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I'm rather evangelistic with regards to the use of likelihood ratios for representing the objective evidence for/against a given phenomenon. However, I recently learned that the Bayes factor serves a similar function in the context of Bayesian methods (i.e. the subjective prior is combined with the objective Bayes factor to yield an objectively updated subjective state of belief). I'm now trying to understand the computational and philosophical differences between a likelihood ratio and a Bayes factor.

At the computational level, I understand that while the likelihood ratio is usually computed using the likelihoods that represent the maximum likelihood for each model's respective parameterization (either estimated by cross validation or penalized according to model complexity using AIC), apparently the Bayes factor somehow uses likelihoods that represent the likelihood of each model integrated over it's entire parameter space (i.e. not just at the MLE). How is this integration actually achieved typically? Does one really just try to calculate the likelihood at each of thousands (millions?) of random samples from the parameter space, or are there analytic methods to integrating the likelihood across the parameter space? Additionally, when computing the Bayes factor, does one apply correction for complexity (automatically via cross-validated estimation of likelihood or analytically via AIC) as one does with the likelihood ratio?

Also, what are the philosophical differences between the likelihood ratio and the Bayes factor (n.b. I'm not asking about the philosophical differences between the likelihood ratio and Bayesian methods in general, but the Bayes factor as a representation of the objective evidence specifically). How would one go about characterizing the meaning of the Bayes factor as compared to the likelihood ratio?

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apparently the Bayes factor somehow uses likelihoods that represent the likelihood of each model integrated over it's entire parameter space (i.e. not just at the MLE). How is this integration actually achieved typically? Does one really just try to calculate the likelihood at each of thousands (millions?) of random samples from the parameter space, or are there analytic methods to integrating the likelihood across the parameter space?

First, any situation where you consider a term such as $P(D|M)$ for data $D$ and model $M$ is considered a likelihood model. This is often the bread and butter of any statistical analysis, frequentist or Bayesian, and this is the portion that your analysis is meant to suggest is either a good fit or a bad fit. So Bayes factors are not doing anything fundamentally different than likelihood ratios.

It's important to put Bayes factors in their right setting. When you have two models, say, and you convert from probabilities to odds, then Bayes factors act like an operator on prior beliefs:

$$ Posterior Odds = Bayes Factor * Prior Odds $$ $$ \frac{P(M_{1}|D)}{P(M_{2}|D)} = B.F. \times \frac{P(M_{1})}{P(M_{2})} $$

The real difference is that likelihood ratios are cheaper to compute and generally conceptually easier to specify. The likelihood at the MLE is just a point estimate of the Bayes factor numerator and denominator, respectively. Like most frequentist constructions, it can be viewed as a special case of Bayesian analysis with a contrived prior that's hard to get at. But mostly it arose because it's analytically tractable and easier to compute (in the era before approximate Bayesian computational approaches arose).

To the point on computation, yes: you will evaluate the different likelihood integrals in the Bayesian setting with a large-scale Monte Carlo procedure in almost any case of practical interest. There are some specialized simulators, such as GHK, that work if you assume certain distributions, and if you make these assumptions, sometimes you can find analytically tractable problems for which fully analytic Bayes factors exist.

But no one uses these; there is no reason to. With optimized Metropolis/Gibbs samplers and other MCMC methods, it's totally tractable to approach these problems in a fully data driven way and compute your integrals numerically. In fact, one will often do this hierarchically and further integrate the results over meta-priors that relate to data collection mechanisms, non-ignorable experimental designs, etc.

I recommend the book Bayesian Data Analysis for more on this. Although, the author, Andrew Gelman, seems not to care too much for Bayes factors. As an aside, I agree with Gelman. If you're going to go Bayesian, then exploit the full posterior. Doing model selection with Bayesian methods is like handicapping them, because model selection is a weak and mostly useless form of inference. I'd rather know distributions over model choices if I can... who cares about quantizing it down to "model A is better than model B" sorts of statements when you do not have to?

