2c:null 2d:["$","title",null,{"children":"Prior distributions for link ratios | Ledger | Observable"}] 32:T63c,Link ratios, also known as loss development factors or age-to-age factors, are one of the most pivotal quantities of any non-life insurance actuarial analysis, transforming observed losses into predictions of future, possibly ultimate, losses and loss ratios. Link ratios are typically derived from a statistical model, such as deterministic or stochastic versions of [chain ladder](https://en.wikipedia.org/wiki/Chain-ladder_method). At Ledger, [we predominantly use Bayesian estimation](https://cms.ledgerinvesting.com/media/files/modeling-philosophy.pdf) to fit stochastic models to price casualty insurance risks for insurance-linked securities. Bayesian methods provide a robust and principled approach to quantifying uncertainty, and probabilistic programming languages, like [Stan](https://mc-stan.org/), give us almost unlimited flexibility in fitting models of arbitrary complexity. Newcomers to Bayesian methods, however, frequently have one common question: *how do we select prior distributions*? Despite the uptake of Bayesian methods in insurance (e.g. [Meyers, 2015](https://www.casact.org/sites/default/files/2021-02/01-Meyers.PDF)), relatively few resources exist discussing the choice of prior distribution for Bayesian chain ladder models. Given that chain ladder is such an important model in actuarial science, this notebook will provide insight into prior distributions and the importance of their selection using the chain ladder model. We assume some familiarity with both insurance data, loss reserving, and Bayes' theorem, but key concepts are explained along the way.33:T12de,## Bayes' theorem and prior probabilities Newcomers to Bayesian methods frequently have trouble understanding and selecting prior distributions, so we will start with a brief overview of Bayes' theorem by way of a simple example. Feel free to skip this section if you feel comfortable with the basics of Bayesian probability. Imagine we're interested in the probability that a new private passenger auto insurance policy holder will make a claim on their policy if they've made a claim for a car accident in the past. Luckily, we have access to a large dataset of historical claims to help us. To be specific, we'll use the variable `C` with outcomes `c1` or `c2` to record whether a claim has been made in the past (`c1`) or not (`c2`), and the variable `A` with outcomes `a1` or `a2` to record whether an accident occurs on the policy (`a1`) or not (`a2`). We can tabulate the probability of different possible outcomes, where `p(A = a, C = c)` refers to the probability of both outcomes. You can think of these probabilities as the number of insureds in the historical dataset who fall in each category, divided by the total number of insureds. Here's the table: | `C = c1` | `C = c2` | `p(A)` -------- | -------- | -------- | ----- **`A = a1`** | `p(a1, c1)` | `p(a1, c2)` | `p(a1)` **`A = a2`** | `p(a2, c1)` | `p(a2, c2)` | `p(a2)` **`p(C)`** | `p(c1)` | `p(c2)` Each cell is, mathematically, the *joint* probability of events `A = a` and `C = c`. That is, the probability of both events happening. For instance, the top-left cell is the probability of someone having an accident and having made a claim previously, whereas the bottom right cell is the joint probability of someone not having an accident and not having made a claim previously. The last column and last row tell us something different, which is the probability of each of the four outcomes, regardless of the other variable. Thus, `p(c1)` is the probability of not having a previous claim, regardless of whether someone will make a claim on their current policy. These values are known as *marginal* probabilities, and are nothing but the sum of the rows and columns, respectively, where `p(C) = p(c1) + p(c2) = 1` and vice versa for `p(A)`. Back to our question, what is the probability of a new policy holder having an accident given they've made a claim in the past? At first, you might think this is `p(a1, c1)`, but note that we are already assuming that `c1`, a previous claim, has occurred. In tabular terms, we are *restricting* our attention to the first column, which has the effect of filtering the dataset we care about from all policy holders to all policy holders that have made a claim previously. In probability terms, this is `p(a1, c1) / p(c1)`, the joint probability divided by (i.e. 'corrected for') the focus on those who have made claims previously. This expression, `p(a1, c1) / p(c1)`, is a *conditional* probability, written as `p(a1 | c1)`, where the vertical bar means "given that" the following variables have occurred. Finally, we can factor the joint probability in the numerator into two components: a conditional probability and a marginal probability, allowing us to write `p(a1, c1) = p(a1 | c1) p(c1)`. Fortunately, joint probabilities are symmetric, so `p(a1, c1) = p(c1 | a1) p(a1)` is also true. In total, this expression is known as [Bayes' theorem](https://en.wikipedia.org/wiki/Bayes%27_theorem). In general, ${tex.block` \begin{aligned} \tag{Bayes' theorem} p(b \mid a) &= \frac{p(a \mid b) p(b)}{p(a)}\\ \mathrm{posterior} &= \frac{\mathrm{likelihood} \times \mathrm{prior}}{\mathrm{evidence}}\\ \end{aligned} `} where the last line uses the typical names for each component. Bayes' theorem is simply a way of computing conditional probabilities, and the posterior probability is proportional to the joint probability, or the likelihood multiplied by the prior. Since any joint distribution is a balancing act of its conditional and marginal probabilities, the posterior is a balancing act of the likelihood and prior. The chance someone has an accident given they've made a claim previously will be small if either of those components are small. Even if people who have accidents during their policy have a very high probability of having previously made claims, `p(c1 | a1)`, the posterior probability of having an accident given a previous claim, `p(a1 | c1)`, will be low if the chance of previous claims at all, `p(c1)`, is very low. This leads to many unintuitive statistical problems known collectively as [the base rate