Linear approximation operators and statistical models

We discuss the approximation properties of sequences of linear operators T_n mapping densities to densities. We give conditions for their convergence, explicit their general form, obtain rates of convergences and generalise the index parameter to obtain nets \{T_n\}_{n \in N}.

Notations. Let (\mathbb{M}, d) be a compact metric space, equipped with a finite measure \mu defined on its Borel \sigma-algebra, and denote by \mathcal{F} \subset L^1 the set of all essentially bounded probability densities on \mathbb{M}. The set \mathcal{F} is then a complete separable metric space under the total variation distance proportional to || f-g ||_1 = \int |f-g| d\mu.

In bayesian statistics, it is of interest to specify a probability measure P on \mathcal{F}, representing uncertainty about which distribution of \mathcal{F} is generating independent observations x_i \in \mathbb{M}. The problem is that \mathcal{F} is usually rather big: by Baire’s category theorem, if \mathbb{M} is not a finite set of points, then \mathcal{F} cannot be written as a countable union of finite dimensional subspaces. To help in prior elicitation, that is to help a statistician specify P, we may decompose \mathcal{F} in simpler parts.

Here, I discuss how to obtain a sequence of approximating finite dimensional sieves \mathcal{S}_n \subset \mathcal{F}, such that \cup_n \mathcal{S}_n is dense in \mathcal{F}. A prior P on \mathcal{F} may then be specified as the countable mixture

P = \sum _{n \geq 1} \alpha_n P_{\mathcal{S}_n}, \quad \alpha_n \geq 0,\, \sum_n \alpha_n = 1,

where P_{\mathcal{S}_n} is a prior on \mathcal{S}_n for all n.

Let me emphasize that the following ideas are elementary.  Some may be found, with more or less generality, in analysis and approximation theory textbooks. It is, however, interesting to recollect the facts relevant in statistical applications.

1. The basics

The finite dimensional sieves \mathcal{S}_n take the form

\mathcal{S}_n =  \left\{ \sum_{i=0}^{m_n} c_i \phi_{i,n} \right\}, \quad m_n \in \mathbb{N}

where the \phi_{i,n} are densities and the coefficients c_i range through some set which we assume contains the simplex \Delta_n = \left\{ (c_i) : \sum c_i = 1,\, c_i \geq 0 \right\}.

The following lemma gives sufficient conditions for \cup_n \mathcal{S}_n to be dense in \mathcal{F}, with the total variation distance.

Lemma 1. Suppose that there exists a measurable partition \{R_{i,n}\} _{i=0}^{m_n} of \mathbb{M}, with \max_i \text{diam}(R_{i,n}) \rightarrow 0, such that:

  1. for all \delta > 0, \sum_{i: d(x, R_{i,n}) > \delta} \mu(R_{i,n}) \phi_{i,n}(x) \rightarrow 0, uniformly in x; and
  2. \sum_i \mu(R_{i,n}) \phi_{i,n} (x) \rightarrow 1, uniformly in x.

Then, \cup_n \mathcal{S}_n is dense in (\mathbb{F},||\cdot||_1). More precisely, the linear operator T_n : f \mapsto \sum_i \int_{R_{i,n}}f d\mu \, \phi_{i,n} maps densities to densities and is such that for all integrable h, ||T_n h - h||_1 \rightarrow 0 and for all continuous g, ||T_n g - g||_\infty \rightarrow 0.

Proof: We first show that ||T_n g - g||_\infty \rightarrow 0, for all continuous g. The method of proof is well-known.

