Deriving the CRM Posterior Update

From Cohort Outcomes to a Posterior

In the one-parameter power CRM, observed dose-limiting toxicities (DLTs) update uncertainty about a single parameter, $\beta$. The update is not a separate rule for each cohort: each cohort contributes evidence through its probability under the model, and the cohort likelihoods multiply. A prior distribution then supplies the starting weight for possible values of $\beta$.

This tutorial derives that calculation for a short hypothetical trial. The sequence will use a fixed dose-toxicity skeleton $q_1,\ldots,q_J$, where each $q_j$ is a probability strictly between 0 and 1. At dose $j$, the power CRM specifies the DLT probability

p_j(\beta)=q_j^{\exp(\beta)}.

In particular, $\beta=0$ gives $p_j(0)=q_j$, because $\exp(0)=1$. This identity is a useful arithmetic check; it does not mean that the observed DLT rate in a cohort must equal its skeleton probability.

Definition: Let cohort $k$ contain $n_k$ patients treated at dose $j_k$, of whom $y_k$ experience a DLT during the specified assessment period. Conditional on $\beta$ and the assigned doses, the cohort likelihood is $$ L_k(\beta)=\binom{n_k}{y_k}p_{j_k}(\beta)^{y_k}\left[1-p_{j_k}(\beta)\right]^{n_k-y_k}. $$ For $K$ cohorts, the likelihood is $L(\beta;D)=\prod_{k=1}^{K}L_k(\beta)$, where $D$ denotes the observed cohort outcomes and assignments.

Assumptions Behind the Cohort Likelihood

The binomial contribution assumes that, conditional on $\beta$ and the assigned dose, each patient's DLT indicator is Bernoulli with probability $p_{j_k}(\beta)$, and that patient outcomes within and across cohorts are conditionally independent. Patients in a cohort must have the same modeled dose probability and the same defined DLT assessment window. A patient without complete DLT follow-up cannot simply be counted as a patient with no DLT.

The dose assignments in a CRM are often adaptive: the next cohort's dose can depend on earlier outcomes. When the assignment rule depends only on the observed history and any randomization used by the rule is independent of $\beta$, inference for $\beta$ conditional on the realized assignments uses the outcome likelihood above. The assignment probabilities do not add a $\beta$-dependent factor. If assignment itself depends directly on the unknown $\beta$, or if the assignment mechanism is part of the modeled data-generating process, that requires a different likelihood specification.

For the worked sequence, take the hypothetical skeleton $q_1=0.05$, $q_2=0.15$, and $q_3=0.30$, and use the illustrative prior $\beta\sim N(0,1)$. Suppose three complete cohorts each have three patients: cohort 1 has 0 DLTs at dose 1, cohort 2 has 1 DLT at dose 2, and cohort 3 has 2 DLTs at dose 3. These observations and the prior are hypothetical and are used only to show the calculation.

Deriving the Posterior Density

For a particular cohort, the $n_k$ Bernoulli outcome probabilities multiply. If only the count $y_k$ is recorded, there are $\binom{n_k}{y_k}$ possible patient-level arrangements with that count, giving the binomial likelihood contribution. Conditional independence across cohorts then gives the product of their contributions.

L(\beta;D)= \prod_{k=1}^{K} \binom{n_k}{y_k} \left[q_{j_k}^{\exp(\beta)}\right]^{y_k} \left[1-q_{j_k}^{\exp(\beta)}\right]^{n_k-y_k}.

Bayes' rule weights this likelihood by the prior density $\pi(\beta)$. The normalizing constant is the integral of that product over all real values of $\beta$. Thus the posterior density is the following ratio:

Theorem: If the prior density is $\pi(\beta)$ and the likelihood is positive on a set with positive prior probability, then the posterior density is $$ \pi(\beta\mid D)= \frac{L(\beta;D)\pi(\beta)} {\int_{-\infty}^{\infty}L(t;D)\pi(t)\,dt}. $$ The denominator is a constant with respect to $\beta$; it makes the posterior density integrate to 1.

Under the illustrative $N(0,1)$ prior, $\pi(\beta)=(2\pi)^{-1/2}\exp(-\beta^2/2)$. Substitution of the three hypothetical cohorts gives the posterior kernel:

\pi(\beta\mid D)\ \propto\ (2\pi)^{-1/2}\exp(-\beta^2/2) [1-p_1(\beta)]^3 \{3p_2(\beta)[1-p_2(\beta)]^2\} \{3p_3(\beta)^2[1-p_3(\beta)]\}.

Here “proportional to” means that the displayed expression still needs division by its integral over $\beta$. The binomial factors 3 are constant in $\beta$, so they may be dropped when calculating a posterior kernel up to proportionality. They are retained here to show the likelihood of the observed counts. This update is generally not a conjugate normal update: the powers $q_j^{\exp(\beta)}$ make the likelihood nonlinear in $\beta$.