fallacy](https://en.wikipedia.org/wiki/Base_rate_fallacy). A famous one is the [Librarian or farmer?](https://www.goodreads.com/quotes/10198601-an-individual-has-been-described-by-a-neighbor-as-follows) problem from Daniel Kahneman's *Thinking Fast and Slow* (2011).34:T62a,## Prior knowledge applied to link ratios When it comes to data analysis, and Bayesian statistical analysis, we have some observed data ${tex`y`} and unknown parameter ${tex`\theta`}, and we want to estimate the conditional probability ${tex`p(\theta \mid y)`}: the probability of the unknown parameter given our data, and for that we can use Bayes' theorem. Those familiar with statistical modeling will be somewhat practiced at choosing the likelihood, ${tex`p(y \mid \theta)`}, whether the term 'likelihood' is used or not, as that is the distribution of the observed data, conditional on the unknown parameters. We also need the prior probability, ${tex`p(\theta)`}, the marginal distribution of the unknown parameter ${tex`\theta`} representing what we expect of ${tex`\theta`} knowing nothing about the actual data. As we saw above, there's nothing particularly special about this quantity mathematically. It follows naturally from basic probability theory. What is our prior knowledge about link ratios? Let's define the link ratios using ${tex`\alpha = \alpha_{1}, \alpha_{2}, ..., \alpha_{M - 1}`}, where we have ${tex`M`} development periods in our data across ${tex`N`} accident or policy periods. Intuitively, we know that link ratios are continuous and positive, and we also expect them to converge, whether quickly or slowly, to a value of 1.0 at higher development lags, as fewer claims and losses are left to emerge. A uniform distribution, with a lower bound of 0, might be a place to start. An example is shown below as a histogram with an upper bound of 50:35:T4d0,The uniform distribution, written ${tex`\mathrm{Uniform}(X \mid a, b)`} meaning the random variable ${tex`X`} has outcomes or *realizations* that are uniformly distributed with lower and upper bounds ${tex`a`} and ${tex`b`}, assigns equal probability to each realization of ${tex`X`}. In other words, ${tex`p(X = x) = 1`} for any value of ${tex`x`}. The above plot shows one million realizations from a ${tex`\mathrm{Uniform}(X \mid a = 0, b = 50)`} distribution split into 20 bins, where the random variation in sampling causes minor variation in the exact frequency of each bin. Is this a sensible distribution to describe our knowledge about link ratios? The choice of ${tex`b = 50`} in this instance is somewhat arbitrary -- big enough to encompass plausibly high values of some link ratios, depending on the context, but not too large to be unrealistic. However, for link ratios at higher development lags, we would not expect equal weight to be applied to link ratios of 50. The truth is that it is often difficult, and sometimes impossible, to understand the implications of our priors through inspection of their individual distributions alone. Instead, a technique called *prior predictive simulation* is highly useful.36:T6b0,## Prior predictive simulations help understanding priors The prior predictive distribution is the denominator of Bayes' theorem, or the *evidence*. If we refer to our data as ${tex`y`} and parameters as ${tex`\theta`}, the prior predictive distribution is the marginal distribution ${tex`p(y)`}. For discrete parameters, like our first example of car accidents, this is a sum of joint probabilities across different parameter values. It is the likelihood of the data across all possible parameter values. But for continuous parameters, the evidence is the often intractable integral ${tex`p(y) = \int p(y \mid \theta) p(\theta) d \theta`}. This marginal distribution is the reason we need probabilistic programming languages, like Stan, to approximate the posterior distribution for more than elementary models, and therefore cannot be calculated easily. However, we can *sample* from the prior predictive distribution easily, which tells us what a model believes about ${tex`y`} regardless of the specific values of ${tex`\theta`}. To make this less abstract, consider we have an initial loss ratio of 20% in a particular accident period. A simple instantiation of the chain ladder model assumes that ${tex`y_{i} = y_{i - 1} \alpha_{i - 1}`}, that is, the next loss ratio is the previous loss ratio multiplied by the link ratio for that development lag. We are ignoring any distributional assumptions on loss ratios directly, thereby ignoring observational error on loss ratios, for simplicity. You can think of this as the mean or expected loss ratios predicted from chain ladder. If we have the link ratios `(2.5, 1.2, 1.15, 1.1, 1.05, 1.03, 1.02, 1.01, 1.0)`, a sequence of loss ratios would look like:37:Ta01,As is probably immediately obvious, the loss ratios look nothing like the previous plot and the scale of the loss ratios on the y-axis is absurd, so much so that it's difficult to pick out any particular dynamic in loss ratios other than that they increase dramatically. This is entirely the result of the prior distribution on the link ratios resulting in implausibly high values. The multiplicative nature of the chain ladder model means that link ratios that are even marginally too large might bias loss ratios too high, too quickly. This ${tex`\mathrm{Uniform}(\alpha \mid 0, 50)`} prior distribution for link ratios is not representative of the data, so it might be surprising to learn that many Bayesian software packages set, or recommended, default prior distributions that are similarly wide in a quest for 'vague' or 'uninformative' distributions (e.g. in an insurance context, see [Meyers, 2012](The Leveled Chain Ladder Model for Stochastic Loss Reserving))). The goal of uninformative priors is for the prior to have as little say as possible, and for the data to say as much as possible, about the posterior distribution of the unknown parameter. Remember that a posterior probability is proportional to the likelihood and prior, and so a uniform distribution that sets all parameter values as equally likely simply drops out of the equation, and the likelihood is given all the weight. Appealing to vagueness is a decent starting point for setting priors, and when there is enough