The fact that T_n is linear and maps densities to densities is easily verified. It follows that T_n monotonous (f < h \Rightarrow T_n f < T_n h). Now, let \varepsilon > 0. By hypothesis (2), we can suppose that T_n 1 = 1. Thus for all x and by the monotonicity of T_n, |T_n g\, (x) -g(x)| = |T_n(g-g(x))\, (x)| \le T_n|g-g(x)|\,(x). Since \mathbb{M} is compact, g is absolutely continuous and there exists \delta > 0 such that d(x, t) < \delta \Rightarrow |g(x)-g(t)| < \varepsilon. Take n sufficienly large so that \max_i \text{diam}(R_{i,n}) < \delta / 2. We have

T_n|g-g(x)|\,(x) = \sum_{i : d(x, R_{i,n}) < \delta/2} \int_{R_{i,n}} |g(t) - g(x)|\mu(dt) \phi_{i,n}(x) + \sum_{i : d(x, R_{i,n} \geq \delta/2} \int_{R_{i,n}} |g(t) - g(x)|\mu(dt) \phi_{i,n}(x).

The first sum is bounded above by \varepsilon, independently of x, and the second sum goes uniformly to 0. Therefore ||T_n g - g||_\infty \rightarrow 0.

We now show that ||T_n h - h||_1 \rightarrow 0 for all integrable h. Let \varepsilon > 0. The space of continuous functions is dense in L^1; there exists a continuous g with ||g-h||_1 < \varepsilon. Therefore, ||T_n h - h||_1 \le ||T_n h -T_n g||_1 + ||T_ng - g||_1 + ||g-h||_1. Because T_n maps densities to densities it is of norm 1 and ||T_n h - T_ng||_1 \le ||h-g||_1. Thus for n sufficiently large so that ||T_n g - g||_\infty < \varepsilon / \mu(\mathbb{M}), we obtain {}||T_n h - h||_1 \le 3\varepsilon.

Finally, T_n(\mathcal{F}) \subset \mathcal{S}_n and the preceding implies \cup_n T_n(\mathcal{F}) is dense in \mathcal{F}. QED.

1.1 Examples.

  1. On the unit interval [0,1] with the Lebesgue measure, the densities {}\phi_{i,n} = (n+1)\mathbb{I}_{\left[\frac{i}{n+1}, \frac{i+1}{n+1}\right)}, with {}\phi_{n,n} = (n+1)\mathbb{I}_{\left[\frac{n}{n+1}, 1 \right]}, obviously satisfy the hypotheses of the preceding lemma. Here, \mathbb{I}_A is the indicator function of the set A.
  2. The indicator functions above may be replaced by the Bernstein polynomial densities \phi_{i,n}(x) = (n+1){n \choose i}x^i(1-x)^{n-i}. The conditions of the lemma are also satisfied and the proof is relatively straightforward.

Note that any operator of the form T_n : f \mapsto \sum_i \int_{R_{i,n}} f\, d\mu \phi_{i,n} may be decomposed as T_n = S_n \circ H_n, where H_n is the histogram operator H_n f = \sum_i \int_{R_{i,n}} f d\mu\, \mu(R_{i,n})^{-1}\mathbb{I}_{R_{i,n}} and S_n\left( \sum_i c_i \, \mu(R_{i,n})^{-1}\mathbb{I}_{R_{i,n}} \right) = \sum_i c_i \phi_{i,n}. In other words, calculating T_n f is the process of reducing f to an associated histogram and then smoothing it.

A dual approximation process.

Consider the histogram operator H_n : f \mapsto \sum_i \int \mathbb{I}_{R_{i,n}} f d\mu \,\mu(R_{i,n})^{-1}\mathbb{I}_{R_{i,n}} and suppose that \sum_i \mu(R_{i,n})\phi_{i,n} = 1. Instead of replacing the densities \mu(R_{i,n})^{-1}\mathbb{I}_{R_{i,n}} on the outside of the integral by \phi_{i,n}, as we did before, we may replace the partition of unity {}\mathbb{I}_{R_{i,n}} inside the integral by the partition of unity {}\mu(R_{i,n})\phi_{i,n}. This yields the following histogram operator \tilde{H}_n:

\tilde{H}_n f = \sum_i \int f \phi_{i,n} d\mu \, \mathbb{I}_{R_{i,n}}.

It can also be extended to act on measures, by letting {}\tilde{H}_n \lambda = \sum_i \int \phi_{i,n} d\lambda \, \mathbb{I}_{R_{i,n}}.

