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Generalized Bayes loss updates and temperature calibration — Examples

1 · Prerequisites

2 · Summary

Finite and countable constructions make the normalizer and temperature conditions explicit. The unit-temperature likelihood example is a finite Bayesian model calculation; rescaling the loss shows the equivalent change in effective temperature. A separate dyadic prior has an infinite normalizer, and a two-state experiment produces a posterior credible set with zero frequentist coverage. Each calculation states its prior, loss and temperature locally; the coverage and likelihood examples also specify their finite sampling experiments. None asserts a general coverage theorem.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)Open item page →

A measurable loss can have an infinite normalizer

Example

Let $Θ = N$ with its power-set sigma-algebra, and put $a_{k} = (1 / 2)^{k + 1}$ for $k \in N$ . Define the prior $π : = \sum_{k = 0}^{\infty} a_{k} δ_{k},$ where $δ_{k}$ is the Dirac measure at $k$ . The geometric-series formula gives $\sum_{k \geq 0} a_{k} = 1$ , so $π$ is a probability measure. Define the real-valued loss $ℓ (k) : = \log a_{k}$ . At temperature $w = 1$ , every unnormalized atom has mass $a_{k} e^{- ℓ (k)} = 1$ , and the normalizer is infinite; therefore the loss update is not defined.

Facts & Assumptions

Given: The von Neumann natural numbers $N$ , their power-set sigma-algebra, weights $a_{k} = (1 / 2)^{k + 1}$ , and Dirac measures $δ_{k}$ .

[F1]

Generalized Bayes loss update and temperature: at fixed $w > 0$ , $Z = \int e^{- w ℓ} d π$ must be finite and positive for the update to be defined.

[F2]

A countable discrete measurable space is standard Borel: every nonempty at-most-countable set with its power-set sigma-algebra is standard Borel.

[F3]

Finite, countably infinite, countable, uncountable: a set equinumerous with $N$ is at-most-countable.

[F4]

The natural numbers $N$ (von Neumann): $0 \in N$ and $N$ is closed under successor.

[F5]

Nonnegative scalar multiples and countable weighted sums of measures are measures: a countable nonnegative weighted sum of measures on one measurable space is a measure.

[F6]

The Dirac set function at a point: $δ_{k} (A) = 1$ when $k \in A$ and is zero otherwise.

[F7]

A Dirac set function is a probability measure: each $δ_{k}$ is a probability measure.

[F8]

Probability measures and probability spaces: a probability measure has total mass one.

[F9]

The principle of mathematical induction: a property holding at zero and preserved by successor holds for all natural numbers.

[F10]

Integer powers $a^{m}$ : $b^{0} = 1$ and $b^{k + 1} = b^{k} b$ for natural $k$ .

[F11]

For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges: for $∣ r ∣ < 1$ , $\sum_{j = 0}^{\infty} r^{j} = 1 / (1 - r)$ .

[F12]

The natural logarithm as the inverse of the exponential function: $\log$ is inverse to $\exp$ on $(0, \infty)$ .

[F13]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $\exp (t) > 0$ and $\exp (- t) = 1 / \exp (t)$ .

[F14]

Nonnegative simple measurable functions: a finite nonnegative linear combination of indicators of measurable sets is a nonnegative simple measurable function.

[F15]

The nonnegative Lebesgue integral: the nonnegative integral is the supremum of the integrals of nonnegative simple minorants, with values in $[0, + \infty]$ .

[F16]

The integral of a nonnegative simple function: the integral of a simple function represented on disjoint measurable sets is the sum of its coefficient times the measure of each set.

[F17]

The nonnegative integral agrees with the simple integral on simple functions: the nonnegative integral of a simple function equals its simple integral.

[F18]

Monotonicity and nonnegative homogeneity of the nonnegative integral: if $0 \leq f \leq g$ , then $\int f d π \leq \int g d π$ .

[F19]

The Cauchy-sequence reals have the least-upper-bound property: $R$ is a complete ordered field.

[F20]

Every complete ordered field is Archimedean: in a complete ordered field, for every $x$ there is a natural $n \geq 1$ with $x < n$ .

[F21]

Absolute value in an ordered field: if $b \geq 0$ , then $∣ b ∣ = b$ .

[F22]

Ordered field: products of positive reals are positive.

