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Saddlepoint approximations, likelihood asymptotics, and approximate conditional inference

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Maximum likelihood methods may be inadequate for parameter estimation in models where many nuisance parameters are present. The modified profile likelihood (MPL) of Barndorff-Nielsen (1983) serves as …

Maximum likelihood methods may be inadequate for parameter estimation in models where many nuisance parameters are present. The modified profile likelihood (MPL) of Barndorff-Nielsen (1983) serves as a highly accurate approximation to the marginal or conditional likelihood, when either exist, and can be viewed as an approximate conditional likelihood when they do not. We examine the modified profile likelihood, its variants, and its connections with Laplace and saddlepoint approximations under both theoretical and pragmatic lenses.


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  • 1. Saddlepoint approximations, likelihoodasymptotics, and approximate conditional inference Jared Tobin (BSc, MAS) Department of Statistics The University of Auckland Auckland, New Zealand February 24, 2011 Jared Tobin Approximate conditional inference
  • 2. ./helloWorld I’m from St. John’s, Canada Jared Tobin Approximate conditional inference
  • 3. ./helloWorldIt’s a charming city known for A street containing the most pubs per square foot in North America What could be the worst weather on the planetthese characteristics are probablyrelated.. Jared Tobin Approximate conditional inference
  • 4. ./tellThemAboutMe I recently completed my Master’s degree in Applied Statistics at Memorial University. I’m also a Senior Research Analyst with the Government of Newfoundland & Labrador. This basically means I do a lot of programming and statistics.. (thank you for R!) Here at Auckland, my main supervisor is Russell. I’m also affiliated with Fisheries & Oceans Canada (DFO) via my co-supervisor, Noel. Jared Tobin Approximate conditional inference
  • 5. ./whatImDoingHere Today I’ll be talking about my Master’s research, as well as what I plan to work on during my PhD studies here in Auckland. So let’s get started. Jared Tobin Approximate conditional inference
  • 6. Likelihood inference Consider a probabilistic model Y ∼ f (y ; ζ), ζ ∈ R, and a sample y of size n from f (y ; ζ). How can we estimate ζ from the sample y? Everybody knows maximum likelihood.. (right?) It’s the cornerstone of frequentist inference, and remains quite popular today. (ask Russell) Jared Tobin Approximate conditional inference
  • 7. Likelihood inference Define L(ζ; y) = n f (yj ; ζ) and call it the likelihood function of j=1 ζ, given the sample y. If we take R to be the (non-extended) real line, then ζ ∈ R lets us ˆ define ζ = argmaxζ L(ζ; y) to be the maximum likelihood estimator (or MLE) of ζ. ˆ We can think of ζ as the value of ζ that maximizes the probability of observing the sample y. Jared Tobin Approximate conditional inference
  • 8. Nuisance parameter models Now consider a probabilistic model Y ∼ f (y ; ζ) where ζ = (θ, ψ), ζ ∈ RP θ∈R ψ ∈ RP−1 If we are only interested in θ, then ψ is called a nuisance parameter, or incidental parameter. It turns out that these things are aptly named.. Jared Tobin Approximate conditional inference
  • 9. The nuisance parameter problem: example Take a very simple example; a H-strata model with two observations per stratum. Yh1 and Yh2 are iid N(µh , σ 2 ) random variables, h = 1, . . . , H, and we are interested in estimating σ 2 . Define µ = (µ1 , . . . , µH ). Assuming σ 2 is known, the log-likelihood for µ is (yh1 − µh )2 + (yh2 − µh )2 l(µ; σ 2 , y) = − log 2πσ 2 − 2σ 2 h and the MLE for the hth stratum, µh , is that stratum’s sample ˆ mean (yh1 + yh2 )/2. Jared Tobin Approximate conditional inference
