Laplace approximation of a beta posterior
Laplace's approximation is one technique we can use to approximate the denominator. (The rest of this post closely follows Reference 1.). Enter marquis de Laplace In my first post on Bayesian data works using grid approximation to arrive at posterior distributions for our. We have seen that the Beta distribution (s+1, n-s+1) provides an estimate of the binomial probability p when we have observed s successes in n independent. RELATIONSHIP BETWEEN FOURIER AND LAPLACE TRANSFORMS TUTORIAL
This provides the computational advantages see Rue and Held that reduce computation time of model fitting. Furthermore, Havard Rue, Martino, and Chopin develop a new approximation to the posterior marginal distributions of the model parameters based on the Laplace approximation see, for example, MacKay Note that some of them may be missing observations. Furthermore, these observations will have an associated likelihood not necessarily from the exponential family.
Observations will be independent given its linear predictor, i. INLA will take advantage of this sparse structure as well as conditional independence properties of GMRFs in order to speed up computations. For the observations with missing values, their predictive distribution can be easily computed by INLA as described in Chapter A Gaussian approximation is possible, but Havard Rue, Martino, and Chopin obtain better approximations by resorting to other methods, such as the Laplace approximation.
The approximation to the marginals of the latent effects requires integrating the hyperparameters out and marginalizing over the latent effects. This is done using a quasi-Newton method. This will provide a set of configurations of the hyperparameters about the posterior mode that can be used in the numerical integration procedures required in INLA.
This is often known as the grid strategy. This can be more efficient that the grid strategy as the dimension of the hyperparameter space increases. Both integration strategies are exemplified in Figure 2. For the grid strategy, the log-density is explored along the axes black dots first. For the CCD approach, a number of points that fill the space are chosen using a response surface approach. Havard Rue, Martino, and Chopin point out that this method worked well in many cases.
What about their pass defense? Is this more or less related to score differential than rushing? A 3 yard rush on 3rd down with 2 yards to go, is a lot more valuable than a 3 yard rush on 3rd down with 15 yards to go. The rushing differential also displays a positive relationship, but not quite as strong.
Remember, all we need to fully describe a Gaussian distribution is the mean and variance. To make this simpler and more realistic from point of view of gaining an advantage: a 0. Meanwhile a 0. But my prior says run the ball! I completely glossed over the prior distributions in setting up this model. The ones I used above are really weak priors, that give you results pretty close to just using good old lm for your standard linear regression. Your team has just hired Brian Schottenheimer to be the offensive coordinator.
Brian loves to run the football 5 , in fact he tells you that from his 20 years of professional coaching experience, he knows that the run game is much more important than the passing game. So how do we account for his prior knowledge or to be blunt, his bias? Priors play an important role in Bayesian data analysis precisely for this reason, they can have a major impact on your results hence why many people prefer to avoid working with them.
Obviously this analysis was pretty simple, and much more work would need to be done for really assessing the importance of passing and rushing in the NFL. But even with this simple approach, we can quantify in terms of points how much more valuable passing is relative to rushing.
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Bernoulli and Beta distributions. However, it is rarely possible to compute the posterior distribution in closed form, hence the need for approximations. Laplace approximation. We consider the Laplace approximation of a Beta random variable, that is a Gaussian with mean at the mode of the original density and variance equal to the inverse of the second derivative of the log-density. Left: densities. Right: negative log-densities translated so that they have matched first two derivatives at their minimum.
Extensions High-order expansion. In more than one dimension, that quickly gets complicated see, e. Stationary phase approximation. This can be further extended to more general complex integrals. Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator. References  Pierre-Simon Laplace. The LaplacesDemon function uses the LaplaceApproximation algorithm to optimize initial values and save time for the user.
Most optimization algorithms assume that the logarithm of the unnormalized joint posterior density is defined and differentiable. Some methods calculate an approximate gradient for each initial value as the difference in the logarithm of the unnormalized joint posterior density due to a slight increase in the parameter. The step size parameter, which is often plural and called rate parameters in other literature, is adapted each iteration with the univariate version of the Robbins-Monro stochastic approximation in Garthwaite The step size shrinks when a proposal is rejected and expands when a proposal is accepted.
Gradient ascent is criticized for sometimes being relatively slow when close to the maximum, and its asymptotic rate of convergence is inferior to other methods. However, compared to other popular optimization algorithms such as Newton-Raphson, an advantage of the gradient ascent is that it works in infinite dimensions, requiring only sufficient computer memory.
Although Newton-Raphson converges in fewer iterations, calculating the inverse of the negative Hessian matrix of second-derivatives is more computationally expensive and subject to singularities. Therefore, gradient ascent takes longer to converge, but is more generalizable. BFGS may be the most efficient and popular quasi-Newton optimiziation algorithm.
As a quasi-Newton algorithm, the Hessian matrix is approximated using rank-one updates specified by approximate gradient evaluations. Since BFGS is very popular, there are many variations of it. This is a version by Nash that has been adapted from the Rvmmin package, and is used in the optim function of base R.
The approximate Hessian is not guaranteed to converge to the Hessian. The BHHH algorithm is a quasi-Newton method that includes a step-size parameter, partial derivatives, and an approximation of a covariance matrix that is calculated as the inverse of the sum of the outer product of the gradient OPG , calculated from each record. The OPG method becomes more costly with data sets with more records.