data, the choice of prior will become less important. Even very diffuse priors that look unreasonable in a prior predictive simulation could result in accurately-estimated posterior distributions when the data is informative enough. However, this relegates priors to a 'be seen and not heard' status, when selecting priors carefully has many practical and theoretical advantages (e.g. see [Gelman et al. 2017](https://www.mdpi.com/1099-4300/19/10/555) for a good discussion). We nearly always know more about what a parameter should look like than is encoded in an uninformative prior distribution, and utilising this prior knowledge can allow fitting better-performing models that could be challenging to successfully estimate with uninformative priors. Indeed, uninformative priors are typically not uninformative or vague, but offer a logically-opposed view of the real world, which can hurt our modeling in the long-term. Selecting priors should start by thinking about the processes generating the data and what parameter values are realistically possible.38:T406,This is another improvement on our prior predictive simulations, in terms of the range of plausible values. Nonetheless, note that there is a non-negligible chance that the link ratios might produce loss ratios that decrease, rather than increase, over development time. This is not implausible if there are recoveries to claim amounts, but we might want to improve on our prior distributions to ensure losses are nearly always monotonically increasing. You can make priors increasingly complex functions of input variables and even other parameters. For instance, we could define a parametric form for the standard deviation following an exponentially-decaying curve, which would more quickly regularize the priors towards 1.0 at higher development lags: ${tex.block` \begin{aligned} \sigma_{i} &= a \exp(-b i) \end{aligned} `} where ${tex`a`} is the starting or base standard deviation when ${tex`b = 0`}, and ${tex`b`} controls the speed of growth. Setting ${tex`a = b = 1`} produces the prior predictive simulation below:39:T418,## A default prior distribution for link ratios If you have strong domain knowledge about what the link ratios should look like, you can select the individual mean and standard deviations for each development lag's link ratio prior distribution. However, this manual approach is harder to apply at scale. One option for a default prior distribution is to use the exponentially decaying function we used above for the standard deviations on both the mean and standard deviations of the lognormal distribution. We would expect that earlier development lags will have higher and more variable link ratios than later development lags, and so applying this parametric form to both mean and standard deviations is useful. Setting the mean and standard deviation to the following functions of development lag produces realistic-looking loss ratios: ${tex.block` \begin{aligned} \mu_{i} &= 1.75 \exp(- i)\\ \sigma_{i} &= 0.5 \exp(- i) \end{aligned} `} This results in the prior predictive simulation below, where we show 1000 different scenarios. 3a:T592,## Summary Setting priors on link ratios is a non-trivial task, even for a relatively simple model like chain ladder. Priors that might commonly be seen in the literature, or priors that might seem weakly informative, can produce unreasonable patterns of loss ratios *a priori*. Although we might have enough data that the influence of the prior distribution is minimal, this is not a given when the volume of business is low or the amount of historical data we have is lacking. When the performance or solvency of an insurance company is dependent on accurate loss reserving models, small differences could have a big effect. To understand how big an effect, prior sensitivity analyses should be run ([Kruschke, 2021](https://www.nature.com/articles/s41562-021-01177-7)), either by re-fitting models with different priors or using approximate methods such as power-scaling (see [Kallioinen et al., 2023](https://link.springer.com/article/10.1007/s11222-023-10366-5)). Priors for link ratio models is an area that should be the target for future research. At Ledger, we tap into large catalogues of industry data to build prior distributions for all our models that are representative of different lines of business and volume. These priors are backtested against plausible alternatives, and their performance compared on out-of-sample data to ensure our preferred models are those most likely to make accurate predictions. 3b:[{"id":"48f9f6170d9a24fa97ad8d9ac97c30870055ecc522387afe846544615603e5b6c36b4cc0765cb198c99706b166f77b0f555bc258a5c15929caf529f1a3aae4b7","href":"https://static.observableusercontent.com/files/48f9f6170d9a24fa97ad8d9ac97c30870055ecc522387afe846544615603e5b6c36b4cc0765cb198c99706b166f77b0f555bc258a5c15929caf529f1a3aae4b7","name":"link-ratio-header@1.png","type":"image/png","lastModified":1721827329897.436,"size":308284}] 30:{"id":"41ae8eae53afda89","slug":null,"title":"Prior distributions for link ratios","thumbnail":"aac20e9b331d01dd0d4c833a992faaa27d5f61c09bb107d8d539aaaa0b831e58","access_level":"public","create_time":"$D2024-07-11T13:31:26.622Z","update_time":"$D2024-08-01T13:20:35.818Z","view_time":null,"first_public_time":"$D2025-11-18T10:34:27.391Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"5b9bd0b917e240d6","login":"ledger","name":"Ledger","avatar_url":"https://avatars-next.observableusercontent.com/c0873f971b4f7860a38fc812d8f161d44fc1110a7470efba6ea5413fef288f74","tier":"pro"},"version":1334,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"78ec1d5b1b7f6c10","login":"cmgoold","name":"cmgoold","avatar_url":"https://avatars-next.observableusercontent.com/d9ec02aadca0e78a24722cfc7426fa986a6379dacd127f0b6a3bcf79faebaefd","tier":null},{"type":"individual","id":"fa49b6e92c82c1bf","login":"mark-shoun","name":"Mark Shoun","avatar_url":"https://avatars-next.observableusercontent.com/3b5baa5aaa42bc683ebf87c3ee2f3cd1738c206b12d4903f51e4a7d01413534e","tier":null}],"members":[{"type":"individual","id":"78ec1d5b1b7f6c10","login":"cmgoold","name":"cmgoold","avatar_url":"https://avatars-next.observableusercontent.com/d9ec02aadca0e78a24722cfc7426fa986a6379dacd127f0b6a3bcf79faebaefd","tier":null,"role":"editor"},{"type":"team","id":"5b9bd0b917e240d6","login":"ledger","name":"Ledger","avatar_url":"https://avatars-next.observableusercontent.com/c0873f971b4f7860a38fc812d8f161d44fc1110a7470efba6ea5413fef288f74","tier":"pro","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Prior