The preceding is of interest in kernel density estimation: given the empirical distribution \lambda_n = \frac{1}{n} \sum_{i=1}^{n} \delta_{x_i} of observed data (x_i), a possible density estimate is \tilde{H}_n \lambda_n. Binning the data through the integral \int \phi_{i,n} d\lambda_n rather than \int_{R_{i,n}} d\lambda_n = \#\{x_i | x_i \in R_{i,n}\}/n can reduce the sensitivity of the density estimate to the choice of bins R_{i,n}.

1.2 The general form of linear operators.

Let T_n: L^1 \rightarrow L^1 be a sequence of positive linear operators mapping densities to densities and such that T_n 1 = 1. Then for each x and n, there exists a random variable Y_n(x) such that T_n f (x) = \text{E}[f(Y_n(x))]. This is a direct consequence of the Riesz representation theorem for positive functionals on \mathcal{C}(\mathbb{M}). In particular, if Y_n(x) admits a density K_n(x, \cdot), then

T_n f (x) = \int f(t) K_n(x,t) \mu(dt).

Note that for general random variables Y_n(x), the function x \mapsto \text{E}[f(Y_n(x))] may not be a density.

A sufficient condition so that \text{E}[f(Y_n(x))] \rightarrow f(x), uniformly in x and for all continuous g, is the following:

  • for all \delta > 0, P(|Y_n(x) - x| > \delta) \rightarrow 0, uniformly in x, meaning that the random variables Y_n(x) satisfy the weak law of large numbers uniformly in x.

Indeed, for all \delta > 0 and continuous f,

\text{E}[|f(Y_n(x)) - f(x)|] = \text{E}[|f(Y_n(x)) - f(x)| \mathbb{I}(|Y_n(x) - x| < \delta)] + \text{E}[|f(Y_n(x)) - f(x)| \mathbb{I}(|Y_n(x) - x| \geq \delta)].

The first mean on the RHS is bounded by any \varepsilon > 0 when \delta is sufficiently small. The second mean is bounded by a constant multiple of P(|Y_n(x) - x| > \delta), which goes to zero as n\rightarrow \infty.

Rates of convergence

Let f be a continuous density and w_f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{\geq 0} be a modulus of continuity, for instance

w_f(\delta) = \sup_{d(x,y) < \delta} |f(x) - f(y)|.

For all \delta > 0, we have

w_f(d(x, t)) \le w_f(\delta)(1+\delta^{-1}d(x,t)).

Therefore, for any sequence \delta_n > 0, a calculation yields

{}||\text{E}[f(Y_n(x))] - f(x)||_\infty \le w_f(\delta_n) \left( 2 + \text{E}\left[ \frac{d(Y_n(x), x)}{\delta_n} \mathbb{I}\left(d(Y_n(x), x) \geq \delta_n\right) \right] \right).

In euclidean space, for example, when \text{E}[Y_n(x)] = x and \text{Var}(Y_n(x)) exists, \sigma_n^2 \geq\text{Var}(Y_n(x)) for all x,  we find

||\text{E}[f(Y_n(x))] - f(x)||_\infty = \mathcal{O}(w_f(\sigma_n)).

1.3 Introducing other parameters

We may index our operators by general parameters \theta \in \Theta, whenever \Theta is a directed set (i.e. for all \theta_1, \theta_2 \in \Theta, there exists \theta_3 \in \Theta such that \theta_1 \le \theta_3 and \theta_2 \le \theta_3). The sequence \{T_n\}_{n \in \mathbb{N}} then becomes the net \{T_\theta\}_{\theta \in \Theta}. We say that \lim ||T_\theta f - f||_\infty = 0 if for all \varepsilon > 0 there exists \theta_\varepsilon such that \theta \geq \theta_\varepsilon implies ||T_\theta f - f||_\infty < \varepsilon.

For example, we can consider the tensor product operator T_{n,m} = T_n \otimes T_m acting on the space of product densities, where \mathbb{N} \times \mathbb{N} is ordered as (n,m) \le (n', m') iff n\le n' and m \le m'.

These extensions are straightforward; the point is that many cases can be treated under the same formalism.

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