  • 10. Example, continued.. To estimate σ 2 , however, we must use that estimate for µ. It is common to use the profile likelihood, defined for this example as l (P) (σ 2 ; µ, y) = sup l(µ, σ 2 ; y) ˆ µ to estimate σ 2 . Maximizing yields 1 (yh1 − µh )2 + (yh2 − µh )2 ˆ ˆ σ2 = ˆ H 2 h as the MLE for σ 2 . Jared Tobin Approximate conditional inference
  • 11. Example, continued.. Let’s check for bias.. let Sh = [(Yh1 − µh )2 + (Yh2 − µh )2 ]/2 and 2 ˆ ˆ note that Sh2 = (Y − Y )2 /4 and σ 2 = H −1 ˆ 2. h1 h2 h Sh Some algebra shows that 1 E [Sh ] = (varYh1 + µ2 + varYh2 + µ2 − 2E [Yh1 Yh2 ]) 2 h h 4 and since Yh1 , Yh2 are independent, E [Yh1 Yh2 ] = µ2 so that h E [Sh ] = σ 2 /2. 2 Jared Tobin Approximate conditional inference
  • 12. Example, continued.. Put it together and we have 1 E [ˆ 2 ] = σ 2 E [Sh ] H h 1 σ2 = H 2 h σ2 = 2 No big deal - everyone knows the MLE for σ 2 is biased.. But notice the implication for consistency.. lim P(|ˆ 2 − σ 2 | < ) = 0 σ n→∞ Jared Tobin Approximate conditional inference
  • 13. Neyman-Scott problems This result isn’t exactly new.. It was described by Neyman & Scott as early as 1948. That’s merit for the name; this type of problem is typically known as a Neyman-Scott problem in the literature. The problem is that one of the required regularity conditions is not met. We usually require that the dimension of (µ, σ 2 ) remain constant for increasing sample size.. H But notice that n = h=1 2, so n → ∞ iff H → ∞ iff dim(µ) → ∞. Jared Tobin Approximate conditional inference
  • 14. The profile likelihood In a general setting where Y ∼ f (y ; ψ, θ) with nuisance parameter −1 ψ, consider the partial information iθθ|ψ = iθθ − iθψ iψψ iψθ . (P) It can be shown that the profile expected information iθθ is first-order equivalent to iθθ .. (P) so iθθ|ψ < iθθ in an asymptotic sense. In other words, the profile likelihood places more weight on information about θ than it ’ought’ to be doing. Jared Tobin Approximate conditional inference
  • 15. Two-index asymptotics for the profile likelihood Take a general stratified model again.. Yh ∼ f (yh ; ψh , θ) with H strata. Remove the per-stratum sample size restriction of nh = 2. Now, if we let both nh and H approach infinity, we get different results depending on the speed at which nh → ∞ and H → ∞. ˆ If nh → ∞ faster than H does, we have that θ − θ = Op (n−1/2 ). −1 If H → ∞ faster, the same difference is of order Op (nh ). In other words, we can probably expect the profile likelihood to make relatively poor estimates of θ if H > nh on average. Jared Tobin Approximate conditional inference
  • 16. Alternatives to using the profile likelihood This type of model comes up a lot in practice.. (example later) What solutions are available to tackle the nuisance parameter problem? For the normal nuisance parameter model, the method of moments estimator is an option (and happens to be unbiased & consistent). Jared Tobin Approximate conditional inference
  • 17. Alternatives to using the profile likelihood But what if the moments themselves involve multiple parameters? ... then it may be difficult or impossible to construct a method of moments estimator. Also, likelihood-based estimators generally have desirable statistical properties, and it would be nice to retain those. There may be a way to patch up the problem with ‘standard’ ML in these models.. hint: there is. my plane ticket was $2300 CAD, so I’d better have something entertaining to tell you.. Jared Tobin Approximate conditional inference
  • 18. Motivating example This research started when looking at a stock assessment problem. Take the waters of the coast of Newfoundland & Labrador, which historically supported the largest fishery of Atlantic cod in the world. (keyword: historically) Jared Tobin Approximate conditional inference