[F23]

The multiplicative identity is positive: $1 > 0$ .

[F24]

Laws of finite sums and finite products: finite sums scale by a real constant, $\sum_{k < n} c u_{k} = c \sum_{k < n} u_{k}$ .

[F25]

Series, partial sums, convergence and the sum, divergence, and the tail series: a convergent series sums to the limit of its finite partial sums.

[F26]

Algebra of limits: sums, scalar multiples, products and quotients: scaling a convergent sequence by a fixed real constant scales its limit by that constant.

[F27]

A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum: for nonnegative terms, the usual convergent series sum is the supremum of its finite partial sums.

[F28]

Series in the nonnegative extended real line: the nonnegative extended sum is the supremum of its finite partial sums.

[F29]

Nonnegative scalar multiples and countable weighted sums of measures: the countable weighted sum is evaluated pointwise by the nonnegative extended sum of its weighted measures.

Verification

1.1F3F4F2given

The identity map shows $N$ is equinumerous with itself, hence at-most-countable by [F3]; [F4] makes it nonempty. Therefore [F2] shows $(Θ, P (Θ))$ is standard Borel.

1.2F21F10F23F22F9F19givenalgebra

Put $b = 1 / 2$ . Ordered-field arithmetic gives $0 < b < 1$ , so [F21] gives $∣ b ∣ = b < 1$ . The powers are positive: $b^{0} = 1 > 0$ by [F10] and [F23], and if $b^{k} > 0$ , then $b^{k + 1} = b^{k} b > 0$ by [F10] and [F22]. Induction [F9] gives $a_{k} = b^{k + 1} > 0$ for every $k$ .

2.1F11F10F24F25F26F27F28step 1.2algebra

By [F11], $\sum_{k = 0}^{\infty} b^{k} = 1 / (1 - b)$ . For every $N$ , recursion [F10] and finite-sum scaling [F24] give $\sum_{k < N + 1} a_{k} = \sum_{k < N + 1} b^{k + 1} = b \sum_{k < N + 1} b^{k} .$ The partial sums of $\sum b^{k}$ converge to $1 / (1 - b)$ by [F11] and [F25]; therefore [F26] gives $\sum_{k = 0}^{\infty} a_{k} = b / (1 - b) = 1$ . By [F27] and [F28], this usual series sum is also the nonnegative extended sum of the coefficients used in the weighted measure.

2.2F12F13step 1.2given

Since $a_{k} > 0$ , $ℓ (k) = \log a_{k}$ is finite. Every inverse image of a Borel subset of $R$ is a subset of $N$ , hence is in $P (N)$ ; therefore $ℓ$ is measurable. By [F12]–[F13], $e^{- ℓ (k)} = 1 / a_{k}$ , so the normalizer is $Z = \int e^{- ℓ} d π$ .

3.1F5F7F6F28F29step 2.1F8algebra

By [F5] and [F7], $π = \sum_{k \geq 0} a_{k} δ_{k}$ is a measure. Its total mass is $π (Θ) = \sum_{k \geq 0} a_{k} δ_{k} (Θ) = \sum_{k \geq 0} a_{k} = 1$ by [F6], [F28], [F29], and step 2.1, so [F8] makes $π$ a probability. For each $k$ , [F6] gives $π ({k}) = a_{k}$ .

4.1F14F13F16F17step 3.1F18F15step 2.2algebra

For each $N \in N$ , set $s_{N} : = \sum_{k = 0}^{N} a_{k}^{- 1} 1_{{k}} .$ It is a nonnegative simple function by [F14], and $0 \leq s_{N} \leq e^{- ℓ}$ : there is equality at $k \leq N$ , and for $k > N$ the left side is zero while $e^{- ℓ (k)} > 0$ by [F13]. By [F16]–[F17] and step 3.1, $\int s_{N} d π = \sum_{k = 0}^{N} a_{k}^{- 1} π ({k}) = N + 1.$ Thus [F18] yields $Z \geq N + 1$ for every $N$ .

5.1F19F20step 4.1F15F1contradiction∎

If $Z$ were finite, [F19]–[F20] would give $N \in N$ , $N \geq 1$ , with $Z < N$ . Step 4.1 gives $Z \geq N + 1$ , a contradiction. Since [F15] places $Z$ in $[0, + \infty]$ , it follows that $Z = + \infty$ . The finite-positive condition in [F1] therefore fails, so no generalized Bayes update is defined.

CounterexampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)Open item page →

A generalized Bayes credible set can have zero coverage

Statement refuted

A positive temperature and a proper generalized Bayes loss update alone guarantee that every posterior $0.95$ -credible set has frequentist coverage at least $0.95$ under the sampling experiment.

Facts & Assumptions

Given: The finite parameter space $Θ = {0, 1}$ , singleton observation space $X = {x_{0}}$ , prior $(1 / 100, 99 / 100)$ , sampling laws $P_{0} = P_{1} = δ_{x_{0}}$ , and the displayed loss at $w = 1$ .

[F1]

A countable discrete measurable space is standard Borel: a nonempty at-most-countable set with its power-set sigma-algebra is standard Borel.

[F2]

Probability measures and probability spaces: a probability measure has total mass one.

[F3]

The Dirac set function at a point: $δ_{x_{0}} (A) = 1$ if $x_{0} \in A$ and is zero otherwise.

[F4]

A Dirac set function is a probability measure: each Dirac set function is a probability measure.

[F5]

Statistical experiments: an experiment consists of a sample measurable space, parameter set and a probability law on the sample space for each parameter.

[F6]

Generalized Bayes loss update and temperature: if $0 < Z = \int e^{- w ℓ_{x}} d π < \infty$ , the update is the normalized measure with that density.

[F7]

A finite-normalizer loss update is a probability measure: each update with a finite positive normalizer is a probability measure.

[F8]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $e^{t} > 0$ and $e^{- t} = 1 / e^{t}$ for real $t$ .

[F9]

The exponential function is strictly increasing: the exponential is strictly increasing on $R$ .

[F10]

Confidence sets and coverage: a confidence-set procedure maps data to parameter subsets with measurable membership sections; coverage at $θ$ is the sampling probability that its set contains $θ$ .

Counterexample

Let $Θ = {0, 1}$ and $X = {x_{0}}$ , each with its power-set sigma-algebra. Use prior $π (0) = 1 / 100$ , $π (1) = 99 / 100$ , and sampling laws $P_{0} = P_{1} = δ_{x_{0}}$ . At the sole observation $x_{0}$ , let $ℓ (0, x_{0}) = 1$ and $ℓ (1, x_{0}) = 0$ , with temperature $w = 1$ . The singleton data-to-set rule $C (x_{0}) = {1}$ is called $0.95$ -credible here when the update $q_{x_{0}}$ satisfies $q_{x_{0}} (C (x_{0})) \geq 0.95$ . It has posterior mass greater than $0.95$ , yet its coverage at true parameter $0$ is zero.

1.1F1F2F3F5F6F8F7F4givenalgebra

The parameter and data spaces are nonempty finite sets, so [F1] makes them standard Borel. The prior masses sum to one, hence [F2] makes $π$ a probability. [F3]–[F5] show the sampling laws define a statistical experiment. [F6]–[F8] give the normalizer $Z = \frac{1}{100} e^{- 1} + \frac{99}{100} e^{0},$ which is finite and positive. By [F7], the resulting update $q$ is a probability measure.

2.1F9F8F6step 1.1algebra

Since $- 1 < 0$ , [F9] gives $e^{- 1} < e^{0}$ . Both are positive by [F8], and $Z = \frac{1}{100} e^{- 1} + \frac{99}{100} e^{0} < e^{0} .$ Therefore $q ({1}) = \frac{(99 / 100) e^{0}}{Z} > \frac{99}{100} > 0.95.$ Thus $C (x_{0}) = {1}$ is a posterior credible set of mass greater than $0.95$ .

Verification

3.1F10step 1.1step 2.1algebra∎

Regard $C$ as the data-to-set procedure with sole value $C (x_{0}) = {1}$ . Its membership section for parameter $0$ is empty, and for parameter $1$ is ${x_{0}}$ ; both are measurable. The coverage definition [F10] therefore applies, and at true parameter $0$ it gives $P_{0} {x : C (x) ∋ 0} = P_{0} (\emptyset) = 0.$ This is below $0.95$ , refuting the proposed universal guarantee.