Worked Example: Constructing the Likelihood for Three Cohorts

For the hypothetical sequence above, write the likelihood from the observed counts before using the prior. The first cohort has 0 DLTs out of 3 at dose 1, the second has 1 out of 3 at dose 2, and the third has 2 out of 3 at dose 3.

The first cohort contributes $\binom{3}{0}p_1(\beta)^0[1-p_1(\beta)]^3=[1-p_1(\beta)]^3$. The second contributes $\binom{3}{1}p_2(\beta)[1-p_2(\beta)]^2$, and the third contributes $\binom{3}{2}p_3(\beta)^2[1-p_3(\beta)]$. Therefore,

L(\beta;D)=[1-p_1(\beta)]^3 \{3p_2(\beta)[1-p_2(\beta)]^2\} \{3p_3(\beta)^2[1-p_3(\beta)]\}.

At $\beta=0$, the probabilities are the skeleton values, $0.05$, $0.15$, and $0.30$. The cohort contributions are $0.95^3=0.857375$, $3(0.15)(0.85)^2=0.325125$, and $3(0.30)^2(0.70)=0.189000$. Multiplying them gives $0.857375(0.325125)(0.189000)=0.0526845$, or approximately $0.05268$.

This is the likelihood of the observed cohort counts evaluated at $\beta=0$, not a posterior probability that $\beta=0$. It includes the count multiplicities for the cohorts with one and two DLTs.

Evaluating the Posterior Kernel at a Second Value

To compare parameter values, calculate the model probabilities at each value, evaluate all cohort contributions, and multiply by the prior density. The posterior kernel gives relative density, not posterior probability at an exact point. For this continuous parameter, the probability of any one exact value is zero.

Worked Example: Checking the Kernel at $\beta=-1$

For the same hypothetical data and prior, evaluate the model probabilities and posterior kernel at $\beta=-1$. Since $\exp(-1)\approx0.367879$, the model gives $p_j(-1)=q_j^{0.367879}$. For the stated skeleton, these probabilities are approximately $0.3322$, $0.4976$, and $0.6422$.

The three cohort likelihood contributions are approximately $(1-0.3322)^3=0.2978$, $3(0.4976)(1-0.4976)^2=0.3768$, and $3(0.6422)^2(1-0.6422)=0.4427$. Their product is about $0.04968$. The $N(0,1)$ prior density at $-1$ is $(2\pi)^{-1/2}\exp(-1/2)\approx0.24197$, so the unnormalized posterior density is approximately

L(-1;D)\pi(-1)\approx0.04968(0.24197)=0.01202.

For comparison, at $\beta=0$ the prior density is $1/\sqrt{2\pi}\approx0.39894$. The kernel there is $0.0526845(0.39894)\approx0.02102$. These two kernel values alone do not give posterior probabilities for neighborhoods; normalization and integration over ranges of $\beta$ are needed for such probabilities.

What the Product Does, and a Useful Derivation Check

The multiplicative form follows directly from conditional independence. For each patient, the probability of a DLT indicator $Y=1$ is $p_j(\beta)$, and the probability of $Y=0$ is $1-p_j(\beta)$. A particular ordered set of $y$ DLTs and $n-y$ non-DLTs therefore has probability $p_j(\beta)^y[1-p_j(\beta)]^{n-y}$. If the data are summarized as a count rather than an ordered sequence, there are $\binom{n}{y}$ arrangements, so the probability of that count is the ordered-sequence probability multiplied by $\binom{n}{y}$. Independence across cohorts gives the product.

This also explains why omitting a binomial coefficient does not change the posterior shape: it is constant in $\beta$, so it appears in both numerator and denominator and cancels. But it does change the numerical likelihood of the count. Keep it when reporting the likelihood of observed counts, and omit it only when explicitly working with a proportional kernel.

Worked Example: Comparing Two Kernel Values Using One Cohort

Isolate the second hypothetical cohort, which had 1 DLT among 3 patients at dose 2. Compare the contribution of this cohort times the prior density at $\beta=0$ and $\beta=-1$. This comparison shows how one cohort's evidence and the prior combine; it is not the ratio of the full posterior kernels for all three cohorts.

At $\beta=0$, the cohort contribution is $3(0.15)(0.85)^2=0.325125$. At $\beta=-1$, $p_2(-1)\approx0.4976$, giving a contribution of approximately $3(0.4976)(0.5024)^2=0.3768$. For the standard normal prior, the prior-density ratio is $\pi(0)/\pi(-1)=\exp(1/2)\approx1.64872$. Consequently, the ratio of the cohort-times-prior kernels at 0 and $-1$ is approximately

\frac{0.325125\,\pi(0)}{0.3768\,\pi(-1)} = \exp(1/2)\frac{0.325125}{0.3768} \approx1.423.