distributions for link ratios","cells":[{"id":0,"value":"# Prior distributions for link ratios","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1294,"value":"linkRatioHeader1 = FileAttachment(\"link-ratio-header@1.png\").image()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":8,"value":"$32","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":400,"value":"$33","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":453,"value":"$34","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":483,"value":"Plot.plot({\n marginLeft: 50,\n marks: [\n Plot.rectY(\n uniform_prior,\n Plot.binX({ y: \"count\" }, { x: \"x\", thresholds: 20 })\n )\n ],\n caption: \"A Uniform(0, 50) distribution.\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":510,"value":"$35","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":526,"value":"$36","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":952,"value":"Plot.plot({\n y: { label: \"Loss ratios (%)\" },\n x: { label: \"Development lag\" },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0.5]),\n Plot.line(f, {\n x: \"lag\",\n y: \"lr\",\n strokeWidth: 2\n }),\n Plot.dot(f, {\n x: \"lag\",\n y: \"lr\",\n r: 5,\n fill: \"white\",\n stroke: \"black\"\n })\n ],\n caption:\n \"Example loss ratios through development time with link ratios (2.5, 1.2, 1.15, 1.1, 1.05, 1.03, 1.02, 1.01, 1.0).\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":946,"value":"Rather than fixed values of link ratios, if we simulate the link ratios from their prior distribution, we can calculate sequences of loss ratios that are representative of our model. By simulating many instances of link ratios from the prior distribution, we are *marginalizing* over the prior distribution, that is, sampling from the marginal distribution. \n\nUsing the uniform prior distribution above, one hundred such sequences for 9 uniformly distributed link ratios are shown below:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":556,"value":"Plot.plot({\n marginLeft: 130,\n y: { label: \"Loss ratios (%)\" },\n x: { label: \"Development lag\" },\n marks: [\n Plot.ruleY([0, -1.7e13]),\n Plot.ruleX([0.5]),\n Plot.line(uniform_pps050, {\n x: \"lag\",\n y: \"lr\",\n z: \"sim\",\n strokeWidth: 0.5\n }),\n Plot.dot(uniform_pps050, {\n x: \"lag\",\n y: \"lr\",\n r: 2,\n fill: \"white\",\n stroke: \"black\"\n })\n ],\n caption:\n \"Loss ratios through development time from the prior predictive distribution, with link ratios distributed as Uniform(0, 50).\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":596,"value":"$37","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":625,"value":"## Weakly-informative priors\n\nRecommendations for setting prior distributions in Bayesian models have advanced, with the most common starting place for default priors being *weakly informative* priors (for instance, see [Gelman et al., 2008](https://projecteuclid.org/journals/annals-of-applied-statistics/volume-2/issue-4/A-weakly-informative-default-prior-distribution-for-logistic-and-other/10.1214/08-AOAS191.full)). Weakly informative priors are diffuse enough to allow a variety of parameter values but put negligible weight on unrealistic parameter values. What about a ${tex`\\mathrm{Uniform}(\\alpha \\mid 0.5, 3)`} prior? This is a much more realistic range for link ratios in general, particularly for larger books of business.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":664,"value":"Plot.plot({\n marginLeft: 70,\n y: { label: \"Loss ratios (%)\" },\n x: { label: \"Development lag\" },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0.5]),\n Plot.line(uniform_pps_weakly, {\n x: \"lag\",\n y: \"lr\",\n z: \"sim\",\n strokeWidth: 0.5\n }),\n Plot.dot(uniform_pps_weakly, {\n x: \"lag\",\n y: \"lr\",\n r: 2,\n fill: \"white\",\n stroke: \"black\"\n })\n ],\n caption:\n \"Loss ratios through development time from the prior predictive distribution, with link ratios distributed as Uniform(0.5, 3).\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":670,"value":"As shown in this figure, even this prior can produce loss ratios that are too high, and have the undesirable property that they do not plateau at higher development lags. One problem is that we might not believe that all values in the range are equally likely: we would expect values closer to 1 to be more likely than values further away. Another candidate distribution that would encode this assumption is a normal distribution, parameterized by its mean ${tex`\\mu`} and standard deviation ${tex`\\sigma`}, and written as ${tex`\\mathrm{Normal}(X \\mid \\mu, \\sigma)`}. Here's a standard normal distribution, with mean and standard deviation of 0 and 1, respectively:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":692,"value":"Plot.plot({\n marks: [\n Plot.rectY(z_prior, Plot.binX({ y: \"count\" }, { x: \"x\", thresholds: 20 }))\n ],\n caption: \"A Normal(0, 1) distribution.\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":702,"value":"However, the normal distribution can produce values across the real line, which does not match our requirement that link ratios remain positive. Instead, we can exponentiate the normal distribution to ensure that values are positive, which defines the lognormal distribution, ${tex`\\mathrm{Lognormal}(Y \\mid \\mu, \\sigma)`}. If ${tex`X`} is normally distributed, then ${tex`Y = \\exp(X)`} has a lognormal distribution, with mean and standard deviation defined on the log-scale for the random variable ${tex`X`}. Here's a ${tex`\\mathrm{Lognormal}(Y \\mid \\mu = 0, \\sigma = 1)`} distribution:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":711,"value":"Plot.plot({\n marginLeft: 50,\n marks: [\n Plot.rectY(\n lognormal_prior,\n Plot.binX({ y: \"count\" }, { x: \"x\", thresholds: 50 })\n )\n ],\n caption: \"A Lognormal(0, 1) distribution.