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)Open item page →

Rescaling loss changes the effective temperature

Example

Let $Θ = {0, 1}$ with its power-set sigma-algebra and prior $π (0) = p$ , $π (1) = 1 - p$ for $0 < p < 1$ . At a fixed observation $x_{0}$ , let $ℓ_{x_{0}} (0) = a$ and $ℓ_{x_{0}} (1) = b$ with $a \neq b$ . For every $w, c > 0$ , replacing this loss section by $c ℓ_{x_{0}}$ at temperature $w$ gives the same update as retaining $ℓ_{x_{0}}$ at temperature $c w$ . At any two distinct positive temperatures, the updates are different.

Facts & Assumptions

Given: The finite parameter space $Θ = {0, 1}$ with its power-set sigma-algebra; an observation $x_{0}$ ; $0 < p < 1$ ; real loss-section values $a \neq b$ ; and positive temperatures $w, c, t_{1}, t_{2}$ with $t_{1} \neq t_{2}$ .

[F1]

A countable discrete measurable space is standard Borel: every nonempty at-most-countable set with its power-set sigma-algebra is standard Borel.

[F2]

Probability measures and probability spaces: a probability measure is a measure of total mass one.

[F3]

Generalized Bayes loss update and temperature: when its normalizer is finite and positive, the update is the measure with density $e^{- t ℓ} / Z_{t}$ relative to the prior.

[F4]

A finite-normalizer loss update is a probability measure: each update with a finite positive normalizer is a probability measure.

[F5]

The exponential function is strictly increasing: the exponential is strictly increasing, hence injective, on $R$ .

[F6]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $e^{u} > 0$ and $e^{- u} = 1 / e^{u}$ for every real $u$ .

Verification

1.1F1F2F6givenalgebra

The set $Θ$ is nonempty and finite, so [F1] makes its power-set measurable space standard Borel. Since $0 < p < 1$ , the masses $p$ and $1 - p$ are positive and sum to one, so [F2] makes $π$ a probability. For every $t > 0$ , the two real loss values give positive finite weights $e^{- t a}$ and $e^{- t b}$ by [F6]; their prior-weighted sum is therefore finite and positive.

2.1F3F4step 1.1algebra

By [F3]–[F4], the update $q_{t} = G_{t, π, x_{0}}$ has masses $q_{t} (0) = \frac{p e^{- t a}}{p e^{- t a} + (1 - p) e^{- t b}}, q_{t} (1) = \frac{(1 - p) e^{- t b}}{p e^{- t a} + (1 - p) e^{- t b}} .$ Consequently its odds satisfy $\frac{q_{t} (1)}{q_{t} (0)} = \frac{1 - p}{p} e^{- t (b - a)} .$ .

2.2step 1.1F3algebra

For $θ \in Θ$ , the unnormalized weight after replacing $ℓ_{x_{0}}$ by $c ℓ_{x_{0}}$ at temperature $w$ is $π (θ) e^{- w c ℓ_{x_{0}} (θ)} = π (θ) e^{- (c w) ℓ_{x_{0}} (θ)}$ . These are exactly the unnormalized masses for $q_{c w}$ . The two updates have the same finite positive normalizer by step 1.1, so their values on both singletons, and hence on every subset of $Θ$ , agree.

3.1F5step 2.1algebra∎

Because $b - a \neq 0$ and $t_{1} \neq t_{2}$ , the real numbers $- t_{1} (b - a)$ and $- t_{2} (b - a)$ differ. Injectivity in [F5] makes their exponentials different. Multiplication by the fixed positive factor $(1 - p) / p$ preserves that difference, so step 2.1 gives distinct odds. On a two-point probability space the odds determine both singleton masses; therefore $q_{t_{1}} \neq q_{t_{2}}$ .

ExampleConstruction: AI-generatedVerification: AI-generatedOpen item page →

The unit-temperature likelihood update is ordinary Bayes in a finite model

Example

Let $Θ = {0, 1}$ have prior masses $π (0) = 1 / 3$ and $π (1) = 2 / 3$ . Let the observation space be $X = {0, 1}$ and define sampling laws by $P_{0} (Y = 1) = 1 / 4$ , $P_{0} (Y = 0) = 3 / 4$ , $P_{1} (Y = 1) = 3 / 4$ , and $P_{1} (Y = 0) = 1 / 4$ . These finite spaces with power-set sigma-algebras are standard Borel, and the displayed laws define a statistical experiment. Take $ℓ (θ, y) = - \log P_{θ} (Y = y)$ for every $(θ, y) \in Θ \times X$ . For observed $y = 1$ at $w = 1$ , the unnormalized masses are $1 / 12$ and $1 / 2$ , so the update is $(1 / 7, 6 / 7)$ . The same values follow by conditioning the joint prior-sampling law on $Y = 1$ .