  • 19. Motivating example (continued) Here’s the model: divide these waters (the stock area) into N equally-sized sampling units, where the j th unit, j = 1, . . . , N contains λj fish. Each sampling unit corresponds to the area over the ocean bottom covered by a standardized trawl tow made at a fixed speed and duration, N Then the total number of fish in the stock is λ = j=1 λj , and we want to estimate this. In practice we estimate a measure of trawlable abundance. We weight λ by the probability of catching a fish on any given tow, q, and estimate µ = qλ. Jared Tobin Approximate conditional inference
  • 20. Motivating example (continued) DFO conducts two research trawl surveys on these waters every year using a stratified random sampling scheme.. Jared Tobin Approximate conditional inference
  • 21. Motivating example (continued) For each stratum h, h = 1, . . . , H, we model an observed catch as Yh ∼ negbin(µh , k). The negative binomial mass function is yh k Γ(yh + k) µh k P(Yh = yh ; µh , k) = Γ(yh + 1)Γ(k) µh + k µh + k and it has mean µh and variance µh + k −1 µ2 . h Jared Tobin Approximate conditional inference
  • 22. Motivating example (continued) We have a nuisance parameter model.. if we want to make interval estimates for µ, we must estimate the dispersion parameter k. Breezing through the literature will suggest any number of increasingly esoteric ways to do this.. method of moments pseudo-likelihood optimal quadratic estimating equations extended quasi-likelihood adjusted extended quasi-likelihood double extended quasi-likelihood etc. WTF Jared Tobin Approximate conditional inference
  • 23. Motivating example (continued) Which of these is best?? (and why are there so many??) Noel and I wrote a paper that tried to answer the first question.. .. unfortunately, we also wound up adding another long-winded estimator to the list, and so increased the scope of the second. We coined something called the ‘adjusted double-extended quasi-likelihood’ or ADEQL estimator, which performed best in our simulations. Jared Tobin Approximate conditional inference
  • 24. Motivating example (fini) When writing my Master’s thesis, I wanted to figure out why this estimator worked as well as it did? And what exactly are all the other ones? This involved looking into functional approximations and likelihood asymptotics.. .. but I managed to uncover some fundamental answers that simplified the whole nuisance parameter estimation mumbojumbo. Jared Tobin Approximate conditional inference
  • 25. Conditional inference For simplicity, recall the general nonstratified nuisance parameter model.. i.e. Y ∼ f (y ; ψ, θ). Start with some theory. Let (t1 , t2 ) be jointly sufficient for (ψ, θ) and let a be ancillary. If we could factorize the likelihood as L(ψ, θ) ≈ L(θ; t2 |a)L(ψ, θ; t1 |t2 , a) or L(ψ, θ) ≈ L(θ; t2 |t1 , a)L(ψ, θ; t1 |a) then we could maximize L(θ; t2 |a), called the marginal likelihood, or L(θ; t2 |t1 , a), the conditional likelihood, to obtain an estimate of θ. Jared Tobin Approximate conditional inference
  • 26. Conditional inference Each of these functions condition on statistics that contain all of the information about θ, and negligible information about ψ. They seek to eliminate the effect of ψ when estimating θ, and thus theoretically solve the nuisance parameter problem. (both the marginal and conditional likelihoods are special cases of Cox’s partial likelihood function) Jared Tobin Approximate conditional inference
  • 27. Approximate conditional inference? Disclaimer: theory vs. practice It is pretty much impossible to show that a factorization like this even exists in practice. The best we can usually do is try to approximate the conditional or marginal likelihood. Jared Tobin Approximate conditional inference
  • 28. Approximations There are many, many ways to approximate functions and integrals.. We’ll briefly touch on an important one.. (recall the title of this talk for a hint) .. the saddlepoint approximation, which is a highly accurate approximation to arbitrary functions. Often it’s capable of outperforming more computationally demanding methods, i.e. Metropolis Hastings/Gibbs MCMC. For a few cases (normal, gamma, inverse gamma densities), it’s even exact. Jared Tobin Approximate conditional inference