\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":751,"value":"These values remain positive but there are still some very large values that we might not want to support. Lowering the standard deviation to 0.5 looks more reasonable:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":745,"value":"Plot.plot({\n marginLeft: 50,\n marks: [\n Plot.rectY(\n rlnorm(1e6, 0, 0.5).map((i) => {\n return { x: i };\n }),\n Plot.binX({ y: \"count\" }, { x: \"x\", thresholds: 50 })\n )\n ],\n caption: \"A Lognormal(0, 0.5) distribution.\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":734,"value":"What does a prior predictive simulation using this distribution for the link ratios look like?","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":738,"value":"Plot.plot({\n marginLeft: 70,\n y: { label: \"Loss ratios (%)\" },\n x: { label: \"Development lag\" },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0.5]),\n Plot.line(lognormal_pps_weakly, {\n x: \"lag\",\n y: \"lr\",\n z: \"sim\",\n strokeWidth: 0.5\n }),\n Plot.dot(lognormal_pps_weakly, {\n x: \"lag\",\n y: \"lr\",\n r: 2,\n fill: \"white\",\n stroke: \"black\"\n })\n ],\n caption:\n \"Loss ratios through development time from the prior predictive distribution, with link ratios distributed as Lognormal(0, 0.5).\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":758,"value":"These loss ratios are starting too look slightly more reasonable, although loss ratios of 1000%, even rarely, are still unrealistic.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":760,"value":"## Prior functions\n\nWe have assumed that each link ratio receives the same prior distribution, but that's unlikely to be true. Link ratios at later development lags should have values closer to 1.0 than earlier lags. A simple method of enforcing this requirement is to make the standard deviation a function of development lag, such as ${tex`\\mathrm{Lognormal}(\\alpha_i \\mid \\mu, \\frac{\\sigma}{i + 1})`}, for ${tex`i = 1, ..., M - 1`}. Then there is less variability in possible priors at later development lags, reflecting our more certain prior knowledge.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":802,"value":"Plot.plot({\n marginLeft: 70,\n y: { label: \"Loss ratios (%)\" },\n x: { label: \"Development lag\" },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0.5]),\n Plot.line(lognormal_pps_regularize, {\n x: \"lag\",\n y: \"lr\",\n z: \"sim\",\n strokeWidth: 0.5\n }),\n Plot.dot(lognormal_pps_regularize, {\n x: \"lag\",\n y: \"lr\",\n r: 2,\n fill: \"white\",\n stroke: \"black\"\n })\n ],\n caption:\n \"Loss ratios through development time from the prior predictive distribution, with link ratios distributed as Lognormal(0, sigma / (lag + 1)).\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":805,"value":"$38","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":817,"value":"Plot.plot({\n marginLeft: 70,\n y: { label: \"Loss ratios (%)\" },\n x: { label: \"Development lag\" },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0.5]),\n Plot.line(lognormal_pps_growth, {\n x: \"lag\",\n y: \"lr\",\n z: \"sim\",\n strokeWidth: 0.5\n }),\n Plot.dot(lognormal_pps_growth, {\n x: \"lag\",\n y: \"lr\",\n r: 2,\n fill: \"white\",\n stroke: \"black\"\n })\n ],\n caption:\n \"Loss ratios through development time from the prior predictive distribution, with link ratios distributed as Lognormal(0, exp(-i)).\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":853,"value":"This is starting to look more realistic. Rather than setting the values of ${tex`a`} and ${tex`b`} manually, we could make these unknowns parameters to be estimated in our model, which is easily added in modern Bayesian modelling software. In that case, our prior distribution would not simply be the marginal distribution ${tex`p(\\alpha)`}, but a joint distribution itself composed of ${tex`p(\\alpha, a, b) = p(\\alpha \\mid a, b) p(a) p(b)`}, where ${tex`p(a)`} and ${tex`p(b)`} are given their own prior distributions encoding our assumptions.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1071,"value":"$39","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1079,"value":"Plot.plot({\n marginLeft: 70,\n y: { label: \"Loss ratios (%)\" },\n x: { label: \"Development lag\" },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0.5]),\n Plot.line(lognormal_pps_default, {\n x: \"lag\",\n y: \"lr\",\n z: \"sim\",\n strokeWidth: 0.1\n }),\n Plot.dot(lognormal_pps_default, {\n x: \"lag\",\n y: \"lr\",\n r: 2,\n fill: \"white\",\n stroke: \"black\"\n })\n ],\n caption:\n \"Simulated mean loss ratios with decaying exponential curves on link ratio priors' means and standard deviations.\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1145,"value":"The range of mean loss ratios looks highly plausible, allowing for some early volatility and some larger values at development lag 10, but the bulk of the loss ratios grow steadily and converge to a loss ratio between 40 and 80%. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":855,"value":"$3a","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1286,"value":"### Interactive plot\n\nBelow is a simple interactive plot for you to play with the prior distribution family (uniform, normal and lognormal) and parameters (upper/lower bounds, or mean and standard deviation). By default, we are showing the lognormal prior distribution [from above](https://observablehq.com/d/41ae8eae53afda89#cell-1071), with values filled in to match closely to the mean and standard deviation equations. Try moving the means or standard deviations below for different link ratios to see how big of a difference small changes can make! ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1182,"value":"viewof distribution = Inputs.select([runif, rnorm, rlnorm], {\n label: \"Distribution\",\n format: (x) => x.name,\n value: rlnorm\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1194,"value":"viewof link_ratio1 = Inputs.form(\n [\n Inputs.range(distribution == rlnorm ? [-10, 10] : [0, 10], {\n label:\n distribution == runif ? \"Link ratio 1 | Lower\" : \"Link ratio 1 | Mean\",\n step: 0.01,\n value: distribution == rlnorm ? 0.64 : 0\n }),\n Inputs.range([0, 10], {\n label: distribution == runif ? \"Upper\" : \"Standard deviation\",\n step: 0.01,\n value: distribution == rlnorm ? 0.18 : 1\n })\n ],\n {\n template: (inputs) => htl.html`