Facts & Assumptions

Given: The parameter and observation spaces ${0, 1}$ with power-set sigma-algebras, prior $(1 / 3, 2 / 3)$ , and the four displayed sampling probabilities.

[F1]

A countable discrete measurable space is standard Borel: a nonempty at-most-countable set with its power-set sigma-algebra is standard Borel.

[F2]

Probability measures and probability spaces: a probability measure has total mass one.

[F3]

Statistical experiments: a statistical experiment consists of a sample measurable space, a parameter set, and a probability law on the sample space for each parameter.

[F4]

Generalized Bayes loss update and temperature: at $w = 1$ the update weights are proportional to $e^{- ℓ (θ, y)}$ and are normalized by their integral against the prior.

[F5]

A finite-normalizer loss update is a probability measure: a finite-positive-normalizer update is a probability measure.

[F6]

The natural logarithm as the inverse of the exponential function: $\log$ is the inverse of $\exp$ on $(0, \infty)$ .

[F7]

Conditional probability $P (A ∣ B)$ for $P (B) > 0$ : in a finite probability space, $\Pr (A ∣ B) = \Pr (A \cap B) / \Pr (B)$ when $\Pr (B) > 0$ .

Verification

1.1F1F2F3givenalgebra

Both parameter and sample spaces are nonempty finite sets, so [F1] makes them standard Borel. The prior masses are nonnegative and sum to one; each pair of sampling probabilities is nonnegative and sums to one. By [F2] these are probability measures, and [F3] therefore gives the stated finite statistical experiment.

2.1F6F4step 1.1algebra

All four likelihoods are positive, so [F6] makes $ℓ$ real-valued on $Θ \times X$ . Each parameter section is measurable because the parameter sigma-algebra is the power set, satisfying the loss hypotheses of [F4] for every $y \in X$ . At $y = 1$ , the inverse relation in [F6] gives $e^{- ℓ (θ, 1)} = e^{\log P_{θ} (Y = 1)} = P_{θ} (Y = 1)$ . Hence the unnormalized weights are $π (0) P_{0} (Y = 1) = \frac{1}{12}, π (1) P_{1} (Y = 1) = \frac{1}{2},$ and their normalizer is $Z = 1 / 12 + 1 / 2 = 7 / 12$ , finite and positive.

3.1F4F5step 2.1algebra

By [F4]–[F5], the update is a probability measure with $q (0) = \frac{1 / 12}{7 / 12} = \frac{1}{7}, q (1) = \frac{1 / 2}{7 / 12} = \frac{6}{7} .$ .

4.1F7step 1.1step 2.1step 3.1algebra∎

Define the joint masses by $\Pr (θ, y) = π (θ) P_{θ} (Y = y)$ . The four masses are $1 / 4, 1 / 12, 1 / 6, 1 / 2$ at $(0, 0), (0, 1), (1, 0), (1, 1)$ , respectively; they sum to one, and $\Pr (Y = 1) = 7 / 12 > 0$ . Applying [F7] to each parameter singleton gives $\Pr (θ = 0 ∣ Y = 1) = \frac{1 / 12}{7 / 12} = \frac{1}{7}, \Pr (θ = 1 ∣ Y = 1) = \frac{1 / 2}{7 / 12} = \frac{6}{7} .$ Thus the loss update equals the ordinary finite-model posterior at $y = 1$ .

Generalized Bayes loss updates and temperature calibration — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

A measurable loss can have an infinite normalizer

Example

Facts & Assumptions

Verification

A generalized Bayes credible set can have zero coverage

Statement refuted

Facts & Assumptions

Counterexample

Verification

Rescaling loss changes the effective temperature

Example

Facts & Assumptions

Verification

The unit-temperature likelihood update is ordinary Bayes in a finite model

Example

Facts & Assumptions

Verification

Sources