  • 29. Laplace approximation Familiar with the Laplace approximation? We need it first. We’re b interested in a f (y ; θ)dy for some a < b, and the idea is to use e −g (y ;θ) , for g (y ; θ) = − log f (y ; θ) to do it. Truncate a Taylor expansion of e −g (y ;θ) about y , where ˆ y = argmaxy g on (a, b). ˆ Then integrate over (a, b).. we wind up integrating the kernel of a N(ˆ , −1/g (ˆ )) density and get y y 1 b 2π 2 f (y ; θ)dy ≈ exp {g (ˆ )} − y a g (ˆ ) y It works because the value of the integral depends mainly on g (ˆ ) y (the function’s maximum on (a, b)) and g (ˆ ) (its curvature at the y maximum). Jared Tobin Approximate conditional inference
  • 30. Saddlepoint approximation We can then do some nifty math in order to refine Taylor’s approximation to a function. Briefly, we want to relate the cumulant generating function to its corresponding density by creating an approximate inverse mapping. For K (t) the cumulant generating function, the moment generating function can be written e K (t) = e ty +log f (y ;θ) dy Y so fix t and let g (t, y ) = −ty − log f (y ; θ). Laplace’s approximation yields 2π e K (t) ≈ e tyt f (yt ; θ) g (t, yt ) where yt solves g (t, yt ) = 0 Jared Tobin Approximate conditional inference
  • 31. Saddlepoint approximation Sparing the nitty gritty, we do some solving and rearranging (particularly involving the saddlepoint equation K (t) = y ), and we come up with −1/2 f (yt ; θ) ≈ 2πK (t) exp {K (t) − tyt } where yt solves the saddlepoint equation. This guy is called the unnormalized saddlepoint approximation to ˆ f , and is typically denoted f . Jared Tobin Approximate conditional inference
  • 32. Saddlepoint approximation We can normalize the saddlepoint approximation by using ˆ c = Y f (y ; θ)dy . ˆ Call f ∗ (y ; θ) = c −1 f (y ; θ) the renormalized saddlepoint approximation. How does it perform relative to Taylor’s approximation? For a sample of size n, Taylor’s approximation is typically accurate to O(n−1/2 ).. the unnormalized saddlepoint approximation is accurate to O(n−1 ), while the renormalized version does even better at O(n−3/2 ). If small samples are involved, the saddlepoint approximation can make a big difference. Jared Tobin Approximate conditional inference
  • 33. The p ∗ formula We could use the saddlepoint approximation to directly approximate the marginal likelihood. (or could we? where would we start?) Best to continue from Barndorff-Nielsen’s idea.. he and Cox did some particularly horrific math in the 80’s and came up with a second-order approximation to the distribution of the MLE. Briefly, it involves taking a regular exponential family and then using an unnormalized saddlepoint approximation to approximate the distribution of a minimal sufficient statistic.. make a particular reparameterization and renormalize, and you get the p ∗ formula: L(θ; y ) p ∗ (θ; θ) = κ(θ)|j(θ)|1/2 ˆ ˆ ˆ L(θ; y ) where κ is a renormalizing constant (depending on θ) and j is the observed information. Jared Tobin Approximate conditional inference
  • 34. Putting likelihood asymptotics to work So how does the p ∗ formula help us approximate the marginal likelihood? Let t = (t1 , t2 ) be a minimal sufficient statistic and u be a statistic ˆ ˆ ˆ such that both (ψ, u) and (ψ, θ) are one-to-one transformations of t, with the distribution of u depending only on θ. Barndorff-Nielsen (who else) showed that the marginal density of u can be written as ˆ ˆ ∂(ψ, θ) ˆ ∂ ψθ f (u; θ) = ˆ ˆ f (ψ, θ; ψ, θ) ˆ / f (ψθ ; ψ, θ|u) ˆ ∂(ψ, u) ∂ψ ˆ Jared Tobin Approximate conditional inference
  • 35. Putting likelihood asymptotics to work It suffices that we know ∂ψ ˆ −1 ˆ ˆ ˆ = jψψ (λθ ) lψ;ψ (λθ ; ψ, u) ˆ ˆ ∂ ψθ ˆ ˆ ˆ ˆ where λθ = (θ, ψθ ) and |lψ;ψ (λθ ; ψ, u)| is the determinant of a ˆ ˆ ˆ sample space derivative, defined as the matrix ∂ 2 l(λ; λ, u)/∂ψ∂ ψ T . ˆ ˆ ˆ We don’t need to worry about the other term |∂(ψ, θ)/∂(ψ, u)|. It doesn’t depend on θ. Jared Tobin Approximate conditional inference