\n ${inputs}\n

`\n }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1223,"value":"viewof link_ratio2 = Inputs.form(\n [\n Inputs.range(distribution == rlnorm ? [-10, 10] : [0, 10], {\n label:\n distribution == runif ? \"Link ratio 2 | Lower\" : \"Link ratio 2 | Mean\",\n step: 0.01,\n value: distribution == rlnorm ? 0.25 : 0\n }),\n Inputs.range([0, 10], {\n label: distribution == runif ? \"Upper\" : \"Standard deviation\",\n step: 0.01,\n value: distribution == rlnorm ? 0.07 : 1\n })\n ],\n {\n template: (inputs) => htl.html`

\n ${inputs}\n

`\n }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1230,"value":"viewof link_ratio3 = Inputs.form(\n [\n Inputs.range(distribution == rlnorm ? [-10, 10] : [0, 10], {\n label:\n distribution == runif ? \"Link ratio 3 | Lower\" : \"Link ratio 3 | Mean\",\n step: 0.01,\n value: distribution == rlnorm ? 0.09 : 0\n }),\n Inputs.range([0, 10], {\n label: distribution == runif ? \"Upper\" : \"Standard deviation\",\n step: 0.01,\n value: distribution == rlnorm ? 0.02 : 1\n })\n ],\n {\n template: (inputs) => htl.html`

\n ${inputs}\n

`\n }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1232,"value":"viewof link_ratio4 = Inputs.form(\n [\n Inputs.range(distribution == rlnorm ? [-10, 10] : [0, 10], {\n label:\n distribution == runif ? \"Link ratio 4 | Lower\" : \"Link ratio 4 | Mean\",\n step: 0.01,\n value: distribution == rlnorm ? 0.03 : 0\n }),\n Inputs.range([0, 10], {\n label: distribution == runif ? \"Upper\" : \"Standard deviation\",\n step: 0.01,\n value: distribution == rlnorm ? 0.01 : 1\n })\n ],\n {\n template: (inputs) => htl.html`

