide 
>  and provide commented, minimal, self-contained, reproducible code.