  • 36. Putting likelihood asympotics to work ˆ ˆ We can use the p ∗ formula to approximate both f (ψ, θ; ψ, θ) and ˆθ ; ψ, θ|u). Doing so, we get f (ψ 1/2 ˆ ˆ j(ψ, θ) ˆ −1 L(ψ, θ) L(ψθ , θ) ˆ ˆ L(θ; u) ∝ j(ψθ , θ) lψ;ψ (ψθ , θ) ˆ ˆ 1/2 ˆ ˆ L(ψ, θ) L(ψ, θ) j(ψθ , θ) 1/2 −1 ˆ ˆ ∝ L(ψθ , θ) j(ψθ , θ) ˆ lψ;ψ (ψθ , θ) ˆ 1/2 −1 ˆ ˆ = L(P) (θ; ψθ ) jψψ (θ, ψθ ) ˆ lψ;ψ (ψθ , θ) ˆ Jared Tobin Approximate conditional inference
  • 37. Modified profile likelihood (MPL) Taking the logarithm, we get ˆ 1 ˆ ˆ l(θ; u) ≈ l (P) (θ; ψθ ) + log jψψ (θ, ψθ ) − log lψ;ψ (θ, ψθ ) . ˆ 2 known as the modified profile likelihood for θ and denoted l (M) (θ). As it’s based on the saddlepoint approximation, it is a highly accurate approximation to the marginal likelihood l(θ; u) and thus (from before) L(θ; t2 |a). In cases where the marginal or conditional likelihood do not exist, it can be thought of as an approximate conditional likelihood for θ. Jared Tobin Approximate conditional inference
  • 38. Two-index asymptotics for the MPL Recall the stratified model with H strata.. how does the modified profile likelihood perform in a two-index asymptotic setting? If nh → ∞ faster than H, we have a similar bound as before: ˆ θ(M) − θ = Op (n−1/2 ) The difference this time is that nh must only increase without bound faster than H 1/3 , which is a much weaker condition. If H → ∞ faster than nh , then we have a boost in performance ˆ −2 over the profile likelihood in that θ(M) − θ = Op (nh ) (as opposed −1 to Op (nh )). Jared Tobin Approximate conditional inference
  • 39. Modified profile likelihood (MPL) ˆ The profile observed information term jψψ (θ, ψθ ) in the MPL corrects the profile likelihood’s habit of putting excess information on θ. What about the sample space derivative term lψ;ψ ? ˆ .. this preserves the structure of the parameterization. If θ and ψ are not parameter orthogonal, this term ensures that parameterization invariance holds. What if θ and ψ are parameter orthogonal? Jared Tobin Approximate conditional inference
  • 40. Adjusted profile likelihood (APL) If θ and ψ are orthogonal, we can do without the sample space derivative.. .. we can define 1 ˆ l (A) (θ) = l (P) (θ) − log jψψ (θ, ψθ ) 2 as the adjusted profile likelihood, which is equivalent to the MPL when θ and ψ are parameter orthogonal. As a special case of the MPL, the APL has comparable performance as long as θ and ψ are approximately orthogonal. Jared Tobin Approximate conditional inference
  • 41. MPL vs APL It’s interesting to note the nature of the difference between the MPL and APL.. While the MPL arises via the p ∗ formula, the APL can actually be derived via a lower-order Laplace approximation to the integrated likelihood L(θ) = L(ψ, θ)dψ R   ˆ  2π  ≈ exp l (P) (θ; ψθ ) − ∂ 2 l(ψ,θ)   2 ∂ψ ˆ ψ=ψθ = L(P) (θ; ψθ ) jψψ (ψθ )−1/2 ˆ ˆ Jared Tobin Approximate conditional inference
  • 42. MPL vs APL In practice we can often get away with using the APL. May require assuming that θ and ψ are parameter orthogonal, but this is often the case anyway (i.e. joint mean/dispersion GLMs, mixed models - i.e. REML). In particular, if θ is a scalar, then an orthogonal reparameterization can always be found. The applicability means that the adjustment term ˆ − 1 log jψψ (θ, ψθ ) can be broadly used in GLMs, quasi-GLMs, 2 HGLMs, etc. Jared Tobin Approximate conditional inference
  • 43. Getting back to the problem.. In mine & Noel’s paper, we compared a bunch of estimators for the negative binomial dispersion parameter k.. the most relevant methods to us are maximum (profile) likelihood (ML) adjusted profile likelihood (AML) extended quasi-likelihood (EQL) adjusted extended quasi-likelihood (AEQL) double extended quasi-likelihood (DEQL) adjusted double extended quasi-likelihood (ADEQL) What insight did the whole likelihood asymptotics exercise shed on this? It showed two branches of estimators and developed a theoretical hierarchy in each.. Jared Tobin Approximate conditional inference