Additionally, when computing the Bayes factor, does one apply correction for complexity (automatically via cross-validated estimation of likelihood or analytically via AIC) as one does with the likelihood ratio?

This is one of the nice things about Bayesian methods. Bayes factors automatically account for model complexity in a technical sense. You can set up a simple scenario with two models, $M_{1}$ and $M_{2}$ with assumed model complexities $d_{1}$ and $d_{2}$, respectively, with $d_{1} < d_{2}$ and a sample size $N$.

Then if $B_{1,2}$ is the Bayes factor with $M_{1}$ in the numerator, under the assumption that $M_{1}$ is true one can prove that as $N\to\infty$, $B_{1,2}$ approaches $\infty$ at a rate that depends on the difference in model complexity, and that the Bayes factor favors the simpler model. More specifically, you can show that under all of the above assumptions, $$ B_{1,2} = \mathcal{O}(N^{\frac{1}{2}(d_{2}-d_{1})}) $$

I'm familiar with this derivation and the discussion from the book Finite Mixture and Markov Switching Models by Sylvia Frühwirth-Schnatter, but there are likely more directly statistical accounts that dive more into the epistemology underlying it.

I don't know the details well enough to give them here, but I believe there are some fairly deep theoretical connections between this and the derivation of AIC. The Information Theory book by Cover and Thomas hinted at this at least.

Also, what are the philosophical differences between the likelihood ratio and the Bayes factor (n.b. I'm not asking about the philosophical differences between the likelihood ratio and Bayesian methods in general, but the Bayes factor as a representation of the objective evidence specifically). How would one go about characterizing the meaning of the Bayes factor as compared to the likelihood ratio?

The Wikipedia article's section on "Interpretation" does a good job of discussing this (especially the chart showing Jeffreys' strength of evidence scale).

Like usual, there's not too much philosophical stuff beyond the basic differences between Bayesian methods and frequentist methods (which you seem already familiar with).

The main thing is that the likelihood ratio is not coherent in a Dutch book sense. You can concoct scenarios where the model selection inference from likelihood ratios will lead one to accept losing bets. The Bayesian method is coherent, but operates on a prior which could be extremely poor and has to be chosen subjectively. Tradeoffs.. tradeoffs...

FWIW, I think this kind of heavily parameterized model selection is not very good inference. I prefer Bayesian methods and I prefer to organize them more hierarchically, and I want the inference to center on the full posterior distribution if it is at all computationally feasible to do so. I think Bayes factors have some neat mathematical properties, but as a Bayesian myself, I am not impressed by them. They conceal the really useful part of Bayesian analysis, which is that it forces you to deal with your priors out in the open instead of sweeping them under the rug, and allows you to do inference on full posteriors.

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  • $\begingroup$ "Like usual, there's not too much philosophical stuff beyond the basic differences between Bayesian methods and frequentist methods (which you seem already familiar with). The main thing is that the likelihood ratio test..." Just a point of clarification, I didn't intent to compare Bayes factors with Likelihood ratio tests, but with likelihood ratios on their own, with no frequentist/null hypothesis testing baggage. $\endgroup$ – Mike Lawrence Apr 30 '12 at 14:14
  • $\begingroup$ Pursuant to my clarification above: Therefore, it seems to me that the big difference between BFs and LRs is that, as you say, the former auto-correct for complexity but require lots of computation while the latter require much less computation but require explicit correction for model complexity (either using AIC, which is computationally quick, or cross-validation, which is rather more computationally costly). $\endgroup$ – Mike Lawrence Apr 30 '12 at 14:16
  • $\begingroup$ Sorry, the likelihood ratio test was a typo, should have just been likelihood ratio. I think you're mostly right, but you're still missing the bigger picture that likelihood ratio is just a point estimate. It's only going to be useful if the underlying probability distributions behave nice up to a quadratic approximation in the neighborhood of the MLE.. Bayes factors doesn't need to care about asymptotic distribution properties like this, so it's specifically more general. It subsumes MLE-based model selection inference. $\endgroup$ – ely Apr 30 '12 at 14:19
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    $\begingroup$ To put it another way, the MLE can be viewed as the maximum a posteriori estimator (MAP), just with an improper prior (when the integration allows for this), and MAP is a more compelling point estimate since it incorporates prior information. Now, instead of just picking the mode of the posterior... why not combine all values of the posterior according to their prior probability? It won't give you a point estimate of the parameters, but most often people don't really want a point estimate. Distributions over parameters are always more useful than point estimates when you can afford to get them $\endgroup$ – ely Apr 30 '12 at 14:21
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In understanding the difference between likelihood ratios and Bayes factors, it is useful to consider one key feature of Bayes factors in more detail:

How do Bayes factors manage to automatically account for the complexity of the underlying models?

One perspective on this question is to consider methods for deterministic approximate inference. Variational Bayes is one such method. It may not only dramatically reduce the computational complexity of stochastic approximations (e.g., MCMC sampling). Variational Bayes also provide an intuitive understanding of what makes up a Bayes factor.

Recall first that a Bayes factor is based on the model evidences of two competing models,

\begin{align} BF_{1,2} = \frac{p(\textrm{data} \mid M_1)}{p(\textrm{data} \mid M_2)}, \end{align}

where the individual model evidences would have to be computed by a complicated integral:

\begin{align} p(\textrm{data} \mid M_i) = \int p(\textrm{data} \mid \theta,M_i ) \ p(\theta \mid M_i) \ \textrm{d}\theta \end{align}

This integral is not only needed to compute a Bayes factor; it is also needed for inference on the parameters themselves, i.e., when computing $p(\theta \mid \textrm{data}, M_i)$.

A fixed-form variational Bayes approach addresses this problem by making a distributional assumption about the conditional posteriors (e.g., a Gaussian assumption). This turns a difficult integration problem into a much easier optimisation problem: the problem of finding the moments of an approximate density $q(\theta)$ that is maximally similar to the true, but unknown, posterior $p(\theta \mid \textrm{data},M_i)$.

Variational calculus tells us that this can be achieved by maximising the so-called negative free-energy $\mathcal{F}$, which is directly related to the log model evidence:

\begin{align} \mathcal{F} = \textrm{log} \; p(\textrm{data} \mid M_i) - \textrm{KL}\left[q(\theta) \; || \; p(\theta \mid \textrm{data},M_i) \right] \end{align}

From this you can see that maximising the negative free-energy does not only provide us with an approximate posterior $q(\theta) \approx p(\theta \mid \textrm{data},M_i)$. Because the Kullback-Leibler divergence is non-negative, $\mathcal{F}$ also provides a lower bound on the (log) model evidence itself.

We can now return to the original question of how a Bayes factor automatically balances goodness of fit and complexity of the involved models. It turns out that the negative free-energy can be rewritten as follows:

\begin{align} \mathcal{F} = \left\langle p(\textrm{data} \mid \theta,M_i) \right\rangle_q - \textrm{KL}\left[ q(\theta) \; || \; p(\theta \mid M_i) \right] \end{align}

The first term is the log-likelihood of the data expected under the approximate posterior; it represents the goodness of fit (or accuracy) of the model. The second term is the KL divergence between the approximate posterior and the prior; it represents the complexity of the model, under the view that a simpler model is one which is more consistent with our prior beliefs, or under the view that a simpler model does not have to be stretched as much to accommodate the data.

The free-energy approximation to the log model evidence shows that the model evidence incorporates a trade-off between modelling the data (i.e., goodness of fit) and remaining consistent with our prior (i.e., simplicity or negative complexity).

A Bayes factor (in contrast to a likelihood ratio) thus says which of two competing models is better at providing a simple yet accurate explanation of the data.

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