For this isolated comparison, the cohort-times-prior kernel at 0 is about 1.423 times its value at $-1$. The other two cohorts would also enter the ratio for the full posterior kernel.

Numerical Normalization, Diagnostics, and Limitations

The denominator in Bayes' rule is an integral that usually has no convenient closed form for the power CRM. It can be evaluated numerically, for example by quadrature over a sufficiently wide range of $\beta$, or the posterior can be represented using an appropriate computational method. Whatever method is used, the calculation should be checked against the defining density: the normalized posterior must be nonnegative and integrate to 1. A grid that is too narrow can miss posterior mass, while a coarse grid can distort its shape.

A stable implementation often evaluates the log likelihood:

\log L(\beta;D)= \sum_{k=1}^{K} \left[ \log\binom{n_k}{y_k} +y_k\log p_{j_k}(\beta) +(n_k-y_k)\log\{1-p_{j_k}(\beta)\} \right].

This avoids multiplying many small likelihood factors directly. Check that every skeleton probability lies strictly between 0 and 1, that each count satisfies $0\le y_k\le n_k$, and that the correct dose and outcome count are paired for every cohort. Common warning signs include a probability outside $[0,1]$, a posterior that fails to normalize, or a result that changes substantially under modest refinement of the numerical grid.

A frequent mistake is to use $q_j^\beta$ instead of $q_j^{\exp(\beta)}$. The latter is the power CRM model; replacing its exponent changes the likelihood and can even produce invalid probabilities for some parameter values. Another mistake is to treat a patient with incomplete DLT follow-up as a non-DLT. The cohort likelihood requires observed outcomes under its defined assessment window; incomplete follow-up needs appropriate handling rather than automatic coding as $Y=0$.

The likelihood-only approach reports how compatible data are with parameter values, but it is not a probability distribution over $\beta$. A Bayesian update combines that likelihood with a prior and normalizes the result. A normal approximation can summarize a posterior in some settings, but it is not the exact update and may be inaccurate when the posterior is skewed or concentrated near a region of low likelihood. The one-parameter power model also remains conditional on its chosen skeleton and assumptions: the posterior update does not remove potential model misspecification.

Key takeaway: For each cohort, use its assigned dose to calculate $p_j(\beta)=q_j^{\exp(\beta)}$, form the binomial likelihood contribution, multiply cohort contributions and the prior density, then normalize over $\beta$. Verify the model probabilities, counts, arithmetic, and numerical normalization.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A hypothetical cohort has 2 DLTs among 4 patients at dose $j$. Write its likelihood contribution in terms of $p_j(\beta)$.
Why does $\beta=0$ provide a useful check on the power CRM probabilities?
For one cohort with 1 DLT among 3 patients and $p=0.20$, calculate the likelihood of the count.
What happens to the posterior kernel if all binomial coefficients are omitted?
Why should a patient with incomplete DLT follow-up not automatically be counted as a non-DLT?

Solutions. (1) The contribution is $\binom{4}{2}p_j(\beta)^2[1-p_j(\beta)]^2=6p_j(\beta)^2[1-p_j(\beta)]^2$. (2) Since $\exp(0)=1$, $p_j(0)=q_j$, so the probabilities at zero must reproduce the skeleton. (3) The count likelihood is $\binom{3}{1}(0.20)(0.80)^2=3(0.20)(0.64)=0.384$. (4) Each coefficient is constant in $\beta$, so omitting them changes the overall likelihood scale but not the normalized posterior shape. (5) The likelihood assumes outcomes are observed over the specified assessment period; incomplete follow-up does not establish that no DLT occurred.

Deriving the CRM Posterior Update

What You'll Learn

From Cohort Outcomes to a Posterior

Assumptions Behind the Cohort Likelihood

Deriving the Posterior Density

Worked Example: Constructing the Likelihood for Three Cohorts

Evaluating the Posterior Kernel at a Second Value

Worked Example: Checking the Kernel at \(\beta=-1\)

What the Product Does, and a Useful Derivation Check

Worked Example: Comparing Two Kernel Values Using One Cohort

Numerical Normalization, Diagnostics, and Limitations

Check Your Understanding

Deriving the CRM Posterior Update

What You'll Learn

From Cohort Outcomes to a Posterior

Assumptions Behind the Cohort Likelihood

Deriving the Posterior Density

Worked Example: Constructing the Likelihood for Three Cohorts

Evaluating the Posterior Kernel at a Second Value

Worked Example: Checking the Kernel at \(\beta=-1\)

What the Product Does, and a Useful Derivation Check

Worked Example: Comparing Two Kernel Values Using One Cohort

Numerical Normalization, Diagnostics, and Limitations

Check Your Understanding

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