  • 44. Insight The EQL function is actually a saddlepoint approximation to an exponential family likelihood.. + yhi + k q(P) (k) = yhi log yh + (yhi + k) log ¯ yh + k ¯ h,i 1 1 yhi − log(yhi + k) + log k − 2 2 12k(yhi + k) .. so it should perform similarly to (but worse than) the MLE. The double extended quasi-likelihood function is actually the EQL function for the strata mean model. And the AEQL function is actually an approximation to the adjusted profile likelihood.. so the adjusted profile likelihood should intuitively perform better. Jared Tobin Approximate conditional inference
  • 45. Insight In our paper, the results didn’t exactly follow this theoretical pattern.. Jared Tobin Approximate conditional inference
  • 46. Insight .. but in that paper we had capped estimates of k at 10k. I decided to throw out (and not resample) nonconverging estimates of k in my thesis. This means I had some information about how many estimates failed to converge, but those estimates didn’t throw off my simulation averages. Surely enough, upon doing that, the estimators performed according to the theoretical hierarchy. Jared Tobin Approximate conditional inference
  • 47. Estimator performance (thesis) Table: Average perfomance measures across all factor combinations, by estimator. ML AML EQL CREQL LNEQL CTEQL Avg. abs. % bias 109.00 30.00 110.00 31.00 33.00 26.00 Avg. MSE 10.21 0.47 10.22 0.47 1.58 0.55 Avg. prop. NC 0.09 0.00 0.09 0.00 0.02 0.00 Table: Ranks of estimator by criterion. Overall rank is calculated as the ranked average of all other ranks. ML AML EQL CREQL LNEQL CTEQL Avg. abs. % bias 5 2 6 3 4 1 Avg. MSE 5 1 6 2 4 3 Avg. prop. NC. 5.5 2 5.5 2 4 2 Overall rank 5 1 6 3 4 2 Jared Tobin Approximate conditional inference
  • 48. End of the story.. The odd estimator out was the one we originally called ADEQL in our paper. It’s the k-root of this guy: yh + k ¯ k yhi (yhi + 2k) 2k log + − − (n − H) = 0 yhi + k yhi + k 6k(yhi + k)2 h,i The whole saddlepoint deal revealed the DEQL part was really just EQL.. so really it’s an adjustment of an approximation to the profile likelihood, and the adjustment itself is degrees-of-freedom based. It performed very well; best in our paper, and second only to the adjusted profile likelihood in my thesis. I since called it the Cadigan-Tobin EQL (or CTEQL) estimator for k. Jared Tobin Approximate conditional inference
  • 49. Future research direction Integrated likelihood Should the adjusted profile likelihood be adopted as a ‘standard’ way to remove nuisance parameters? How does the adjusted profile likelihood compare to the MPL if we depart from parameter orthogonality? If we do find poor performance under parameter non-orthogonality, how difficult is it to approximate a sample space derivative in general? Can autodiff assist with this? Or is there some neat saddlepoint-like approximation that will do the trick? Jared Tobin Approximate conditional inference
  • 50. Acronym treadmill..Russell calls integrated likelihood’GREML’, for Generalized REstrictedMaximum Likelihood.Tacking on ‘INference’ shows thedangers of acronymization..We’re oh-so-close to GREMLIN.. (hopefully that won’t be my greatest contribution to statistics.. ) Jared Tobin Approximate conditional inference
  • 51. Future research direction Other interests Machine learning Information geometry and asymptotics Quantum information theory and L2 -norm probability (?) I would be happy to work with anyone in any of these areas! Jared Tobin Approximate conditional inference
  • 52. ContactEmail: jared@jtobin.caSkype: jahredtobinWebsite/Blog:http://jtobin.caMAS Thesis:http://jtobin.ca/jTobin MAS thesis.pdf Jared Tobin Approximate conditional inference