\n ${inputs}\n

`\n }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1234,"value":"viewof link_ratio5 = Inputs.form(\n [\n Inputs.range(distribution == rlnorm ? [-10, 10] : [0, 10], {\n label:\n distribution == runif ? \"Link ratio 5 | Lower\" : \"Link ratio 5 | Mean\",\n step: 0.01,\n value: distribution == rlnorm ? 0.01 : 0\n }),\n Inputs.range([0, 10], {\n label: distribution == runif ? \"Upper\" : \"Standard deviation\",\n step: 0.01,\n value: distribution == rlnorm ? 0 : 1\n })\n ],\n {\n template: (inputs) => htl.html`

\n ${inputs}\n

`\n }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1236,"value":"viewof link_ratio6 = Inputs.form(\n [\n Inputs.range(distribution == rlnorm ? [-10, 10] : [0, 10], {\n label:\n distribution == runif ? \"Link ratio 6 | Lower\" : \"Link ratio 6 | Mean\",\n step: 0.01,\n value: 0\n }),\n Inputs.range([0, 10], {\n label: distribution == runif ? \"Upper\" : \"Standard deviation\",\n step: 0.01,\n value: distribution == rlnorm ? 0 : 1\n })\n ],\n {\n template: (inputs) => htl.html`

\n ${inputs}\n

`\n }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1238,"value":"viewof link_ratio7 = Inputs.form(\n [\n Inputs.range(distribution == rlnorm ? [-10, 10] : [0, 10], {\n label:\n distribution == runif ? \"Link ratio 7 | Lower\" : \"Link ratio 7 | Mean\",\n step: 0.01,\n value: 0\n }),\n Inputs.range([0, 10], {\n label: distribution == runif ? \"Upper\" : \"Standard deviation\",\n step: 0.01,\n value: distribution == rlnorm ? 0 : 1\n })\n ],\n {\n template: (inputs) => htl.html`

\n ${inputs}\n

`\n }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1240,"value":"viewof link_ratio8 = Inputs.form(\n [\n Inputs.range(distribution == rlnorm ? [-10, 10] : [0, 10], {\n label:\n distribution == runif ? \"Link ratio 8 | Lower\" : \"Link ratio 8 | Mean\",\n step: 0.01,\n value: 0\n }),\n Inputs.range([0, 10], {\n label: distribution == runif ? \"Upper\" : \"Standard deviation\",\n step: 0.01,\n value: distribution == rlnorm ? 0 : 1\n })\n ],\n {\n template: (inputs) => htl.html`

\n ${inputs}\n

`\n }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1241,"value":"viewof link_ratio9 = Inputs.form(\n [\n Inputs.range(distribution == rlnorm ? [-10, 10] : [0, 10], {\n label:\n distribution == runif ? \"Link ratio 9 | Lower\" : \"Link ratio 9 | Mean\",\n step: 0.01,\n value: 0\n }),\n Inputs.range([0, 10], {\n label: distribution == runif ? \"Upper\" : \"Standard deviation\",\n step: 0.01,\n value: distribution == rlnorm ? 0 : 1\n })\n ],\n {\n template: (inputs) => htl.html`

