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The product response law has the product density

Statement

In the setup of Explicit exponential-family response contracts, let $P_{θ} = ⨂_{i = 1}^{n} P_{i, θ_{i}, ϕ}$ and $ν = ⨂_{i = 1}^{n} ν_{i}$ . Then the joint sampling law has density

$q_{θ} (y_{1}, \dots, y_{n}) = \prod_{i = 1}^{n} f_{i} (y_{i}; θ_{i}, ϕ)$

relative to $ν$ . At any $y$ for which every factor $f_{i} (y_{i}; θ_{i}, ϕ)$ is positive, the log likelihood is $\log q_{θ} (y) = \sum_{i} \log f_{i} (y_{i}; θ_{i}, ϕ)$ .

Facts & Assumptions

Given: The finite collection of sigma-finite component spaces, normalized densities, fixed supports, and product sampling law in Explicit exponential-family response contracts.

[F1]

Explicit exponential-family response contracts: each $f_{i}$ is a nonnegative measurable probability density relative to $ν_{i}$ , and the joint sampling law is the finite product of the component laws.

[F2]

The product sigma-algebra and its finite iterates: the finite product sigma algebra is generated recursively by measurable rectangles.

[F3]

For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique: for sigma-finite factors, the product measure is a measure and satisfies $(ν_{1} \times ν_{2}) (A_{1} \times A_{2}) = ν_{1} (A_{1}) ν_{2} (A_{2})$ .

[F4]

Arithmetic and lattice operations preserve measurability whenever they are defined: finite pointwise products of measurable real-valued functions are measurable.

[F5]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: for nonnegative product-measurable functions on sigma-finite product spaces, the integral equals either iterated integral.

[F6]

The measure with density $f$ relative to $μ$ : if $q$ is nonnegative measurable, $q d ν$ is the measure $A \mapsto \int_{A} q d ν$ .

[F7]

Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system: measures agreeing on a generating pi-system and on an increasing finite-measure exhaustion from that pi-system are equal.

Proof

1.1F1F2F4

For each coordinate projection $π_{i} (y) = y_{i}$ , the preimage of a measurable set is a measurable rectangle, so $f_{i} \circ π_{i}$ is product-measurable by [F2]. Their finite product $q_{θ} = \prod_{i} (f_{i} \circ π_{i})$ is product-measurable by [F4].

2.1F1F3F5step 1.1

Construct $ν$ and $P_{θ}$ by repeated finite products using [F3]. Repeated Tonelli [F5], applied to the nonnegative factors, gives $\int_{\prod_{i} A_{i}} q_{θ} d ν = \prod_{i} \int_{A_{i}} f_{i} d ν_{i} = \prod_{i} P_{i, θ_{i}, ϕ} (A_{i}) = P_{θ} (\prod_{i} A_{i})$ for every measurable rectangle. Taking the rectangle to be the whole product and using normalization from [F1] gives $\int q_{θ} d ν = 1$ .

3.1F1F2F6F7step 2.1algebra∎

By [F6], $Q (A) : = \int_{A} q_{θ} d ν$ is a probability measure, and step 2.1 shows that $Q$ and $P_{θ}$ agree on the rectangle pi-system. That system contains the whole product, which is a one-set increasing finite-measure exhaustion for both probability measures, so [F2] and [F7] give $Q = P_{θ}$ . Whenever every factor is positive, taking the logarithm of the product gives the stated sum.

Depends on

Used by

Generalized linear model with explicit conditioning Definition
Likelihood deviance for nested models on common support Definition
Canonical-link score and Hessian Theorem
General-link score and Fisher information Theorem

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Sources

Philippe Rigollet, MIT 18.650 Statistics for Applications, Lecture 10: Generalized Linear Models, PDF pp. 16–21, 30–32 (standard reference, not scraped)
Ryan Tibshirani, Generalized Linear Models, CMU Advanced Methods for Data Analysis, PDF pp. 1–5 (standard reference, not scraped)