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For some classes of likelihood functions, one can prove that the likelihood is log-concave, i.e. that the log-likelihood has second derivatives $<=0$ everywhere, which makes life much easier (e.g. you can often prove the existence of unique global maxima, use specialized optimization methods ...) For example,

• this CV question shows that the exponential-family likelihood with the canonical link function is log-concave
• this paper "Concavity of the Log Likelihood" Pratt 1981, JASA proves log-concavity for a class of models with ordinal responses.

There are certainly counterexamples as well (likelihoods that are provably non-log-concave). For example, any log-likelihood that is bi- or multimodal is non-log-concave ... e.g.

• "A note on bimodality in the log-likelihood function for penalized spline mixed models", Welham and Thompson 2009 CS&DA
• "Flat and Multimodal Likelihoods and Model Lack of Fit in Curved Exponential Families", Sundberg 2010 Scand J Stat
• "Problems with Likelihood Estimation of Covariance Functions of Spatial Gaussian Processes", Warnes and Ripley 1987 Biometrika

For some classes of likelihood functions, one can prove that the likelihood is log-concave, i.e. that the log-likelihood has second derivatives $\leq 0$ everywhere, which makes life much easier (e.g. you can often prove the existence of unique global maxima, use specialized optimization methods ...) For example,

• this CV question shows that the exponential-family likelihood with the canonical link function is log-concave
• this paper "Concavity of the Log Likelihood" Pratt 1981, JASA proves log-concavity for a class of models with ordinal responses.

There are certainly counterexamples as well (likelihoods that are provably non-log-concave). For example, any log-likelihood that is bi- or multimodal is non-log-concave ... e.g.

• "A note on bimodality in the log-likelihood function for penalized spline mixed models", Welham and Thompson 2009 CS&DA
• "Flat and Multimodal Likelihoods and Model Lack of Fit in Curved Exponential Families", Sundberg 2010 Scand J Stat
• "Problems with Likelihood Estimation of Covariance Functions of Spatial Gaussian Processes", Warnes and Ripley 1987 Biometrika

For some classes of likelihood functions, one can prove that the likelihood is log-concave, i.e. that the log-likelihood has second derivatives $<=0$ everywhere, which makes life much easier (e.g. you can often prove the existence of unique global maxima, use specialized optimization methods ...) For example,

• this CV question shows that the exponential-family likelihood with the canonical link function is log-concave
• this paper "Concavity of the Log Likelihood" Pratt 1981, JASA proves log-concavity for a class of models with ordinal responses.

There are certainly counterexamples as well (likelihoods that are provably non-log-concave). For example, any log-likelihood that is bi- or multimodal is non-log-concave ... e.g.

• "A note on bimodality in the log-likelihood function for penalized spline mixed models", Welham and Thompson 2009 CS&DA
• "Flat and Multimodal Likelihoods and Model Lack of Fit in Curved Exponential Families", Sundberg 2010 Scand J Stat
• "Problems with Likelihood Estimation of Covariance Functions of Spatial Gaussian Processes", Warnes and Ripley 1987 Biometrika

For some classes of likelihood functions, one can prove that the likelihood is log-concave, i.e. that the log-likelihood has second derivatives $<=0$ everywhere, which makes life much easier (e.g. you can often prove the existence of unique global maxima, use specialized optimization methods ...) For example,

• this CV question shows that the exponential-family likelihood with the canonical link function is log-concave
• this paper "Concavity of the Log Likelihood" Pratt 1981, JASA proves log-concavity for a class of models with ordinal responses.

There are certainly counterexamples as well (likelihoods that are provably non-log-concave). For example, any log-likelihood that is bi- or multimodal is non-log-concave ... e.g.

• "A note on bimodality in the log-likelihood function for penalized spline mixed models", Welham and Thompson 2009 CS&DA
• "Flat and Multimodal Likelihoods and Model Lack of Fit in Curved Exponential Families", Sundberg 2010 Scand J Stat
• "Problems with Likelihood Estimation of Covariance Functions of Spatial Gaussian Processes", Warnes and Ripley 1987 Biometrika

For some classes of likelihood functions, one can prove that the likelihood is log-concave, i.e. that the log-likelihood has second derivatives $<=0$ everywhere, which makes life much easier (e.g. you can often prove the existence of unique global maxima, use specialized optimization methods ...) For example,

• this CV question shows that the exponential-family likelihood with the canonical link function is log-concave
• this paper "Concavity of the Log Likelihood" Pratt 1981, JASA proves log-concavity for a class of models with ordinal responses.

For some classes of likelihood functions, one can prove that the likelihood is log-concave, i.e. that the log-likelihood has second derivatives $<=0$ everywhere, which makes life much easier (e.g. you can often prove the existence of unique global maxima, use specialized optimization methods ...) For example,

• this CV question shows that the exponential-family likelihood with the canonical link function is log-concave
• this paper "Concavity of the Log Likelihood" Pratt 1981, JASA proves log-concavity for a class of models with ordinal responses.

There are certainly counterexamples as well (likelihoods that are provably non-log-concave). For example, any log-likelihood that is bi- or multimodal is non-log-concave ... e.g.

• "A note on bimodality in the log-likelihood function for penalized spline mixed models", Welham and Thompson 2009 CS&DA
• "Flat and Multimodal Likelihoods and Model Lack of Fit in Curved Exponential Families", Sundberg 2010 Scand J Stat
• "Problems with Likelihood Estimation of Covariance Functions of Spatial Gaussian Processes", Warnes and Ripley 1987 Biometrika