\n ${inputs}\n

`\n }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1174,"value":"{\n let f = prior_predictive(\n 1000,\n distribution,\n [\n link_ratio1[0],\n link_ratio2[0],\n link_ratio3[0],\n link_ratio4[0],\n link_ratio5[0],\n link_ratio6[0],\n link_ratio7[0],\n link_ratio8[0],\n link_ratio9[0]\n ],\n [\n link_ratio1[1],\n link_ratio2[1],\n link_ratio3[1],\n link_ratio4[1],\n link_ratio5[1],\n link_ratio6[1],\n link_ratio7[1],\n link_ratio8[1],\n link_ratio9[1]\n ]\n );\n return Plot.plot({\n marginLeft: 70,\n y: { label: \"Loss ratios (%)\" },\n x: { label: \"Development lag\" },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0.5]),\n Plot.line(f, {\n x: \"lag\",\n y: \"lr\",\n z: \"sim\",\n strokeWidth: 0.1\n }),\n Plot.dot(f, {\n x: \"lag\",\n y: \"lr\",\n r: 2,\n fill: \"white\",\n stroke: \"black\"\n })\n ],\n caption: \"Simulated mean loss ratios.\"\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":471,"value":"function runif(n = 1, a = 0, b = 1) {\n /**\n * Generate random variates from the uniform distribution.\n */\n const x = [];\n if (!(a instanceof Array)) a = new Array(n).fill(a);\n if (!(b instanceof Array)) b = new Array(n).fill(b);\n for (let i = 0; i < n; i++) {\n let u = Math.max(1e-10, Math.random());\n x.push(a[i] + (b[i] - a[i]) * u);\n }\n if (x.length == 1) return x[0];\n else return x;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":69,"value":"function rnorm1(mu, sigma) {\n /**\n * This is the Box-Muller transform.\n\n * Math.random() returns a uniform number in [0, 1),\n * but we need random uniforms in (0, 1), hence the\n * Math.max() below.\n */\n const u1 = runif();\n const u2 = runif();\n const R = Math.sqrt(-2 * Math.log(u1));\n const theta = 2 * Math.PI * u2;\n const z = R * Math.cos(theta);\n return z * sigma + mu;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":88,"value":"function rnorm(n = 1, mu = 0, sigma = 1) {\n /**\n * Generate random variates from the normal distribution.\n */\n const x = [];\n if (!(mu instanceof Array)) mu = new Array(n).fill(mu);\n if (!(sigma instanceof Array)) sigma = new Array(n).fill(sigma);\n for (let i = 0; i < n; i++) {\n x.push(rnorm1(mu[i], sigma[i]));\n }\n if (x.length == 1) return x[0];\n else return x;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":715,"value":"function rlnorm(n = 1, mu = 0, sigma = 1){\n /**\n * Generate random variates from the lognormal distribution.\n */\n const x = rnorm(n, mu, sigma);\n const y = [];\n for (let i = 0; i < n; i++){\n y.push(Math.exp(x[i]));\n }\n if (y.length == 1) return y[0];\n else return y; \n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":142,"value":"function mean(x) {\n const N = x.length;\n const sum = x.reduce((a, b) => a + b);\n return N ** -1 * sum;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":154,"value":"function variance(x) {\n const N = x.length;\n const x_bar = mean(x);\n const ses = x.map((a) => (N - 1) ** -1 * (a - x_bar) ** 2);\n return ses.reduce((a, b) => a + b);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":150,"value":"function std(x) {\n return Math.sqrt(variance(x));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":13,"value":"class ChainLadderPriors {\n constructor(mu, sigma) {\n this.mu = mu;\n this.sigma = sigma;\n this.M = 1e5;\n }\n\n atas(n) {\n const atas = [];\n for (let i = 0; i < n; i++) {\n const samples = [];\n for (let m = 0; m < this.M; m++) {\n samples.push(Math.exp(rnorm(1, this.mu, this.sigma)));\n }\n atas.push(samples);\n }\n return atas;\n }\n\n summary(n, z=1.96) {\n const out = [];\n const atas_mu = this.atas(n).map(mean)\n const atas_sd = this.atas(n).map(std)\n const lowers = atas_mu.map((a, i) => Math.max(0.0, a - z * atas_sd[i]));\n const uppers = atas_mu.map((a, i) => a + z * atas_sd[i]);\n for (let i = 0; i < n; i++) {\n out.push({\n \"lag\": i + 1, \n \"ata\": atas_mu[i],\n \"lower\": lowers[i],\n \"upper\": uppers[i],\n });\n }\n return out;\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"f = [1.0, 2.5, 1.2, 1.15, 1.1, 1.05, 1.03, 1.02, 1.01, 1.0].map(\n function (ata, i) {\n return { lag: i + 1, lr: (this.lr *= ata) };\n },\n { lag: 0, lr: 20 }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":485,"value":"uniform_prior = runif(1e6, 0, 50).map((i) => {\n return { x: i };\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":696,"value":"z_prior = rnorm(1e6).map((i) => {\n return { x: i };\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":723,"value":"lognormal_prior = rlnorm(1e6).map((i) => {\n return { x: i };\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":545,"value":"function sim_loss_ratios(initial, atas) {\n const M = atas.length + 1;\n const loss_ratios = new Array(M);\n loss_ratios[0] = { lag: 1, lr: initial };\n for (let i = 1; i < M; i++) {\n loss_ratios[i] = { lag: i + 1, lr: loss_ratios[i - 1][\"lr\"] * atas[i - 1] };\n }\n return loss_ratios;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":550,"value":"function prior_predictive(N, distr, arg1, arg2) {\n const out = new Array(N);\n let count = 0;\n for (let i = 0; i < N; i++) {\n let lrs = sim_loss_ratios(20, distr(9, arg1, arg2));\n for (let j = 0; j < lrs.length; j++) {\n out[count] = {\n sim: i,\n lag: lrs[j][\"lag\"],\n lr: lrs[j][\"lr\"]\n };\n count += 1;\n }\n }\n return out;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":634,"value":"uniform_pps050 = prior_predictive(100, runif, 0, 50)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":649,"value":"uniform_pps_weakly = prior_predictive(100, runif, 0.5, 3)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":736,"value":"lognormal_pps_weakly = prior_predictive(100, rlnorm, 0, 0.5)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":791,"value":"lognormal_pps_regularize = prior_predictive(\n 100,\n rlnorm,\n 0,\n new Array(9).fill(1).map((x, i) => x / (i + 2))\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":812,"value":"lognormal_pps_growth = prior_predictive(\n 100,\n rlnorm,\n 0,\n new Array(9).fill(1).map((x, i) => Math.exp(-(i + 1)))\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1077,"value":"lognormal_pps_default = prior_predictive(\n 1000,\n rlnorm,\n new Array(9).fill(1).map((x, i) => 1.75 * Math.exp(1 * -(i + 1))),\n new Array(9).fill(1).map((x, i) => 0.5 * Math.exp(1 * -(i + 1)))\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W3b","stdlib":"1","titling":"auto"}} 31:"$30:body"