# Maximum-likelihood Estimate Get ML estimate for product-binomial/multinomial model with linear inequality constraints. ```r ml_binom(k, n, A, b, map, strategy, n.fit = 3, start, progress = FALSE, ...) ml_multinom(k, options, A, b, V, n.fit = 3, start, progress = FALSE, ...) ``` ## Arguments - `k`: vector of observed response frequencies. - `n`: the number of choices per item type. If `k=n=0`, Bayesian inference is relies on the prior distribution only. - `A`: a matrix with one row for each linear inequality constraint and one column for each of the free parameters. The parameter space is defined as all probabilities `x` that fulfill the order constraints `A*x <= b`. - `b`: a vector of the same length as the number of rows of `A`. - `map`: optional: numeric vector of the same length as `k` with integers mapping the frequencies `k` to the free parameters/columns of `A`/`V`, thereby allowing for equality constraints (e.g., `map=c(1,1,2,2)`). Reversed probabilities `1-p` are coded by negative integers. Guessing probabilities of .50 are encoded by zeros. The default assumes different parameters for each item type: `map=1:ncol(A)` - `strategy`: a list that defines the predictions of a strategy, see`strategy_multiattribute`. - `n.fit`: number of calls to constrOptim . - `start`: only relevant if `steps` is defined or `cmin>0`: a vector with starting values in the interior of the polytope. If missing, an approximate maximum-likelihood estimate is used. - `progress`: whether a progress bar should be shown (if `cpu=1`). - `...`: further arguments passed to the function `constrOptim`. To ensure high accuracy, the number of maximum iterations should be sufficiently large (e.g., by setting `control = list(maxit = 1e6, reltol=.Machine$double.eps^.6),outer.iterations = 1000`. - `options`: number of observable categories/probabilities for each item type/multinomial distribution, e.g., `c(3,2)` for a ternary and binary item. - `V`: a matrix of vertices (one per row) that define the polytope of admissible parameters as the convex hull over these points (if provided, `A` and `b` are ignored). Similar as for `A`, columns of `V` omit the last value for each multinomial condition (e.g., a1,a2,a3,b1,b2 becomes a1,a2,b1). Note that this method is comparatively slow since it solves linear-programming problems to test whether a point is inside a polytope (Fukuda, 2004) or to run the Gibbs sampler. ## Returns the list returned by the optimizer `constrOptim`, including the input arguments (e.g., `k`, `options`, `A`, `V`, etc.). * If the Ab-representation was used, `par` provides the ML estimate for the probability vector $\theta$. * If the V-representation was used, `par` provides the estimates for the (usually not identifiable) mixture weights $\alpha$ that define the convex hull of the vertices in $V$, while `p` provides the ML estimates for the probability parameters. Because the weights must sum to one, the $\alpha$-parameter for the last row of the matrix $V$ is dropped. If the unconstrained ML estimate is inside the convex hull, the mixture weights $\alpha$ are not estimated and replaced by missings (`NA`). ## Details First, it is checked whether the unconstrained maximum-likelihood estimator (e.g., for the binomial: `k/n`) is inside the constrained parameter space. Only if this is not the case, nonlinear optimization with convex linear-inequality constrained is used to estimate (A) the probability parameters $\theta$ for the Ab-representation or (B) the mixture weights $\alpha$ for the V-representation. ## Examples ```r # predicted linear order: p1 < p2 < p3 < .50 # (cf. WADDprob in ?strategy_multiattribute) A <- matrix( c( 1, -1, 0, 0, 1, -1, 0, 0, 1 ), ncol = 3, byrow = TRUE ) b <- c(0, 0, .50) ml_binom(k = c(4, 1, 23), n = 40, A, b)[1:2] ml_multinom( k = c(4, 36, 1, 39, 23, 17), options = c(2, 2, 2), A, b )[1:2] # probabilistic strategy: A,A,A,B [e1

ml_binom function