Since partial derivatives must be calculated per record of data, the list of data has special requirements with this method, and must include design matrix X, and dependent variable y or Y. Records must be row-wise. An advantage of BHHH over NR see below is that the covariance matrix is necessarily positive definite, and gauranteed to provide an increase in LP each iteration given a small enough step-size , even in convex areas. The covariance matrix is better approximated with larger data sample sizes, and when closer to the maximum of LP.
CG uses partial derivatives, but does not use the Hessian matrix or any approximation of it. CG usually requires more iterations to reach convergence than other algorithms that use the Hessian or an approximation. CG was originally developed by Hestenes and Stiefel , though this version is adapted from the Rcgminu function in package Rcgmin.
DFP was the first popular, multidimensional, quasi-Newton optimization algorithm. The DFP update of an approximate Hessian matrix maintains symmetry and positive-definiteness. When DFP is used, the approximate Hessian is not used to calculate the final covariance matrix. The length parameter is adapted each iteration with the univariate version of the Robbins-Monro stochastic approximation in Garthwaite The length shrinks when a proposal is rejected and expands when a proposal is accepted. This was adapted from the HJK algorithm in package dfoptim.
Hooke-Jeeves is a derivative-free, direct search method. Each iteration involves two steps: an exploratory move and a pattern move. The exploratory move explores local behavior, and the pattern move takes advantage of pattern direction.
It is sometimes described as a hill-climbing algorithm. If the solution improves, it accepts the move, and otherwise rejects it. Step size decreases with each iteration. The decreasing step size can trap it in local maxima, where it gets stuck and convergences erroneously. Users are encouraged to attempt again after what seems to be convergence, starting from the latest point. Although getting stuck at local maxima can be problematic, the Hooke-Jeeves algorithm is also attractive because it is simple, fast, does not depend on derivatives, and is otherwise relatively robust.
LM uses partial derivatives and approximates the Hessian with outer-products. It is suitable for nonlinear optimization up to a few hundred parameters, but loses its efficiency in larger problems due to matrix inversion. LM is considered between the Gauss-Newton algorithm and gradient descent.
When far from the solution, LM moves slowly like gradient descent, but is guaranteed to converge. This was adapted from the lsqnonlin algorithm in package pracma. Nelder-Mead is a derivative-free, direct search method that is known to become inefficient in large-dimensional problems. As the dimension increases, the search direction becomes increasingly orthogonal to the steepest ascent usually descent direction.
However, in smaller dimensions, it is a popular algorithm. At each iteration, three steps are taken to improve a simplex: reflection, extension, and contraction. Newton-Raphson uses derivatives and a Hessian matrix. The algorithm is included for its historical significance, but is known to be problematic when starting values are far from the targets, and calculating and inverting the Hessian matrix can be computationally expensive.
As programmed here, when the Hessian is problematic, it tries to use only the derivatives, and when that fails, a jitter is applied. A swarm of particles is moved according to velocity, neighborhood, and the best previous solution. The neighborhood for each particle is a set of informing particles. PSO is derivative-free. PSO has been adapted from the psoptim function in package pso. A weight element in a weight vector is associated with each approximate gradient. A weight element is multiplied by 1.
The weight vector is the step size, and is constrained to the interval [0. This algorithm has special requirements for the Model specification function and the Data list. The leader does not move in each iteration, and a line-search is used for each non-leader, up to three times the difference in parameter values between each non-leader and leader.
This algorithm is derivative-free and often considered in the family of evolution algorithms. Numerous model evaluations are performed per non-leader per iteration. This algorithm was adapted from package soma. SPG is a non-monotone algorithm that is suitable for high-dimensional models. The approximate gradient is used, but the Hessian matrix is not. SPG has been adapted from the spg function in package BB.
SR1 is a quasi-Newton algorithm, and the Hessian matrix is approximated, often without being positive-definite. At the posterior modes, the true Hessian is usually positive-definite, but this is often not the case during optimization when the parameters have not yet reached the posterior modes.
Other restrictions, including constraints, often result in the true Hessian being indefinite at the solution. When SR1 is used, the approximate Hessian is not used to calculate the final covariance matrix. The TR algorithm attempts to reach its objective in the fewest number of iterations, is therefore very efficient, as well as safe. The efficiency of TR is attractive when model evaluations are expensive.
The Hessian is approximated each iteration, making TR best suited to models with small to medium dimensions, say up to a few hundred parameters. TR has been adapted from the trust function in package trust. References Azevedo-Filho, A. Bernardo, J. Berndt, E. Annals of Economic and Social Measurement, 3, p. Broyden, C. The New Algorithm". Journal of the Institute of Mathematics and its Applications, 6, p. Fletcher, R. Computer Journal, 13 3 , p. Garthwaite, P. Mathematics of Computation, 24 , p.
Hestenes, M. Journal of Research of the National Bureau of Standards, 49 6 , p. Hooke, R. Journal of the Association for Computing Machinery, 8 2 , p. Kass, R. Journal of the American Statistical Association, 90 , p. Laplace, P. English translation by S. ISBN , translated from the French 6th ed. Levenberg, K. Quarterly of Applied Mathematics, 2, p. Lewis, S. Journal of the American Statistical Association, 92, p.
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